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Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2013 Article ID 798549 25 pageshttpdxdoiorg1011552013798549
Review ArticleFoundations of the Theory of Semilinear StochasticPartial Differential Equations
Stefan Tappe
Leibniz Universitat Hannover Institut fur Mathematische Stochastik Welfengarten 1 30167 Hannover Germany
Correspondence should be addressed to Stefan Tappe tappestochastikuni-hannoverde
Received 28 May 2013 Accepted 16 August 2013
Academic Editor Hong-Kun Xu
Copyright copy 2013 Stefan TappeThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations Inparticular we provide a detailed study of the concepts of strong weak and mild solutions establish their connections and reviewa standard existence and uniqueness result The proof of the existence result is based on a slightly extended version of the Banachfixed point theorem
1 Introduction
Semilinear stochastic partial differential equations (SPDEs)have a broad spectrum of applications including naturalsciences and economics The goal of this review article is toprovide a survey on the foundations of SPDEs which havebeen presented in the monographs [1ndash3] It may be beneficialfor students who are already aware about stochastic calculusin finite dimensions and who wish to have survey materialaccompanying the aforementioned references In particularwe review the relevant results from functional analysis aboutunbounded operators in Hilbert spaces and strongly contin-uous semigroups
A large part of this paper is devoted to a detailed study ofthe concepts of strong weak and mild solutions to SPDEsto establish their connections and to review and prove astandard existence and uniqueness result The proof of theexistence result is based on a slightly extended version of theBanach fixed point theorem
In the last part of this paper we study invariant manifoldsfor weak solutions to SPDEsThis topic does not belong to thegeneral theory of SPDEs but it uses and demonstrates manyof the results and techniques of the previous sections It arisesfrom the natural desire to express the solutions of SPDEswhich generally live in an infinite dimensional state spaceby means of a finite dimensional state process and thus toensure larger analytical tractability
This paper should also serve as an introductory studyto the general theory of SPDEs and it should enable thereader to learn about further topics and generalizations in thisfield Possible further directions are the study of martingalesolutions (see eg [1 3]) SPDEs with jumps (see eg [4] forSPDEs driven by the Levy processes and [5ndash8] for SPDEsdriven by Poisson random measures) and support theoremsas well as further invariance results for SPDEs see forexample [9 10]
The remainder of this paper is organized as follows InSections 2 and 3 we review the required results from func-tional analysis In particular we collect the relevant materialabout unbounded operators and strongly continuous semi-groups In Section 4 we review stochastic processes in infinitedimension In particular we recall the definition of a traceclass Wiener process and outline the construction of theIto integral In Section 5 we present the solution concepts forSPDEs and study their various connections In Section 6 wereview results about the regularity of stochastic convolutionintegrals which is essential for the study of mild solutionsto SPDEs In Section 7 we review a standard existence anduniqueness result Finally in Section 8 we deal with invariantmanifolds for weak solutions to SPDEs
2 Unbounded Operators in the Hilbert Spaces
In this section we review the relevant properties aboutunbounded operators We will start with operators in Banach
2 International Journal of Stochastic Analysis
spaces and focus on operators in Hilbert spaces later on Thereader can find the proofs of the upcoming results in anytextbook about functional analysis such as [11] or [12]
Let 119883 and 119884 be Banach spaces For a linear operator 119860
119883 sup D(119860) rarr 119884 defined on some subspace D(119860) of 119883 wecallD(119860) the domain of 119860
Definition 1 A linear operator 119860 119883 sup D(119860) rarr 119884 is calledclosed if for every sequence (119909
119899)119899isinN sub D(119860) such that the
limits 119909 = lim119899rarrinfin
119909119899
isin 119883 and 119910 = lim119899rarrinfin
119860119909119899
isin 119884 existone has 119909 isin D(119860) and 119860119909 = 119910
Definition 2 A linear operator 119860 119883 sup D(119860) rarr 119884 iscalled densely defined if its domainD(119860) is dense in 119883 thatisD(119860) = 119883
Definition 3 Let 119860 119883 sup D(119860) rarr 119883 be a linear operator
(1) The resolvent set of 119860 is defined as
120588 (119860) = 120582 isin C 120582 minus 119860 D (119860) 997888rarr 119883 is bijective and
(120582 minus 119860)minus1
isin 119871 (119883)
(1)
(2) The spectrum of 119860 is defined as 120590(119860) = C 120588(119860)(3) For 120582 isin 120588(119860) one defines the resolvent 119877(120582 119860) isin
119871(119883) as
119877 (120582 119860) = (120582 minus 119860)minus1
(2)
Now we will introduce the adjoint operator of a denselydefined operator in a Hilbert space Recall that for a boundedlinear operator 119879 isin 119871(119867
1 119867
2) mapping between two Hilbert
spaces1198671and119867
2 the adjoint operator is the unique bounded
linear operator 119879lowast
isin 119871(1198672 119867
1) such that
⟨119879119909 119910⟩1198672
= ⟨119909 119879lowast
119910⟩1198671
forall119909 isin 1198671 119910 isin 119867
2 (3)
In order to extend this definition to unbounded operatorsone recalls the following extension result for linear operators
Proposition 4 Let 119883 be a normed space let 119884 be a Banachspace let119863 sub 119883 be a dense subspace and letΦ 119863 rarr 119884 be acontinuous linear operatorThen there exists a unique continu-ous extension Φ 119883 rarr 119884 that is a continuous linear operatorwith Φ|
119863= Φ Moreover one has Φ = Φ
Now let119867 be aHilbert spaceWe recall the representationtheorem of Frechet-Riesz In the sequel the space119867
1015840 denotesthe dual space of 119867
Theorem 5 For every 1199091015840
isin 1198671015840 there exists a unique element
119909 isin 119867 with ⟨1199091015840
∙⟩ = ⟨119909 ∙⟩ In addition one has 119909 = 1199091015840
Let 119860 119867 sup D(119860) rarr 119867 be a densely defined operatorOne defines the subspace
D (119860lowast
) = 119910 isin 119867 119909 997891997888rarr ⟨119860119909 119910⟩ is continuous
on D (119860)
(4)
Let 119910 isin D(119860lowast
) be arbitrary By virtue of the extension resultfor linear operators (Proposition 4) the operator
D (119860) 997891997888rarr R 119909 997891997888rarr ⟨119860119909 119910⟩ (5)
has a unique extension to a linear functional 1199111015840 isin 1198671015840 By the
representation theorem of Frechet-Riesz (Theorem 5) thereexists a unique element 119911 isin 119867 with ⟨119911
1015840
∙⟩ = ⟨119911 ∙⟩ Thisimplies that
⟨119860119909 119910⟩ = ⟨119909 119911⟩ forall119909 isin D (119860) (6)
Setting 119860lowast
119910 = 119911 this defines a linear operator 119860lowast
119867 sup
D(119860lowast
) rarr 119867 and one has
⟨119860119909 119910⟩ = ⟨119909 119860lowast
119910⟩ forall119909 isin D (119860) 119910 isin D (119860lowast
) (7)
Definition 6 The operator 119860lowast
119867 sup D(119860lowast
) rarr 119867 is calledthe adjoint operator of 119860
Proposition 7 Let 119860 119867 sup D(119860) rarr 119867 be densely definedand closed Then 119860
lowast is densely defined and one has 119860 = 119860lowastlowast
Lemma 8 Let 119867 be a separable Hilbert space and let 119860
119867 sup D(119860) rarr 119867 be a closed operator Then the domain(D(119860) sdot D(119860)
) endowed with the graph norm
119909D(119860)= (119909
2
+ 1198601199092
)12 (8)
is a separable Hilbert space too
3 Strongly Continuous Semigroups
In this section we present the required results about stronglycontinuous semigroups Concerning the proofs of theupcoming results the reader is referred to any textbook aboutfunctional analysis such as [11] or [12] Throughout this sec-tion let 119883 be a Banach space
Definition 9 Let (119878119905)119905ge0
be a family of continuous linearoperators 119878
119905 119883 rarr 119883 119905 ge 0
(1) The family (119878119905)119905ge0
is a called a strongly continuoussemigroup (or 119862
0-semigroup) if the following condi-
tions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905119909 = 119909 for all 119909 isin 119883
(2) The family (119878119905)119905ge0
is called a norm continuous semi-group if the following conditions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905minus Id = 0
Note that every norm continuous semigroup is also a 1198620-
semigroup The following growth estimate (9) will often beused when dealing with SPDEs
International Journal of Stochastic Analysis 3
Lemma 10 Let (119878119905)119905ge0
be a 1198620-semigroup Then there are
constants 119872 ge 1 and 120596 isin R such that1003817100381710038171003817119878119905
1003817100381710038171003817 le 119872119890120596119905
forall119905 ge 0 (9)
Definition 11 Let (119878119905)119905ge0
be a 1198620-semigroup
(1) The semigroup (119878119905)119905ge0
is called a semigroup of contrac-tions (or contractive) if
10038171003817100381710038171198781199051003817100381710038171003817 le 1 forall119905 ge 0 (10)
that is the growth estimate (9) is satisfied with119872 = 1
and 120596 = 0(2) The semigroup (119878
119905)119905ge0
is called a semigroup of pseu-docontractions (or pseudocontractive) if there exists aconstant 120596 isin R such that
10038171003817100381710038171198781199051003817100381710038171003817 le 119890
120596119905
forall119905 ge 0 (11)
that is the growth estimate (9) is satisfiedwith119872 = 1
If (119878119905)119905ge0
is a semigroup of pseudocontractions withgrowth estimate (11) then (119879
119905)119905ge0
given by
119879119905= 119890
minus120596119905
119878119905 119905 ge 0 (12)
is a semigroup of contractions Hence every pseudocon-tractive semigroup can be transformed into a semigroup ofcontractions which explains the term pseudocontractive
Lemma 12 Let (119878119905)119905ge0
be a 1198620-semigroup Then the following
statements are true
(1) The mapping
R+times 119883 997888rarr 119883 (119905 119909) 997891997888rarr 119878
119905119909 (13)
is continuous(2) For all 119909 isin 119883 and 119879 ge 0 the mapping
[0 119879] 997888rarr 119883 119905 997891997888rarr 119878119905119909 (14)
is uniformly continuous
Definition 13 Let (119878119905)119905ge0
be a1198620-semigroupThe infinitesimal
generator (in short generator) of (119878119905)119905ge0
is the linear operator119860 119883 sup D(119860) rarr 119883 which is defined on the domain
D (119860) = 119909 isin 119883 lim119905rarr0
119878119905119909 minus 119909
119905exists (15)
and given by
119860119909 = lim119905rarr0
119878119905119909 minus 119909
119905 (16)
Note that the domain D(119860) is indeed a subspace of 119883The following result gives some properties of the infinitesimalgenerator of a119862
0-semigroup Recall thatwe have provided the
required concepts in Definitions 1 and 2
Proposition 14 The infinitesimal generator 119860 119883 sup
D(119860) rarr 119883 of a 1198620-semigroup (119878
119905)119905ge0
is densely defined andclosed
We proceed with some examples of 1198620-semigroups and
their generators
Example 15 For every bounded linear operator119860 isin 119871(119883) thefamily (119890
119905119860
)119905ge0
given by
119890119905119860
=
infin
sum
119899=0
119905119899
119860119899
119899(17)
is a norm continuous semigroup with generator 119860 In partic-ular one hasD(119860) = 119883
Example 16 We consider the separable Hilbert space 119883 =
1198712
(R) Let (119878119905)119905ge0
be the shift semigroup that is defined as
119878119905119891 = 119891 (119905 + ∙) 119905 ge 0 (18)
Then (119878119905)119905ge0
is a semigroup of contractions with generator119860
1198712
(R) sup D(119860) rarr 1198712
(R) given by
D (119860) = 119891 isin 1198712
(R) 119891 is absolutely continuous
and 1198911015840
isin 1198712
(R)
119860119891 = 1198911015840
(19)
Example 17 On the separable Hilbert space 119883 = 1198712
(R119889
) wedefine the heat semigroup (119878
119905)119905ge0
by 1198780= Id and
(119878119905119891) (119909) =
1
(4120587119905)1198892
intR119889
exp(minus
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
4119905)119891 (119910) 119889119910
119905 gt 0
(20)
that is 119878119905119891 arises as the convolution of 119891 with the density of
the normal distribution 119873(0 2119905) Then (119878119905)119905ge0
is a semigroupof contractions with generator 119860 119871
2
(R119889
) sup D(119860) rarr
1198712
(R119889
) given by
D (119860) = 1198822
(R119889
) 119860119891 = Δ119891 (21)
Here 1198822
(R119889
) denotes the Sobolev space
1198822
(R119889
) = 119891 isin 1198712
(R119889
) 119863(120572)
119891 isin 1198712
(R119889
) exists
forall120572 isin N119889
0with |120572| le 2
(22)
and Δ the Laplace operator
Δ =
119889
sum
119894=1
1205972
1205971199092
119894
(23)
We proceedwith some results regarding calculations withstrongly continuous semigroups and their generators
4 International Journal of Stochastic Analysis
Lemma 18 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesimal
generator 119860 Then the following statements are true
(1) For every 119909 isin D(119860) the mapping
R+997888rarr 119883 119905 997891997888rarr 119878
119905119909 (24)
belongs to class 1198621
(R+ 119883) and for all 119905 ge 0 one has
119878119905119909 isin D(119860) and
119889
119889119905119878119905119909 = 119860119878
119905119909 = 119878
119905119860119909 (25)
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
120588(119860) and1003817100381710038171003817119877(120582 119860)
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2
] = E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] (39)
(3) By localization we extend the Ito integral int1199050
119883119904119889119882
119904
for every predictable 1198710
2(119867)-valued process119883 satisfy-
ing
P(int
119905
0
1003817100381710038171003817Φ119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (40)
The Ito integral (int119905
0
119883119904119889119882
119904)119905ge0
is an 119867-valued continuouslocal martingale and we have the series expansion
int
119905
0
119883119904119889119882
119904= sum
119895isinN
int
119905
0
119883119895
119904119889120573
119895
119904 119905 ge 0 (41)
where 119883119895
= radic120582119895119883119890
119895for each 119895 isin N An indispensable tool
for stochastic calculus in infinite dimensions is Itorsquos formulawhich we will recall here
Theorem 26 (Itorsquos formula) Let 119864 be another separableHilbert space let 119891 isin 119862
12loc119887
(R+times 119867 119864) be a function and
let 119883 be an 119867-valued Ito process of the form
119883119905= 119883
0+ int
119905
0
119884119904119889119904 + int
119905
0
119885119904119889119882
119904 119905 ge 0 (42)
Then (119891(119905 119883119905))119905ge0
is an 119864-valued Ito process and one has P-almost surely
119891 (119905 119883119905)
= 119891 (0 1198830) + int
119905
0
(119863119904119891 (119904 119883
119904) + 119863
119909119891 (119904 119883
119904) 119884
119904
+1
2sum
119895isinN
119863119909119909
119891 (119904 119883119904) (119885
119895
119904 119885
119895
119904))119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 119885
119904119889119882
119904 119905 ge 0
(43)
where one uses the notation 119885119895
= radic120582119895119885119890
119895for each 119895 isin N
Proof This result is a consequence of [3 Theorem 29]
5 Solution Concepts for SPDEs
In this section we present the concepts of strong mild andweak solutions to SPDEs and discuss their relations
Let 119867 be a separable Hilbert space and let (119878119905)119905ge0
be a1198620-semigroup on 119867 with infinitesimal generator 119860 Further-
more let119882 be a trace classWiener process on some separableHilbert space H We consider the SPDE
119889119883119905= (119860119883
119905+ 120572 (119905 119883
119905)) 119889119905 + 120590 (119905 119883
119905) 119889119882
119905
1198830= ℎ
0
(44)
Here 120572 R+
times 119867 rarr 119867 and 120590 R+
times 119867 rarr 1198710
2(119867) are
measurable mappings
6 International Journal of Stochastic Analysis
Definition 27 Let ℎ0 Ω rarr 119867 be aF
0-measurable random
variable and let 120591 gt 0 be a strictly positive stopping time Fur-thermore let119883 = 119883
(ℎ0) be an119867-valued continuous adapted
process such that
P(int
119905and120591
0
(1003817100381710038171003817119883119904
1003817100381710038171003817 +1003817100381710038171003817120572 (119904 119883
119904)1003817100381710038171003817 +
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
) 119889119904 lt infin)
= 1 forall119905 ge 0
(45)
(1) 119883 is called a local strong solution to (44) if
119883119905and120591
isin D (119860) forall119905 ge 0 P-almost surely (46)
P(int
119905and120591
0
1003817100381710038171003817119860119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (47)
and P-almost surely one has
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(48)
(2) 119883 is called a local weak solution to (44) if for all 120577 isin
D(119860lowast
) the following equation is fulfilled P-almostsurely
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(49)
(3) 119883 is called a local mild solution to (44) if P-almostsurely one has
119883119905and120591
= 119878119905and120591
ℎ0+ int
119905and120591
0
119878(119905and120591)minus119904
120572 (119904 119883119904) 119889119904
+ int
119905and120591
0
119878(119905and120591)minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(50)
One calls 120591 the lifetime of 119883 If 120591 equiv infin then one calls 119883 astrong weak ormild solution to (44) respectively
Remark 28 Note that the concept of a strong solution israther restrictive because condition (46) has to be fulfilled
For what follows we fix a F0-measurable random vari-
able ℎ0 Ω rarr 119867 and a strictly positive stopping time 120591 gt 0
Proposition 29 Every local strong solution 119883 to (44) withlifetime 120591 is also a local weak solution to (44) with lifetime 120591
Proof Let 119883 be a local strong solution to (44) with lifetime120591 Furthermore let 120577 isin D(119860
lowast
) be arbitrary Then we haveP-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩119889119882
119904
(51)
showing that 119883 is also a local weak solution to (44) withlifetime 120591
Proposition 30 Let 119883 be a stochastic process with 1198830
= ℎ0
Then the following statements are equivalent
(1) The process 119883 is a local strong solution to (44) withlifetime 120591
(2) The process 119883 is a local weak solution to (44) withlifetime 120591 and one has (46) (47)
Proof (1)rArr(2) This implication is a direct consequence ofProposition 29
(2)rArr(1) Let 120577 isin D(119860lowast
) be arbitrary Then we have P-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
(52)
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Stochastic Analysis
spaces and focus on operators in Hilbert spaces later on Thereader can find the proofs of the upcoming results in anytextbook about functional analysis such as [11] or [12]
Let 119883 and 119884 be Banach spaces For a linear operator 119860
119883 sup D(119860) rarr 119884 defined on some subspace D(119860) of 119883 wecallD(119860) the domain of 119860
Definition 1 A linear operator 119860 119883 sup D(119860) rarr 119884 is calledclosed if for every sequence (119909
119899)119899isinN sub D(119860) such that the
limits 119909 = lim119899rarrinfin
119909119899
isin 119883 and 119910 = lim119899rarrinfin
119860119909119899
isin 119884 existone has 119909 isin D(119860) and 119860119909 = 119910
Definition 2 A linear operator 119860 119883 sup D(119860) rarr 119884 iscalled densely defined if its domainD(119860) is dense in 119883 thatisD(119860) = 119883
Definition 3 Let 119860 119883 sup D(119860) rarr 119883 be a linear operator
(1) The resolvent set of 119860 is defined as
120588 (119860) = 120582 isin C 120582 minus 119860 D (119860) 997888rarr 119883 is bijective and
(120582 minus 119860)minus1
isin 119871 (119883)
(1)
(2) The spectrum of 119860 is defined as 120590(119860) = C 120588(119860)(3) For 120582 isin 120588(119860) one defines the resolvent 119877(120582 119860) isin
119871(119883) as
119877 (120582 119860) = (120582 minus 119860)minus1
(2)
Now we will introduce the adjoint operator of a denselydefined operator in a Hilbert space Recall that for a boundedlinear operator 119879 isin 119871(119867
1 119867
2) mapping between two Hilbert
spaces1198671and119867
2 the adjoint operator is the unique bounded
linear operator 119879lowast
isin 119871(1198672 119867
1) such that
⟨119879119909 119910⟩1198672
= ⟨119909 119879lowast
119910⟩1198671
forall119909 isin 1198671 119910 isin 119867
2 (3)
In order to extend this definition to unbounded operatorsone recalls the following extension result for linear operators
Proposition 4 Let 119883 be a normed space let 119884 be a Banachspace let119863 sub 119883 be a dense subspace and letΦ 119863 rarr 119884 be acontinuous linear operatorThen there exists a unique continu-ous extension Φ 119883 rarr 119884 that is a continuous linear operatorwith Φ|
119863= Φ Moreover one has Φ = Φ
Now let119867 be aHilbert spaceWe recall the representationtheorem of Frechet-Riesz In the sequel the space119867
1015840 denotesthe dual space of 119867
Theorem 5 For every 1199091015840
isin 1198671015840 there exists a unique element
119909 isin 119867 with ⟨1199091015840
∙⟩ = ⟨119909 ∙⟩ In addition one has 119909 = 1199091015840
Let 119860 119867 sup D(119860) rarr 119867 be a densely defined operatorOne defines the subspace
D (119860lowast
) = 119910 isin 119867 119909 997891997888rarr ⟨119860119909 119910⟩ is continuous
on D (119860)
(4)
Let 119910 isin D(119860lowast
) be arbitrary By virtue of the extension resultfor linear operators (Proposition 4) the operator
D (119860) 997891997888rarr R 119909 997891997888rarr ⟨119860119909 119910⟩ (5)
has a unique extension to a linear functional 1199111015840 isin 1198671015840 By the
representation theorem of Frechet-Riesz (Theorem 5) thereexists a unique element 119911 isin 119867 with ⟨119911
1015840
∙⟩ = ⟨119911 ∙⟩ Thisimplies that
⟨119860119909 119910⟩ = ⟨119909 119911⟩ forall119909 isin D (119860) (6)
Setting 119860lowast
119910 = 119911 this defines a linear operator 119860lowast
119867 sup
D(119860lowast
) rarr 119867 and one has
⟨119860119909 119910⟩ = ⟨119909 119860lowast
119910⟩ forall119909 isin D (119860) 119910 isin D (119860lowast
) (7)
Definition 6 The operator 119860lowast
119867 sup D(119860lowast
) rarr 119867 is calledthe adjoint operator of 119860
Proposition 7 Let 119860 119867 sup D(119860) rarr 119867 be densely definedand closed Then 119860
lowast is densely defined and one has 119860 = 119860lowastlowast
Lemma 8 Let 119867 be a separable Hilbert space and let 119860
119867 sup D(119860) rarr 119867 be a closed operator Then the domain(D(119860) sdot D(119860)
) endowed with the graph norm
119909D(119860)= (119909
2
+ 1198601199092
)12 (8)
is a separable Hilbert space too
3 Strongly Continuous Semigroups
In this section we present the required results about stronglycontinuous semigroups Concerning the proofs of theupcoming results the reader is referred to any textbook aboutfunctional analysis such as [11] or [12] Throughout this sec-tion let 119883 be a Banach space
Definition 9 Let (119878119905)119905ge0
be a family of continuous linearoperators 119878
119905 119883 rarr 119883 119905 ge 0
(1) The family (119878119905)119905ge0
is a called a strongly continuoussemigroup (or 119862
0-semigroup) if the following condi-
tions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905119909 = 119909 for all 119909 isin 119883
(2) The family (119878119905)119905ge0
is called a norm continuous semi-group if the following conditions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905minus Id = 0
Note that every norm continuous semigroup is also a 1198620-
semigroup The following growth estimate (9) will often beused when dealing with SPDEs
International Journal of Stochastic Analysis 3
Lemma 10 Let (119878119905)119905ge0
be a 1198620-semigroup Then there are
constants 119872 ge 1 and 120596 isin R such that1003817100381710038171003817119878119905
1003817100381710038171003817 le 119872119890120596119905
forall119905 ge 0 (9)
Definition 11 Let (119878119905)119905ge0
be a 1198620-semigroup
(1) The semigroup (119878119905)119905ge0
is called a semigroup of contrac-tions (or contractive) if
10038171003817100381710038171198781199051003817100381710038171003817 le 1 forall119905 ge 0 (10)
that is the growth estimate (9) is satisfied with119872 = 1
and 120596 = 0(2) The semigroup (119878
119905)119905ge0
is called a semigroup of pseu-docontractions (or pseudocontractive) if there exists aconstant 120596 isin R such that
10038171003817100381710038171198781199051003817100381710038171003817 le 119890
120596119905
forall119905 ge 0 (11)
that is the growth estimate (9) is satisfiedwith119872 = 1
If (119878119905)119905ge0
is a semigroup of pseudocontractions withgrowth estimate (11) then (119879
119905)119905ge0
given by
119879119905= 119890
minus120596119905
119878119905 119905 ge 0 (12)
is a semigroup of contractions Hence every pseudocon-tractive semigroup can be transformed into a semigroup ofcontractions which explains the term pseudocontractive
Lemma 12 Let (119878119905)119905ge0
be a 1198620-semigroup Then the following
statements are true
(1) The mapping
R+times 119883 997888rarr 119883 (119905 119909) 997891997888rarr 119878
119905119909 (13)
is continuous(2) For all 119909 isin 119883 and 119879 ge 0 the mapping
[0 119879] 997888rarr 119883 119905 997891997888rarr 119878119905119909 (14)
is uniformly continuous
Definition 13 Let (119878119905)119905ge0
be a1198620-semigroupThe infinitesimal
generator (in short generator) of (119878119905)119905ge0
is the linear operator119860 119883 sup D(119860) rarr 119883 which is defined on the domain
D (119860) = 119909 isin 119883 lim119905rarr0
119878119905119909 minus 119909
119905exists (15)
and given by
119860119909 = lim119905rarr0
119878119905119909 minus 119909
119905 (16)
Note that the domain D(119860) is indeed a subspace of 119883The following result gives some properties of the infinitesimalgenerator of a119862
0-semigroup Recall thatwe have provided the
required concepts in Definitions 1 and 2
Proposition 14 The infinitesimal generator 119860 119883 sup
D(119860) rarr 119883 of a 1198620-semigroup (119878
119905)119905ge0
is densely defined andclosed
We proceed with some examples of 1198620-semigroups and
their generators
Example 15 For every bounded linear operator119860 isin 119871(119883) thefamily (119890
119905119860
)119905ge0
given by
119890119905119860
=
infin
sum
119899=0
119905119899
119860119899
119899(17)
is a norm continuous semigroup with generator 119860 In partic-ular one hasD(119860) = 119883
Example 16 We consider the separable Hilbert space 119883 =
1198712
(R) Let (119878119905)119905ge0
be the shift semigroup that is defined as
119878119905119891 = 119891 (119905 + ∙) 119905 ge 0 (18)
Then (119878119905)119905ge0
is a semigroup of contractions with generator119860
1198712
(R) sup D(119860) rarr 1198712
(R) given by
D (119860) = 119891 isin 1198712
(R) 119891 is absolutely continuous
and 1198911015840
isin 1198712
(R)
119860119891 = 1198911015840
(19)
Example 17 On the separable Hilbert space 119883 = 1198712
(R119889
) wedefine the heat semigroup (119878
119905)119905ge0
by 1198780= Id and
(119878119905119891) (119909) =
1
(4120587119905)1198892
intR119889
exp(minus
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
4119905)119891 (119910) 119889119910
119905 gt 0
(20)
that is 119878119905119891 arises as the convolution of 119891 with the density of
the normal distribution 119873(0 2119905) Then (119878119905)119905ge0
is a semigroupof contractions with generator 119860 119871
2
(R119889
) sup D(119860) rarr
1198712
(R119889
) given by
D (119860) = 1198822
(R119889
) 119860119891 = Δ119891 (21)
Here 1198822
(R119889
) denotes the Sobolev space
1198822
(R119889
) = 119891 isin 1198712
(R119889
) 119863(120572)
119891 isin 1198712
(R119889
) exists
forall120572 isin N119889
0with |120572| le 2
(22)
and Δ the Laplace operator
Δ =
119889
sum
119894=1
1205972
1205971199092
119894
(23)
We proceedwith some results regarding calculations withstrongly continuous semigroups and their generators
4 International Journal of Stochastic Analysis
Lemma 18 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesimal
generator 119860 Then the following statements are true
(1) For every 119909 isin D(119860) the mapping
R+997888rarr 119883 119905 997891997888rarr 119878
119905119909 (24)
belongs to class 1198621
(R+ 119883) and for all 119905 ge 0 one has
119878119905119909 isin D(119860) and
119889
119889119905119878119905119909 = 119860119878
119905119909 = 119878
119905119860119909 (25)
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
120588(119860) and1003817100381710038171003817119877(120582 119860)
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2
] = E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] (39)
(3) By localization we extend the Ito integral int1199050
119883119904119889119882
119904
for every predictable 1198710
2(119867)-valued process119883 satisfy-
ing
P(int
119905
0
1003817100381710038171003817Φ119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (40)
The Ito integral (int119905
0
119883119904119889119882
119904)119905ge0
is an 119867-valued continuouslocal martingale and we have the series expansion
int
119905
0
119883119904119889119882
119904= sum
119895isinN
int
119905
0
119883119895
119904119889120573
119895
119904 119905 ge 0 (41)
where 119883119895
= radic120582119895119883119890
119895for each 119895 isin N An indispensable tool
for stochastic calculus in infinite dimensions is Itorsquos formulawhich we will recall here
Theorem 26 (Itorsquos formula) Let 119864 be another separableHilbert space let 119891 isin 119862
12loc119887
(R+times 119867 119864) be a function and
let 119883 be an 119867-valued Ito process of the form
119883119905= 119883
0+ int
119905
0
119884119904119889119904 + int
119905
0
119885119904119889119882
119904 119905 ge 0 (42)
Then (119891(119905 119883119905))119905ge0
is an 119864-valued Ito process and one has P-almost surely
119891 (119905 119883119905)
= 119891 (0 1198830) + int
119905
0
(119863119904119891 (119904 119883
119904) + 119863
119909119891 (119904 119883
119904) 119884
119904
+1
2sum
119895isinN
119863119909119909
119891 (119904 119883119904) (119885
119895
119904 119885
119895
119904))119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 119885
119904119889119882
119904 119905 ge 0
(43)
where one uses the notation 119885119895
= radic120582119895119885119890
119895for each 119895 isin N
Proof This result is a consequence of [3 Theorem 29]
5 Solution Concepts for SPDEs
In this section we present the concepts of strong mild andweak solutions to SPDEs and discuss their relations
Let 119867 be a separable Hilbert space and let (119878119905)119905ge0
be a1198620-semigroup on 119867 with infinitesimal generator 119860 Further-
more let119882 be a trace classWiener process on some separableHilbert space H We consider the SPDE
119889119883119905= (119860119883
119905+ 120572 (119905 119883
119905)) 119889119905 + 120590 (119905 119883
119905) 119889119882
119905
1198830= ℎ
0
(44)
Here 120572 R+
times 119867 rarr 119867 and 120590 R+
times 119867 rarr 1198710
2(119867) are
measurable mappings
6 International Journal of Stochastic Analysis
Definition 27 Let ℎ0 Ω rarr 119867 be aF
0-measurable random
variable and let 120591 gt 0 be a strictly positive stopping time Fur-thermore let119883 = 119883
(ℎ0) be an119867-valued continuous adapted
process such that
P(int
119905and120591
0
(1003817100381710038171003817119883119904
1003817100381710038171003817 +1003817100381710038171003817120572 (119904 119883
119904)1003817100381710038171003817 +
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
) 119889119904 lt infin)
= 1 forall119905 ge 0
(45)
(1) 119883 is called a local strong solution to (44) if
119883119905and120591
isin D (119860) forall119905 ge 0 P-almost surely (46)
P(int
119905and120591
0
1003817100381710038171003817119860119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (47)
and P-almost surely one has
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(48)
(2) 119883 is called a local weak solution to (44) if for all 120577 isin
D(119860lowast
) the following equation is fulfilled P-almostsurely
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(49)
(3) 119883 is called a local mild solution to (44) if P-almostsurely one has
119883119905and120591
= 119878119905and120591
ℎ0+ int
119905and120591
0
119878(119905and120591)minus119904
120572 (119904 119883119904) 119889119904
+ int
119905and120591
0
119878(119905and120591)minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(50)
One calls 120591 the lifetime of 119883 If 120591 equiv infin then one calls 119883 astrong weak ormild solution to (44) respectively
Remark 28 Note that the concept of a strong solution israther restrictive because condition (46) has to be fulfilled
For what follows we fix a F0-measurable random vari-
able ℎ0 Ω rarr 119867 and a strictly positive stopping time 120591 gt 0
Proposition 29 Every local strong solution 119883 to (44) withlifetime 120591 is also a local weak solution to (44) with lifetime 120591
Proof Let 119883 be a local strong solution to (44) with lifetime120591 Furthermore let 120577 isin D(119860
lowast
) be arbitrary Then we haveP-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩119889119882
119904
(51)
showing that 119883 is also a local weak solution to (44) withlifetime 120591
Proposition 30 Let 119883 be a stochastic process with 1198830
= ℎ0
Then the following statements are equivalent
(1) The process 119883 is a local strong solution to (44) withlifetime 120591
(2) The process 119883 is a local weak solution to (44) withlifetime 120591 and one has (46) (47)
Proof (1)rArr(2) This implication is a direct consequence ofProposition 29
(2)rArr(1) Let 120577 isin D(119860lowast
) be arbitrary Then we have P-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
(52)
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 3
Lemma 10 Let (119878119905)119905ge0
be a 1198620-semigroup Then there are
constants 119872 ge 1 and 120596 isin R such that1003817100381710038171003817119878119905
1003817100381710038171003817 le 119872119890120596119905
forall119905 ge 0 (9)
Definition 11 Let (119878119905)119905ge0
be a 1198620-semigroup
(1) The semigroup (119878119905)119905ge0
is called a semigroup of contrac-tions (or contractive) if
10038171003817100381710038171198781199051003817100381710038171003817 le 1 forall119905 ge 0 (10)
that is the growth estimate (9) is satisfied with119872 = 1
and 120596 = 0(2) The semigroup (119878
119905)119905ge0
is called a semigroup of pseu-docontractions (or pseudocontractive) if there exists aconstant 120596 isin R such that
10038171003817100381710038171198781199051003817100381710038171003817 le 119890
120596119905
forall119905 ge 0 (11)
that is the growth estimate (9) is satisfiedwith119872 = 1
If (119878119905)119905ge0
is a semigroup of pseudocontractions withgrowth estimate (11) then (119879
119905)119905ge0
given by
119879119905= 119890
minus120596119905
119878119905 119905 ge 0 (12)
is a semigroup of contractions Hence every pseudocon-tractive semigroup can be transformed into a semigroup ofcontractions which explains the term pseudocontractive
Lemma 12 Let (119878119905)119905ge0
be a 1198620-semigroup Then the following
statements are true
(1) The mapping
R+times 119883 997888rarr 119883 (119905 119909) 997891997888rarr 119878
119905119909 (13)
is continuous(2) For all 119909 isin 119883 and 119879 ge 0 the mapping
[0 119879] 997888rarr 119883 119905 997891997888rarr 119878119905119909 (14)
is uniformly continuous
Definition 13 Let (119878119905)119905ge0
be a1198620-semigroupThe infinitesimal
generator (in short generator) of (119878119905)119905ge0
is the linear operator119860 119883 sup D(119860) rarr 119883 which is defined on the domain
D (119860) = 119909 isin 119883 lim119905rarr0
119878119905119909 minus 119909
119905exists (15)
and given by
119860119909 = lim119905rarr0
119878119905119909 minus 119909
119905 (16)
Note that the domain D(119860) is indeed a subspace of 119883The following result gives some properties of the infinitesimalgenerator of a119862
0-semigroup Recall thatwe have provided the
required concepts in Definitions 1 and 2
Proposition 14 The infinitesimal generator 119860 119883 sup
D(119860) rarr 119883 of a 1198620-semigroup (119878
119905)119905ge0
is densely defined andclosed
We proceed with some examples of 1198620-semigroups and
their generators
Example 15 For every bounded linear operator119860 isin 119871(119883) thefamily (119890
119905119860
)119905ge0
given by
119890119905119860
=
infin
sum
119899=0
119905119899
119860119899
119899(17)
is a norm continuous semigroup with generator 119860 In partic-ular one hasD(119860) = 119883
Example 16 We consider the separable Hilbert space 119883 =
1198712
(R) Let (119878119905)119905ge0
be the shift semigroup that is defined as
119878119905119891 = 119891 (119905 + ∙) 119905 ge 0 (18)
Then (119878119905)119905ge0
is a semigroup of contractions with generator119860
1198712
(R) sup D(119860) rarr 1198712
(R) given by
D (119860) = 119891 isin 1198712
(R) 119891 is absolutely continuous
and 1198911015840
isin 1198712
(R)
119860119891 = 1198911015840
(19)
Example 17 On the separable Hilbert space 119883 = 1198712
(R119889
) wedefine the heat semigroup (119878
119905)119905ge0
by 1198780= Id and
(119878119905119891) (119909) =
1
(4120587119905)1198892
intR119889
exp(minus
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
2
4119905)119891 (119910) 119889119910
119905 gt 0
(20)
that is 119878119905119891 arises as the convolution of 119891 with the density of
the normal distribution 119873(0 2119905) Then (119878119905)119905ge0
is a semigroupof contractions with generator 119860 119871
2
(R119889
) sup D(119860) rarr
1198712
(R119889
) given by
D (119860) = 1198822
(R119889
) 119860119891 = Δ119891 (21)
Here 1198822
(R119889
) denotes the Sobolev space
1198822
(R119889
) = 119891 isin 1198712
(R119889
) 119863(120572)
119891 isin 1198712
(R119889
) exists
forall120572 isin N119889
0with |120572| le 2
(22)
and Δ the Laplace operator
Δ =
119889
sum
119894=1
1205972
1205971199092
119894
(23)
We proceedwith some results regarding calculations withstrongly continuous semigroups and their generators
4 International Journal of Stochastic Analysis
Lemma 18 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesimal
generator 119860 Then the following statements are true
(1) For every 119909 isin D(119860) the mapping
R+997888rarr 119883 119905 997891997888rarr 119878
119905119909 (24)
belongs to class 1198621
(R+ 119883) and for all 119905 ge 0 one has
119878119905119909 isin D(119860) and
119889
119889119905119878119905119909 = 119860119878
119905119909 = 119878
119905119860119909 (25)
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
120588(119860) and1003817100381710038171003817119877(120582 119860)
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2
] = E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] (39)
(3) By localization we extend the Ito integral int1199050
119883119904119889119882
119904
for every predictable 1198710
2(119867)-valued process119883 satisfy-
ing
P(int
119905
0
1003817100381710038171003817Φ119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (40)
The Ito integral (int119905
0
119883119904119889119882
119904)119905ge0
is an 119867-valued continuouslocal martingale and we have the series expansion
int
119905
0
119883119904119889119882
119904= sum
119895isinN
int
119905
0
119883119895
119904119889120573
119895
119904 119905 ge 0 (41)
where 119883119895
= radic120582119895119883119890
119895for each 119895 isin N An indispensable tool
for stochastic calculus in infinite dimensions is Itorsquos formulawhich we will recall here
Theorem 26 (Itorsquos formula) Let 119864 be another separableHilbert space let 119891 isin 119862
12loc119887
(R+times 119867 119864) be a function and
let 119883 be an 119867-valued Ito process of the form
119883119905= 119883
0+ int
119905
0
119884119904119889119904 + int
119905
0
119885119904119889119882
119904 119905 ge 0 (42)
Then (119891(119905 119883119905))119905ge0
is an 119864-valued Ito process and one has P-almost surely
119891 (119905 119883119905)
= 119891 (0 1198830) + int
119905
0
(119863119904119891 (119904 119883
119904) + 119863
119909119891 (119904 119883
119904) 119884
119904
+1
2sum
119895isinN
119863119909119909
119891 (119904 119883119904) (119885
119895
119904 119885
119895
119904))119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 119885
119904119889119882
119904 119905 ge 0
(43)
where one uses the notation 119885119895
= radic120582119895119885119890
119895for each 119895 isin N
Proof This result is a consequence of [3 Theorem 29]
5 Solution Concepts for SPDEs
In this section we present the concepts of strong mild andweak solutions to SPDEs and discuss their relations
Let 119867 be a separable Hilbert space and let (119878119905)119905ge0
be a1198620-semigroup on 119867 with infinitesimal generator 119860 Further-
more let119882 be a trace classWiener process on some separableHilbert space H We consider the SPDE
119889119883119905= (119860119883
119905+ 120572 (119905 119883
119905)) 119889119905 + 120590 (119905 119883
119905) 119889119882
119905
1198830= ℎ
0
(44)
Here 120572 R+
times 119867 rarr 119867 and 120590 R+
times 119867 rarr 1198710
2(119867) are
measurable mappings
6 International Journal of Stochastic Analysis
Definition 27 Let ℎ0 Ω rarr 119867 be aF
0-measurable random
variable and let 120591 gt 0 be a strictly positive stopping time Fur-thermore let119883 = 119883
(ℎ0) be an119867-valued continuous adapted
process such that
P(int
119905and120591
0
(1003817100381710038171003817119883119904
1003817100381710038171003817 +1003817100381710038171003817120572 (119904 119883
119904)1003817100381710038171003817 +
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
) 119889119904 lt infin)
= 1 forall119905 ge 0
(45)
(1) 119883 is called a local strong solution to (44) if
119883119905and120591
isin D (119860) forall119905 ge 0 P-almost surely (46)
P(int
119905and120591
0
1003817100381710038171003817119860119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (47)
and P-almost surely one has
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(48)
(2) 119883 is called a local weak solution to (44) if for all 120577 isin
D(119860lowast
) the following equation is fulfilled P-almostsurely
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(49)
(3) 119883 is called a local mild solution to (44) if P-almostsurely one has
119883119905and120591
= 119878119905and120591
ℎ0+ int
119905and120591
0
119878(119905and120591)minus119904
120572 (119904 119883119904) 119889119904
+ int
119905and120591
0
119878(119905and120591)minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(50)
One calls 120591 the lifetime of 119883 If 120591 equiv infin then one calls 119883 astrong weak ormild solution to (44) respectively
Remark 28 Note that the concept of a strong solution israther restrictive because condition (46) has to be fulfilled
For what follows we fix a F0-measurable random vari-
able ℎ0 Ω rarr 119867 and a strictly positive stopping time 120591 gt 0
Proposition 29 Every local strong solution 119883 to (44) withlifetime 120591 is also a local weak solution to (44) with lifetime 120591
Proof Let 119883 be a local strong solution to (44) with lifetime120591 Furthermore let 120577 isin D(119860
lowast
) be arbitrary Then we haveP-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩119889119882
119904
(51)
showing that 119883 is also a local weak solution to (44) withlifetime 120591
Proposition 30 Let 119883 be a stochastic process with 1198830
= ℎ0
Then the following statements are equivalent
(1) The process 119883 is a local strong solution to (44) withlifetime 120591
(2) The process 119883 is a local weak solution to (44) withlifetime 120591 and one has (46) (47)
Proof (1)rArr(2) This implication is a direct consequence ofProposition 29
(2)rArr(1) Let 120577 isin D(119860lowast
) be arbitrary Then we have P-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
(52)
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Stochastic Analysis
Lemma 18 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesimal
generator 119860 Then the following statements are true
(1) For every 119909 isin D(119860) the mapping
R+997888rarr 119883 119905 997891997888rarr 119878
119905119909 (24)
belongs to class 1198621
(R+ 119883) and for all 119905 ge 0 one has
119878119905119909 isin D(119860) and
119889
119889119905119878119905119909 = 119860119878
119905119909 = 119878
119905119860119909 (25)
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
120588(119860) and1003817100381710038171003817119877(120582 119860)
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2
] = E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] (39)
(3) By localization we extend the Ito integral int1199050
119883119904119889119882
119904
for every predictable 1198710
2(119867)-valued process119883 satisfy-
ing
P(int
119905
0
1003817100381710038171003817Φ119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (40)
The Ito integral (int119905
0
119883119904119889119882
119904)119905ge0
is an 119867-valued continuouslocal martingale and we have the series expansion
int
119905
0
119883119904119889119882
119904= sum
119895isinN
int
119905
0
119883119895
119904119889120573
119895
119904 119905 ge 0 (41)
where 119883119895
= radic120582119895119883119890
119895for each 119895 isin N An indispensable tool
for stochastic calculus in infinite dimensions is Itorsquos formulawhich we will recall here
Theorem 26 (Itorsquos formula) Let 119864 be another separableHilbert space let 119891 isin 119862
12loc119887
(R+times 119867 119864) be a function and
let 119883 be an 119867-valued Ito process of the form
119883119905= 119883
0+ int
119905
0
119884119904119889119904 + int
119905
0
119885119904119889119882
119904 119905 ge 0 (42)
Then (119891(119905 119883119905))119905ge0
is an 119864-valued Ito process and one has P-almost surely
119891 (119905 119883119905)
= 119891 (0 1198830) + int
119905
0
(119863119904119891 (119904 119883
119904) + 119863
119909119891 (119904 119883
119904) 119884
119904
+1
2sum
119895isinN
119863119909119909
119891 (119904 119883119904) (119885
119895
119904 119885
119895
119904))119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 119885
119904119889119882
119904 119905 ge 0
(43)
where one uses the notation 119885119895
= radic120582119895119885119890
119895for each 119895 isin N
Proof This result is a consequence of [3 Theorem 29]
5 Solution Concepts for SPDEs
In this section we present the concepts of strong mild andweak solutions to SPDEs and discuss their relations
Let 119867 be a separable Hilbert space and let (119878119905)119905ge0
be a1198620-semigroup on 119867 with infinitesimal generator 119860 Further-
more let119882 be a trace classWiener process on some separableHilbert space H We consider the SPDE
119889119883119905= (119860119883
119905+ 120572 (119905 119883
119905)) 119889119905 + 120590 (119905 119883
119905) 119889119882
119905
1198830= ℎ
0
(44)
Here 120572 R+
times 119867 rarr 119867 and 120590 R+
times 119867 rarr 1198710
2(119867) are
measurable mappings
6 International Journal of Stochastic Analysis
Definition 27 Let ℎ0 Ω rarr 119867 be aF
0-measurable random
variable and let 120591 gt 0 be a strictly positive stopping time Fur-thermore let119883 = 119883
(ℎ0) be an119867-valued continuous adapted
process such that
P(int
119905and120591
0
(1003817100381710038171003817119883119904
1003817100381710038171003817 +1003817100381710038171003817120572 (119904 119883
119904)1003817100381710038171003817 +
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
) 119889119904 lt infin)
= 1 forall119905 ge 0
(45)
(1) 119883 is called a local strong solution to (44) if
119883119905and120591
isin D (119860) forall119905 ge 0 P-almost surely (46)
P(int
119905and120591
0
1003817100381710038171003817119860119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (47)
and P-almost surely one has
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(48)
(2) 119883 is called a local weak solution to (44) if for all 120577 isin
D(119860lowast
) the following equation is fulfilled P-almostsurely
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(49)
(3) 119883 is called a local mild solution to (44) if P-almostsurely one has
119883119905and120591
= 119878119905and120591
ℎ0+ int
119905and120591
0
119878(119905and120591)minus119904
120572 (119904 119883119904) 119889119904
+ int
119905and120591
0
119878(119905and120591)minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(50)
One calls 120591 the lifetime of 119883 If 120591 equiv infin then one calls 119883 astrong weak ormild solution to (44) respectively
Remark 28 Note that the concept of a strong solution israther restrictive because condition (46) has to be fulfilled
For what follows we fix a F0-measurable random vari-
able ℎ0 Ω rarr 119867 and a strictly positive stopping time 120591 gt 0
Proposition 29 Every local strong solution 119883 to (44) withlifetime 120591 is also a local weak solution to (44) with lifetime 120591
Proof Let 119883 be a local strong solution to (44) with lifetime120591 Furthermore let 120577 isin D(119860
lowast
) be arbitrary Then we haveP-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩119889119882
119904
(51)
showing that 119883 is also a local weak solution to (44) withlifetime 120591
Proposition 30 Let 119883 be a stochastic process with 1198830
= ℎ0
Then the following statements are equivalent
(1) The process 119883 is a local strong solution to (44) withlifetime 120591
(2) The process 119883 is a local weak solution to (44) withlifetime 120591 and one has (46) (47)
Proof (1)rArr(2) This implication is a direct consequence ofProposition 29
(2)rArr(1) Let 120577 isin D(119860lowast
) be arbitrary Then we have P-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
(52)
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2
] = E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] (39)
(3) By localization we extend the Ito integral int1199050
119883119904119889119882
119904
for every predictable 1198710
2(119867)-valued process119883 satisfy-
ing
P(int
119905
0
1003817100381710038171003817Φ119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (40)
The Ito integral (int119905
0
119883119904119889119882
119904)119905ge0
is an 119867-valued continuouslocal martingale and we have the series expansion
int
119905
0
119883119904119889119882
119904= sum
119895isinN
int
119905
0
119883119895
119904119889120573
119895
119904 119905 ge 0 (41)
where 119883119895
= radic120582119895119883119890
119895for each 119895 isin N An indispensable tool
for stochastic calculus in infinite dimensions is Itorsquos formulawhich we will recall here
Theorem 26 (Itorsquos formula) Let 119864 be another separableHilbert space let 119891 isin 119862
12loc119887
(R+times 119867 119864) be a function and
let 119883 be an 119867-valued Ito process of the form
119883119905= 119883
0+ int
119905
0
119884119904119889119904 + int
119905
0
119885119904119889119882
119904 119905 ge 0 (42)
Then (119891(119905 119883119905))119905ge0
is an 119864-valued Ito process and one has P-almost surely
119891 (119905 119883119905)
= 119891 (0 1198830) + int
119905
0
(119863119904119891 (119904 119883
119904) + 119863
119909119891 (119904 119883
119904) 119884
119904
+1
2sum
119895isinN
119863119909119909
119891 (119904 119883119904) (119885
119895
119904 119885
119895
119904))119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 119885
119904119889119882
119904 119905 ge 0
(43)
where one uses the notation 119885119895
= radic120582119895119885119890
119895for each 119895 isin N
Proof This result is a consequence of [3 Theorem 29]
5 Solution Concepts for SPDEs
In this section we present the concepts of strong mild andweak solutions to SPDEs and discuss their relations
Let 119867 be a separable Hilbert space and let (119878119905)119905ge0
be a1198620-semigroup on 119867 with infinitesimal generator 119860 Further-
more let119882 be a trace classWiener process on some separableHilbert space H We consider the SPDE
119889119883119905= (119860119883
119905+ 120572 (119905 119883
119905)) 119889119905 + 120590 (119905 119883
119905) 119889119882
119905
1198830= ℎ
0
(44)
Here 120572 R+
times 119867 rarr 119867 and 120590 R+
times 119867 rarr 1198710
2(119867) are
measurable mappings
6 International Journal of Stochastic Analysis
Definition 27 Let ℎ0 Ω rarr 119867 be aF
0-measurable random
variable and let 120591 gt 0 be a strictly positive stopping time Fur-thermore let119883 = 119883
(ℎ0) be an119867-valued continuous adapted
process such that
P(int
119905and120591
0
(1003817100381710038171003817119883119904
1003817100381710038171003817 +1003817100381710038171003817120572 (119904 119883
119904)1003817100381710038171003817 +
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
) 119889119904 lt infin)
= 1 forall119905 ge 0
(45)
(1) 119883 is called a local strong solution to (44) if
119883119905and120591
isin D (119860) forall119905 ge 0 P-almost surely (46)
P(int
119905and120591
0
1003817100381710038171003817119860119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (47)
and P-almost surely one has
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(48)
(2) 119883 is called a local weak solution to (44) if for all 120577 isin
D(119860lowast
) the following equation is fulfilled P-almostsurely
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(49)
(3) 119883 is called a local mild solution to (44) if P-almostsurely one has
119883119905and120591
= 119878119905and120591
ℎ0+ int
119905and120591
0
119878(119905and120591)minus119904
120572 (119904 119883119904) 119889119904
+ int
119905and120591
0
119878(119905and120591)minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(50)
One calls 120591 the lifetime of 119883 If 120591 equiv infin then one calls 119883 astrong weak ormild solution to (44) respectively
Remark 28 Note that the concept of a strong solution israther restrictive because condition (46) has to be fulfilled
For what follows we fix a F0-measurable random vari-
able ℎ0 Ω rarr 119867 and a strictly positive stopping time 120591 gt 0
Proposition 29 Every local strong solution 119883 to (44) withlifetime 120591 is also a local weak solution to (44) with lifetime 120591
Proof Let 119883 be a local strong solution to (44) with lifetime120591 Furthermore let 120577 isin D(119860
lowast
) be arbitrary Then we haveP-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩119889119882
119904
(51)
showing that 119883 is also a local weak solution to (44) withlifetime 120591
Proposition 30 Let 119883 be a stochastic process with 1198830
= ℎ0
Then the following statements are equivalent
(1) The process 119883 is a local strong solution to (44) withlifetime 120591
(2) The process 119883 is a local weak solution to (44) withlifetime 120591 and one has (46) (47)
Proof (1)rArr(2) This implication is a direct consequence ofProposition 29
(2)rArr(1) Let 120577 isin D(119860lowast
) be arbitrary Then we have P-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
(52)
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Stochastic Analysis
Definition 27 Let ℎ0 Ω rarr 119867 be aF
0-measurable random
variable and let 120591 gt 0 be a strictly positive stopping time Fur-thermore let119883 = 119883
(ℎ0) be an119867-valued continuous adapted
process such that
P(int
119905and120591
0
(1003817100381710038171003817119883119904
1003817100381710038171003817 +1003817100381710038171003817120572 (119904 119883
119904)1003817100381710038171003817 +
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
) 119889119904 lt infin)
= 1 forall119905 ge 0
(45)
(1) 119883 is called a local strong solution to (44) if
119883119905and120591
isin D (119860) forall119905 ge 0 P-almost surely (46)
P(int
119905and120591
0
1003817100381710038171003817119860119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (47)
and P-almost surely one has
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(48)
(2) 119883 is called a local weak solution to (44) if for all 120577 isin
D(119860lowast
) the following equation is fulfilled P-almostsurely
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(49)
(3) 119883 is called a local mild solution to (44) if P-almostsurely one has
119883119905and120591
= 119878119905and120591
ℎ0+ int
119905and120591
0
119878(119905and120591)minus119904
120572 (119904 119883119904) 119889119904
+ int
119905and120591
0
119878(119905and120591)minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(50)
One calls 120591 the lifetime of 119883 If 120591 equiv infin then one calls 119883 astrong weak ormild solution to (44) respectively
Remark 28 Note that the concept of a strong solution israther restrictive because condition (46) has to be fulfilled
For what follows we fix a F0-measurable random vari-
able ℎ0 Ω rarr 119867 and a strictly positive stopping time 120591 gt 0
Proposition 29 Every local strong solution 119883 to (44) withlifetime 120591 is also a local weak solution to (44) with lifetime 120591
Proof Let 119883 be a local strong solution to (44) with lifetime120591 Furthermore let 120577 isin D(119860
lowast
) be arbitrary Then we haveP-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩119889119882
119904
(51)
showing that 119883 is also a local weak solution to (44) withlifetime 120591
Proposition 30 Let 119883 be a stochastic process with 1198830
= ℎ0
Then the following statements are equivalent
(1) The process 119883 is a local strong solution to (44) withlifetime 120591
(2) The process 119883 is a local weak solution to (44) withlifetime 120591 and one has (46) (47)
Proof (1)rArr(2) This implication is a direct consequence ofProposition 29
(2)rArr(1) Let 120577 isin D(119860lowast
) be arbitrary Then we have P-almost surely for all 119905 ge 0 the identities
⟨120577 119883119905and120591
⟩ = ⟨120577 ℎ0⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905and120591
0
⟨120577 119860119883119904+ 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905and120591
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904⟩
(52)
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 7
By Proposition 7 the domainD(119860lowast
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
119863119905119891 (119905 119909) = minus119860119878
119905minus119904119909
119863119909119891 (119905 119909) = 119878
119905minus119904
119863119909119909
119891 (119905 119909) = 0
(56)
Hence by Itorsquos formula (see Theorem 26) and Lemma 18 weobtain P-almost surely
119883119905= 119891 (119905 119883
119905)
= 119891 (0 ℎ0) + int
119905
0
(119863119904119891 (119904 119883
119904)
+119863119909119891 (119904 119883
119904) (119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119863119909119891 (119904 119883
119904) 120590 (119904 119883
119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
(minus119860119878119905minus119904
119883119904+ 119878
119905minus119904(119860119883
119904+ 120572 (119904 119883
119904))) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
(57)
Thus 119883 is also a mild solution to (44)
We recall the following technical auxiliary result withoutproof and refer for example to [3 Section 31]
Lemma 33 Let 119879 ge 0 be arbitrary Then the linear space
119880119879
= lin 119892120577 119892 isin 1198621
([0 119879] R) 120577 isin D (119860lowast
) (58)
is dense in 1198621
([0 119879]D(119860lowast
)) where (D(119860lowast
) sdot D(119860lowast)) is
endowed with the graph norm
Lemma 34 Let119883 be a weak solution to (44) Then for all 119879 ge
0 and all 119891 isin 1198621
([0 119879]D(119860lowast
)) one has P-almost surely
⟨119891 (119905) 119883119905⟩
= ⟨119891 (0) ℎ0⟩
+ int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩ + ⟨119891 (119904) 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩119889119882
119904 119905 isin [0 119879]
(59)
Proof By virtue of Lemma 33 it suffices to prove formula (59)for all 119891 isin 119880
119879 Let 119891 isin 119880
119879be arbitrary Then there are
1198921 119892
119899isin 119862
1
([0 119879]R) and 1205771 120577
119899isin D(119860
lowast
) for some119899 isin N such that
119891 (119905) =
119899
sum
119894=1
119892119894(119905) 120577
119894 119905 isin [0 119879] (60)
We define the function
119865 [0 119879] times R119899
997888rarr R 119865 (119905 119909) =
119899
sum
119894=1
119892119894(119905) 119909
119894 (61)
Then we have119865 isin 11986212
([0 119879]timesR119899
R)with partial derivatives
119863119905119865 (119905 119909) =
119899
sum
119894=1
1198921015840
119894(119905) 119909
119894
119863119909119865 (119905 119909) = ⟨119892 (119905) ∙⟩
R119899
119863119909119909
119865 (119905 119909) = 0
(62)
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Stochastic Analysis
Since 119883 is a weak solution to (44) the R119899-valued process
⟨120577 119883⟩ = ⟨120577119894 119883⟩
119894=1119899 (63)
is an Ito process with representation
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904 119905 ge 0
(64)
By Itorsquos formula (Theorem 26) we obtain P-almost surely
⟨119891 (119905) 119883119905⟩ = ⟨
119899
sum
119894=1
119892119894(119905) 120577
119894 119883
119905⟩ =
119899
sum
119894=1
119892119894(119905) ⟨120577
119894 119883
119905⟩
= 119865 (119905 ⟨120577 119883119905⟩) = 119865 (0 ⟨120577 ℎ
0⟩)
+ int
119905
0
(119863119904119865 (119904 ⟨120577 119883
119904⟩) + 119863
119909119865 (119904 ⟨120577 119883
119904⟩)
times (⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩)) 119889119904
+ int
119905
0
119863119909119865 (119904 ⟨120577 119883
119904⟩) ⟨120577 120590 (119904 119883
119904)⟩ 119889119882
119904
=
119899
sum
119894=1
119892119894(0) ⟨120577
119894 ℎ
0⟩
+ int
119905
0
(
119899
sum
119894=1
1198921015840
119894(119905) ⟨120577
119894 119883
119904⟩
+
119899
sum
119894=1
119892119894(119905) (⟨119860
lowast
120577119894 119883
119904⟩
+ ⟨120577119894 120572 (119904 119883
119904)⟩))119889119904
+ int
119905
0
(
119899
sum
119894=1
119892119894(119904) ⟨120577
119894 120590 (119904 119883
119904)⟩)119889119882
119904
119905 isin [0 119879]
(65)
and hence
⟨119891 (119905) 119883119905⟩
= ⟨
119899
sum
119894=1
119892119894(0) 120577
119894 ℎ
0⟩
+ int
119905
0
(⟨
119899
sum
119894=1
1198921015840
119894(119904) 120577
119894 119883
119904⟩ + ⟨119860
lowast
(
119899
sum
119894=1
119892119894(119904) 120577
119894) 119883
119904⟩
+⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120572 (119904 119883
119904)⟩)119889119904
+ int
119905
0
⟨
119899
sum
119894=1
119892119894(119904) 120577
119894 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨119891 (0) ℎ0⟩ + int
119905
0
(⟨1198911015840
(119904) + 119860lowast
119891 (119904) 119883119904⟩
+ ⟨119891 (119904) 120572 (119904 119883119904)⟩ ) 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904 119905 isin [0 119879]
(66)
This concludes the proof
Proposition 35 Every weak solution 119883 to (44) is also a mildsolution to (44)
Proof By Proposition 24 the family (119878lowast
119905)119905ge0
is a 1198620-semi-
group with generator 119860lowast Thus Proposition 23 yields that
the family of restrictions (119878lowast
119905|D(119860lowast))119905ge0
is a 1198620-semigroup
on (D(119860lowast
) sdot D(119860lowast)) with generator 119860
lowast
D((119860lowast
)2
) sub
D(119860lowast
) rarr D((119860lowast
)2
)Now let 119905 ge 0 and 120577 isin D((119860
lowast
)2
) be arbitrary We definethe function
119891 [0 119905] 997888rarr D (119860lowast
) 119891 (119904) = 119878lowast
119905minus119904120577 (67)
By Lemma 18 we have 119891 isin 1198621
([0 119905]D(119860lowast
)) with derivative
1198911015840
(119904) = minus119860lowast
119878lowast
119905minus119904120577 = minus119860
lowast
119891 (119904) (68)
Using Lemma 34 we obtain P-almost surely
⟨120577 119883119905⟩ = ⟨119891 (119905) 119883
119905⟩
= ⟨119891 (0) ℎ0⟩ + int
119905
0
⟨119891 (119904) 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119891 (119904) 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨119878lowast
119905120577 ℎ
0⟩ + int
119905
0
⟨119878lowast
119905minus119904120577 120572 (119904 119883
119904)⟩ 119889119904
+ int
119905
0
⟨119878lowast
119905minus119904120577 120590 (119904 119883
119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0⟩ + int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(69)
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 9
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119879 ge 0 (72)
Then 119883 is also a weak solution to (44)
Proof Let 119905 ge 0 and 120577 isin D(119860lowast
) be arbitraryUsing Lemma 18we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
= ⟨119860lowast
120577 int
119905
0
119878119904ℎ0119889119904
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩ = ⟨120577119860(int
119905
0
119878119904ℎ0119889119904)⟩
= ⟨120577 119878119905ℎ0minus ℎ
0⟩ = ⟨120577 119878
119905ℎ0⟩ minus ⟨120577 ℎ
0⟩
(73)
By Fubinirsquos theorem for Bochner integrals (see [3 Section 11page 21]) and Lemma 18 we obtain P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120572 (119906119883119906) 119889119904⟩119889119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120572 (119904 119883
119904) 119889119906)⟩119889119904
= int
119905
0
⟨120577 119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
= ⟨119860lowast
120577 int
119905
0
(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904⟩
= ⟨119860lowast
120577 int
119905
0
(int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906⟩
= int
119905
0
⟨119860lowast
120577 int
119905
119906
119878119904minus119906
120590 (119906119883119906) 119889119904⟩119889119882
119906
= int
119905
0
⟨119860lowast
120577 int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
isinD(119860)
⟩119889119882119904
= int
119905
0
⟨120577119860(int
119905minus119904
0
119878119906120590 (119904 119883
119904) 119889119906)⟩119889119882
119904
= int
119905
0
⟨120577 119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)⟩119889119882
119904
= ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩ minus int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(75)
Therefore and since 119883 is a mild solution to (44) we obtainP-almost surely
⟨120577 119883119905⟩
= ⟨120577 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
= ⟨120577 119878119905ℎ0⟩ + ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩
+ ⟨120577 int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
10 International Journal of Stochastic Analysis
= ⟨120577 ℎ0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904
+ int
119905
0
⟨119860lowast
120577 int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906⟩119889119904
+ int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(76)
and hence
⟨120577 119883119905⟩ = ⟨120577 ℎ
0⟩ + int
119905
0
⟨119860lowast
120577 119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
⟩119889119904
+ int
119905
0
⟨120577 120572 (119904 119883119904)⟩ 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
= ⟨120577 ℎ0⟩ + int
119905
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119904 119883
119904)⟩) 119889119904 + int
119905
0
⟨120577 120590 (119904 119883119904)⟩ 119889119882
119904
(77)
Consequently the process 119883 is also a weak solution to (44)
Next we provide conditions which ensure that a mildsolution to (44) is also a strong solution
Proposition 38 Let119883 be a mild solution to (44) such that P-almost surely one has
119883119904 120572 (119904 119883
119904) isin D (119860) 120590 (119904 119883
119904) isin 119871
0
2(D (119860)) forall119904 ge 0
(78)
as well as
P(int
119905
0
(1003817100381710038171003817119883119904
1003817100381710038171003817D(119860)+
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817D(119860)
) 119889119904 lt infin) = 1 forall119905 ge 0
(79)
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(D(119860))
119889119904] lt infin forall119879 ge 0 (80)
Then 119883 is also a strong solution to (44)
Proof By hypotheses (78) and (79) we have (46) and (47)Let 119905 ge 0 be arbitrary By Lemma 18 we have
119878119905ℎ0minus ℎ
0= int
119905
0
119860119878119904ℎ0119889119904 (81)
Furthermore by Lemma 18 and Fubinirsquos theorem for Bochnerintegrals (see [3 Section 11 page 21]) we haveP-almost surely
int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120572 (119904 119883
119904) 119889119906)119889119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120572 (119906119883119906) 119889119904) 119889119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
(82)
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
= int
119905
0
(int
119905minus119904
0
119860119878119906120590 (119904 119883
119904) 119889119906)119889119882
119904
= int
119905
0
(int
119905
119906
119860119878119904minus119906
120590 (119906119883119906) 119889119904) 119889119882
119906
= int
119905
0
(int
119904
0
119860119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
= int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(83)
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 11
Since 119883 is a mild solution to (44) we have P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ (119878119905ℎ0minus ℎ
0) + int
119905
0
(119878119905minus119904
120572 (119904 119883119904) minus 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
119904)) 119889119882
119904
(84)
and hence combining the latter identities we obtain P-almost surely
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860119878119904ℎ0119889119904 + int
119905
0
119860(int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906)119889119904
+ int
119905
0
119860(int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)119889119904
(85)
which implies that
119883119905= ℎ
0+ int
119905
0
120572 (119904 119883119904) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
+ int
119905
0
119860(119878119904ℎ0+ int
119904
0
119878119904minus119906
120572 (119906119883119906) 119889119906 + int
119904
0
119878119904minus119906
120590 (119906119883119906) 119889119882
119906)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
119889119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(86)
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
119883119905= 119890
119905119860
ℎ0+ int
119905
0
119890(119905minus119904)119860
120572 (119904 119883119904) 119889119904
+ int
119905
0
119890(119905minus119904)119860
120590 (119904 119883119904) 119889119882
119904
= 119890119905119860
ℎ0+ 119890
119905119860
int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904
+ 119890119905119860
int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(87)
Let 119884 be the Ito process
119884119905= int
119905
0
119890minus119904119860
120572 (119904 119883119904) 119889119904 + int
119905
0
119890minus119904119860
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(88)
Then we have P-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905) 119905 ge 0 (89)
and by Lemma 18 we have
119890119905119860
ℎ0minus ℎ
0= int
119905
0
119860119890119904119860
ℎ0119889119904 (90)
Defining the function
119891 R+times 119867 997888rarr 119867 119891 (119904 119910) = 119890
119904119860
119910 (91)
by Lemma 18 we have 119891 isin 11986212loc119887
(R+times 119867119867) with partial
derivatives
119863119904119891 (119904 119910) = 119860119890
119904119860
119910
119863119910119891 (119904 119910) = 119890
119904119860
119863119910119910
119891 (119904 119910) = 0
(92)
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 International Journal of Stochastic Analysis
By Itorsquos formula (Theorem 26) we get P-almost surely
119890119905119860
119884119905= 119891 (119905 119884
119905)
= 119891 (0 0)
+ int
119905
0
(119863119904119891 (119904 119884
119904) + 119863
119910119891 (119904 119884
119904) 119890
minus119904119860
120572 (119904 119883119904)) 119889119904
+ int
119905
0
119863119910119891 (119904 119884
119904) 119890
minus119904119860
120590 (119904 119883119904) 119889119882
119904
= int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
(93)
Combining the previous identities we obtainP-almost surely
119883119905= 119890
119905119860
(ℎ0+ 119884
119905)
= ℎ0+ (119890
119905119860
ℎ0minus ℎ
0) + 119890
119905119860
119884119905
= ℎ0+ int
119905
0
119860119890119904119860
ℎ0119889119904 + int
119905
0
(119860119890119904119860
119884119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119890119904119860
(ℎ0+ 119884
119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119883119904
+ 120572 (119904 119883119904))119889119904
+ int
119905
0
120590 (119904 119883119904) 119889119882
119904
= ℎ0+ int
119905
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904 + int
119905
0
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(94)
proving that 119883 is a strong solution to (44)
6 Stochastic Convolution Integrals
In this section we deal with the regularity of stochasticconvolution integrals which occur when dealing with mildsolutions to SPDEs of the type (44)
Let 119864 be a separable Banach space and let (119878119905)119905ge0
be a 1198620-
semigroup on 119864 We start with the drift term
Lemma 40 Let 119891 R+
rarr 119864 be a measurable mapping suchthat
int
119905
0
1003817100381710038171003817119891 (119904)1003817100381710038171003817 119889119904 lt infin forall119905 ge 0 (95)
Then the mapping
119865 R+997888rarr 119864 119865 (119905) = int
119905
0
119878119905minus119904
119891 (119904) 119889119904 (96)
is continuous
Proof Let 119905 isin R+be arbitrary It suffices to prove that 119865 is
right-continuous and left-continuous in 119905
(1) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899darr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
0
119878119905119899minus119904119891 (119904) 119889119904 minus int
119905119899
119905
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
le int
119905
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
+ int
119905119899
119905
10038171003817100381710038171003817119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904
(97)
By Lemma 12 the mapping
R+times 119864 997888rarr 119864 (119906 119909) 997891997888rarr 119878
119906119909 (98)
is continuous Thus taking into account estimate (9)from Lemma 10 by Lebesguersquos dominated conver-gence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (99)
(2) Let (119905119899)119899isinN sub R
+be a sequence such that 119905
119899uarr 119905 Then
for every 119899 isin N we have1003817100381710038171003817119865 (119905) minus 119865 (119905
119899)1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
10038171003817100381710038171003817100381710038171003817
=
100381710038171003817100381710038171003817100381710038171003817
int
119905119899
0
119878119905minus119904
119891 (119904) 119889119904 minus int
119905
119905119899
119878119905minus119904
119891 (119904) 119889119904 minus int
119905119899
0
119878119905119899minus119904119891 (119904) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le int
119905119899
0
10038171003817100381710038171003817119878119905minus119904
119891 (119904) minus 119878119905119899minus119904119891 (119904)
10038171003817100381710038171003817119889119904 + int
119905
119905119899
1003817100381710038171003817119878119905minus119904119891 (119904)1003817100381710038171003817 119889119904
(100)
Proceeding as in the previous situation by Lebesguersquosdominated convergence theorem we obtain
1003817100381710038171003817119865 (119905) minus 119865 (119905119899)1003817100381710038171003817 997888rarr 0 for 119899 997888rarr infin (101)
This completes the proof
Proposition 41 Let 119883 be a progressively measurable processsatisfying
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904 lt infin) = 1 forall119905 ge 0 (102)
Then the process 119884 defined as
119884119905= int
119905
0
119878119905minus119904
119883119904119889119904 119905 ge 0 (103)
is continuous and adapted
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 13
Proof The continuity of 119884 is a consequence of Lemma 40Moreover 119884 is adapted because 119883 is progressively measur-able
Now we will deal with stochastic convolution integralsdriven by the Wiener processes Let119867 be a separable Hilbertspace and let (119878
119905)119905ge0
be a 1198620-semigroup on 119867 Moreover let
119882 be a trace class Wiener process on some separable Hilbertspace H
Definition 42 Let 119883 be a 1198710
2(119867)-valued predictable process
such that
P(int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904 lt infin) = 1 forall119905 ge 0 (104)
One defines the stochastic convolution 119883 ⋆ 119882 as
(119883 ⋆ 119882)119905= int
119905
0
119878119905minus119904
119883119904119889119882
119904 119905 ge 0 (105)
One recalls the following result concerning the regularityof stochastic convolutions
Proposition 43 Let 119883 be a 1198710
2(119867)-valued predictable process
such that one of the following two conditions is satisfied
(1) There exists a constant 119901 gt 1 such that
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (106)
(2) The semigroup (119878119905)119905ge0
is a semigroup of pseudocontrac-tions and one has
E [int
119905
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin forall119905 ge 0 (107)
Then the stochastic convolution 119883 ⋆ 119882 has a continuousversion
Proof See [3 Lemma 33]
7 Existence and Uniqueness Results for SPDEs
In this section we will present results concerning existenceand uniqueness of solutions to the SPDE (44)
First we recall the Banach fixed point theoremwhichwillbe a basic result for proving the existence of mild solutions to(44)
Definition 44 Let (119864 119889) be ametric space and letΦ 119864 rarr 119864
be a mapping
(1) ThemappingΦ is called a contraction if for some con-stant 0 le 119871 lt 1 one has
119889 (Φ (119909) Φ (119910)) le 119871 sdot 119889 (119909 119910) forall119909 119910 isin 119864 (108)
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
Φ119899
(119910) = Φ119899minus1
(Φ (119910)) = Φ119899minus1
(119910) = sdot sdot sdot = Φ (119910) = 119910 (111)
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
+be a nonnegative continuous mapping and
let 120573 ge 0 be a constant such that
119891 (119905) le 120573int
119905
0
119891 (119904) 119889119904 forall119905 isin [0 119879] (112)
Then one has 119891 equiv 0
The following result shows that local Lipschitz continuityof 120572 and 120590 ensures the uniqueness of mild solutions to theSPDE (44)
Theorem 48 One supposes that for every 119899 isin N there exists aconstant 119871
119899ge 0 such that1003817100381710038171003817120572 (119905 ℎ
1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
1003817100381710038171003817120590 (119905 ℎ1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 119871119899
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(113)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 with ℎ
1 ℎ
2 le 119899 Let ℎ
0 119892
0
Ω rarr 119867 be twoF0-measurable random variables let 120591 gt 0 be
a strictly positive stopping time and let 119883 119884 be two local mild
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 119892
0and lifetime 120591
Then one has up to indistinguishability
119883120591
1ℎ0=1198920= 119884
120591
1ℎ0=1198920
(114)
(Two processes 119883 and 119884 are called indistinguishable if the set120596 isin Ω 119883
119905(120596) = 119884
119905(120596) for some 119905 isin R
+ is a P-nullset)
Proof Defining the stopping times (120591119899)119899isinN as
120591119899= 120591 and inf 119905 ge 0
1003817100381710038171003817119883119905
1003817100381710038171003817 ge 119899 and inf 119905 ge 0 1003817100381710038171003817119884119905
1003817100381710038171003817 ge 119899
(115)
we have P(120591119899
rarr 120591) = 1 Let 119899 isin N and 119879 ge 0 be arbitraryand set
Γ = ℎ0= 119892
0 isin F
0 (116)
The mapping
119891 [0 119879] 997888rarr R 119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
(117)
is nonnegative and it is continuous by Lebesguersquos dominatedconvergence theorem For all 119905 isin [0 119879] we have
119891 (119905) = E [1Γ
10038171003817100381710038171003817119883119905and120591119899
minus 119884119905and120591119899
10038171003817100381710038171003817
2
]
le 3E [1Γ
10038171003817100381710038171003817119878119905and120591119899
(ℎ0minus 119892
0)10038171003817100381710038171003817
2
]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[1Γ
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
= 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2
]
+ 3E[
10038171003817100381710038171003817100381710038171003817
int
119905and120591119899
0
1Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2
]
(118)
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
119891 (119905)
le 3119879E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904))
10038171003817100381710038171003817
2
119889119904]
+ 3E [int
119905and120591119899
0
100381710038171003817100381710038171Γ119878(119905and120591119899)minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904))
10038171003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3119879(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2
119889119904]
+ 3(119872119890120596119879
)2
E [int
119905and120591119899
0
1Γ
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
E [1Γ
10038171003817100381710038171003817119883119904and120591119899
minus 119884119904and120591119899
10038171003817100381710038171003817
2
] 119889119904
= 3 (119879 + 1) (119872119890120596119879
)2
1198712
119899int
119905
0
119891 (119904) 119889119904
(119)
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
bea 119871
0
2(119867)-valued predictable process such that
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (122)
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 15
Then for every 119901 ge 1 one has
E[
100381710038171003817100381710038171003817100381710038171003817
int
119879
0
119883119904119889119882
119904
100381710038171003817100381710038171003817100381710038171003817
2119901
] le 119862119901E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
(123)
where the constant 119862119901gt 0 is given by
119862119901= (119901 (2119901 minus 1))
119901
(2119901
2119901 minus 1)
21199012
(124)
Proof See [3 Lemma 31]
Theorem 50 Suppose that there exists a constant 119871 ge 0 suchthat
1003817100381710038171003817120572 (119905 ℎ1) minus 120572 (119905 ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 (125)1003817100381710038171003817120590 (119905 ℎ
1) minus 120590 (119905 ℎ
2)100381710038171003817100381711987102(119867)
le 1198711003817100381710038171003817ℎ1 minus ℎ
2
1003817100381710038171003817 (126)
for all 119905 ge 0 and all ℎ1 ℎ
2isin 119867 and suppose that there exists a
constant 119870 ge 0 such that
120572 (119905 ℎ) le 119870 (1 + ℎ) (127)
120590 (119905 ℎ)1198710
2(119867)
le 119870 (1 + ℎ) (128)
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
E [int
119879
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817 119889119904]
le 119872119890120596119879
119870E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817) 119889119904]
= 119872119890120596119879
119870(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817 119889119904])
le 119872119890120596119879
119870(119879 + 1198791minus12119901
times E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
12119901
) lt infin
(132)
Furthermore by the growth estimate (9) the linear growthcondition (128) and Holderrsquos inequality we have
E [int
119879
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
)2
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
le (119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 2(119872119890120596119879
119870)2
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 2(119872119890120596119879
119870)2
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 2(119872119890120596119879
119870)2
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
]
1119901
) lt infin
(133)
The previous two estimates show that Φ is a well-definedmapping on 119871
2119901
119879(119867)
Step 12 Next we show that themappingΦmaps 1198712119901119879(119867) into
itself that is we haveΦ 1198712119901
119879(119867) rarr 119871
2119901
119879(119867) Indeed let119883 isin
1198712119901
119879(119867) be arbitrary Defining the processesΦ
120572119883 andΦ
120590119883 as
(Φ120572119883)
119905= int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 isin [0 119879]
(Φ120590119883)
119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(134)
we have
(Φ119883)119905= 119878
119905ℎ0+ (Φ
120572119883)
119905+ (Φ
120590119883)
119905 119905 isin [0 119879] (135)
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 International Journal of Stochastic Analysis
By the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817119878119905ℎ01003817100381710038171003817
2119901
119889119905] le (119872119890120596119879
)2119901
119879E [1003817100381710038171003817ℎ0
1003817100381710038171003817
2119901
] lt infin (136)
By Holderrsquos inequality and the growth estimate (9) we have
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
le 1199052119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817119878119905minus119904120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119879
0
int
119905
0
1003817100381710038171003817120572 (119904 119883119904)1003817100381710038171003817
2119901
119889119904 119889119905]
(137)
and hence by the linear growth condition (127) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120572119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 1198792119901minus1
(119872119890120596119879
119870)2119901
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le (2119879)2119901minus1
(119872119890120596119879
119870)2119901
(1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(138)
Furthermore by Lemma 49 and the growth estimate (9) wehave
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
= E[int
119879
0
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
119889119905]
= int
119879
0
E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]119889119905
le 119862119901int
119879
0
E[int
119905
0
1003817100381710038171003817119878119905minus119904120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
le 119862119901(119872119890
120596119879
)2119901
int
119879
0
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
119889119905
(139)
and hence by the linear growth condition (128) and Holderrsquosinequality we obtain
E [int
119879
0
1003817100381710038171003817(Φ120590119883)
119905
1003817100381710038171003817
2119901
119889119905]
le 119862119901(119872119890
120596119879
)2119901
119905119901minus1
E [int
119879
0
int
119905
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
119879119901minus1
22119901minus1
E [int
119879
0
int
119905
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904 119889119905]
le 119862119901(119872119890
120596119879
119870)2119901
2119901
(2119879)119901minus1
times (1198792
2+ 119879E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(140)
The previous three estimates show that Φ119883 isin 1198712119901
(119867)Consequently the mapping Φ maps 1198712119901
119879(119867) into itself
Step 13 Now we show that for some index 119899 isin N the map-ping Φ
119899 is a contraction on 1198712119901
119879(119867) Let 119883119884 isin 119871
2119901
119879(119867) and
119905 isin [0 119879] be arbitrary By Holderrsquos inequality the growthestimate (9) and the Lipschitz condition (125) we have
E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 minus int
119905
0
119878119905minus119904
120572 (119904 119884119904) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120572 (119904 119883119904) minus 120572 (119904 119884
119904)) 119889119904
10038171003817100381710038171003817100381710038171003817
2119901
]
le 1199052119901minus1
E [int
119905
0
1003817100381710038171003817119878119905minus119904 (120572 (119904 119883119904) minus 120572 (119904 119884
119904))1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
)2119901
E [int
119905
0
1003817100381710038171003817120572 (119904 119883119904) minus 120572 (119904 119884
119904)1003817100381710038171003817
2119901
119889119904]
le 1198792119901minus1
(119872119890120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(141)
Furthermore by Lemma 49 the growth estimate (9) theLipschitz condition (126) and Holderrsquos inequality we obtain
E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904minus int
119905
0
119878119905minus119904
120590 (119904 119884119904) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
= E[
10038171003817100381710038171003817100381710038171003817
int
119905
0
119878119905minus119904
(120590 (119904 119883119904) minus 120590 (119904 119884
119904)) 119889119882
119904
10038171003817100381710038171003817100381710038171003817
2119901
]
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 17
le 119862119901E[int
119905
0
1003817100381710038171003817119878119905minus119904 (120590 (119904 119883119904) minus 120590 (119904 119884
119904))1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
)2119901
E[int
119905
0
1003817100381710038171003817120590 (119904 119883119904) minus 120590 (119904 119884
119904)1003817100381710038171003817
2
1198710
2(119867)
119889119904]
119901
le 119862119901(119872119890
120596119879
119871)2119901
int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(142)
Therefore defining the constant
119862 = 22119901minus1
(1198792119901minus1
(119872119890120596119879
119871)2119901
+ 119862119901(119872119890
120596119879
119871)2119901
) (143)
by Holderrsquos inequality we get
E [1003817100381710038171003817(Φ119883)
119905minus (Φ119884)
119905
1003817100381710038171003817
2119901
]
le 22119901minus1
(E [1003817100381710038171003817(Φ120572
119883)119905minus (Φ
120572119884)
119905
1003817100381710038171003817
2119901
]
+E [1003817100381710038171003817(Φ120590
119883)119905minus (Φ
120590119884)
119905
1003817100381710038171003817
2119901
])
le 119862int
119905
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904
(144)
Thus by induction for every 119899 isin N we obtain
1003817100381710038171003817Φ119899
119883 minus Φ119899
11988410038171003817100381710038171198712119901
119879(119867)
= (int
119879
0
E [10038171003817100381710038171003817(Φ
119899
119883)1199051
minus (Φ119899
119884)1199051
10038171003817100381710038171003817
2119901
] 1198891199051)
12119901
le (119862int
119879
0
(int
1199051
0
E [100381710038171003817100381710038171003817(Φ
119899minus1
119883)1199052
minus(Φ119899minus1
119884)1199052
100381710038171003817100381710038171003817
2119901
] 1198891199052)119889119905
1)
12119901
le sdot sdot sdot le (119862119899
int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
(int
119905119899
0
E [1003817100381710038171003817119883119904
minus 119884119904
1003817100381710038171003817
2119901
] 119889119904)
times 119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
12119901
le (119862119899
(int
119879
0
int
1199051
0
sdot sdot sdot int
119905119899minus1
0
1119889119905119899sdot sdot sdot 119889119905
2119889119905
1)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119879119899119899
times E [int
119879
0
1003817100381710038171003817119883119904minus 119884
119904
1003817100381710038171003817
2119901
119889119904])
12119901
= ((119862119879)
119899
119899)
12119901
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
rarr0 for 119899rarrinfin
119883 minus 1198841198712119901
119879(119867)
(145)
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
2119901
119879(119867) the following estimate
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2119901
1198710
2(119867)
119889119904]
le 1198702119901
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2119901
119889119904]
le 1198702119901
22119901minus1
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
) 119889119904]
= 119870(2119870)2119901minus1
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]) lt infin
(149)
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
E [int
119879
0
1003817100381710038171003817120590 (119904 119883119904)1003817100381710038171003817
2
1198710
2(119867)
]
le 1198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817)2
119889119904]
le 21198702
E [int
119879
0
(1 +1003817100381710038171003817119883119904
1003817100381710038171003817
2
) 119889119904]
= 21198702
(119879 + E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
119889119904])
le 21198702
(119879 + 1198791minus1119901
E[int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2119901
119889119904]
1119901
) lt infin
(159)
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
119867 = 119879ℎM oplus ⟨120577
1 120577
119898⟩perp
(166)
(3) For every ℎ isin 119880 cap M the mapping
Πℎ= 119863120601 (119910) (⟨120577 ∙⟩) 119867 997888rarr 119879
ℎM 119908ℎ119890119903119890 119910 = ⟨120577 ℎ⟩
(167)
is the corresponding projection according to (166) from119867 onto 119879
ℎM that is we have
Πℎisin 119871 (119867) Π
2
ℎ= Π
ℎ ran (Π
ℎ) = 119879
ℎM
ker (Πℎ) = ⟨120577
1 120577
119898⟩perp
(168)
Proof See [16 Lemma 613]
From now on we assume that M is an 119898-dimensional1198622-submanifold of 119867
Proposition 63 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Furthermore let 120590 isin 119862
1
(119867) be a mappingsuch that
120590 (ℎ) isin 119879ℎM forallℎ isin 119880 cap M (169)
Then for every ℎ isin 119880 cap M the direct sum decomposition of119863120590(ℎ)120590(ℎ) according to (166) is given by
119863120590 (ℎ) 120590 (ℎ)
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(170)
where 119910 = 120601minus1
(ℎ)
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 International Journal of Stochastic Analysis
Proof Since119881 is an open subset ofR119898 there exists 120598 gt 0 suchthat
119910 + 119905119863120601(119910)minus1
120590 (ℎ) isin 119881 forall119905 isin (minus120598 120598) (171)
Therefore the curve
119888 (minus120598 120598) 997888rarr 119880 cap M 119888 (119905) = 120601 (119910 + 119905119863120601(119910)minus1
120590 (ℎ))
(172)
is well defined and we have 119888 isin 1198621
((minus120598 120598)119867) with 119888(0) = ℎ
and 1198881015840
(0) = 120590(ℎ) Hence we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120590 (ℎ) 120590 (ℎ) (173)
Moreover by condition (169) and Proposition 62 we have
119889
119889119905120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905Π119888(119905)
120590 (119888 (119905))
10038161003816100381610038161003816100381610038161003816119905=0
=119889
119889119905119863120601 (⟨120577 119888 (119905)⟩) (⟨120577 120590 (119888 (119905))⟩)
10038161003816100381610038161003816100381610038161003816119905=0
= 119863120601 (119910) (⟨120577 119863120590 (ℎ) 120590 (ℎ)⟩)
+ 1198632
120601 (119910) (⟨120577 120590 (ℎ)⟩ ⟨120577 120590 (ℎ)⟩)
(174)
The latter two identities prove the desired decomposition(170)
After these preliminaries we will study invariant mani-folds for time-homogeneous SPDEs of the form
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + 120590 (119883
119905) 119889119882
119905
1198830= ℎ
0
(175)
with measurable mappings 120572 119867 rarr 119867 and 120590 119867 rarr
1198710
2(119867) As in the previous sections the operator 119860 is the
infinitesimal generator of a 1198620-semigroup (119878
119905)119905ge0
on 119867 Notethat by (41) the SPDE (175) can be rewritten equivalently as
119889119883119905= (119860119883
119905+ 120572 (119883
119905)) 119889119905 + sum
119895isinN
120590119895
(119883119905) 119889120573
119895
119905
1198830= ℎ
0
(176)
where (120573119895
)119895isinN denotes the sequence of real-valued inde-
pendent standard Wiener processes defined in (33) and themappings 120590119895 119867 rarr 119867 119895 isin N are given by 120590
119895
= radic120582119895120590119890
119895
For the rest of this section we assume that there exist aconstant 119871 ge 0 such that
1003817100381710038171003817120572 (ℎ1) minus 120572 (ℎ
2)1003817100381710038171003817 le 119871
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (177)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817le 120581
119895
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817 ℎ1 ℎ
2isin 119867 (178)
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817le 120581
119895(1 + ℎ) ℎ isin 119867 (179)
Proposition 64 For every ℎ0
isin 119867 there exists a (up toindistinguishability) unique weak solution to (176)
Proof By (178) for all ℎ1 ℎ
2isin 119867 we have
1003817100381710038171003817120590 (ℎ1) minus 120590 (ℎ
2)100381710038171003817100381711987102(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
1003817100381710038171003817ℎ1 minus ℎ2
1003817100381710038171003817
(180)
Moreover by (177) for every ℎ isin 119867 we have
120572 (ℎ) le 120572 (ℎ) minus 120572 (0) + 120572 (0)
le 119871 ℎ + 120572 (0)
le max 119871 120572 (0) (1 + ℎ)
(181)
and by (179) we obtain
120590 (ℎ)1198710
2(119867)
= (sum
119895isinN
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
2
)
12
le (sum
119895isinN
1205812
119895)
12
(1 + ℎ)
(182)
Therefore conditions (125)ndash(128) are fulfilled and henceapplying Corollary 53 completes the proof
Recall that M denotes a finite dimensional 1198622-
submanifold of 119867
Definition 65 The submanifold M is called locally invariantfor (176) if for every ℎ
0isin M there exists a local weak solution
119883 to (176) with some lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-almost surely (183)
In order to investigate local invariance of M we willassume from now on that 120590119895 isin 119862
1
(119867) for all 119895 isin N
Lemma 66 The following statements are true
(1) For every ℎ isin 119867 one has
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817lt infin (184)
(2) The mapping
119867 997888rarr 119867 ℎ 997891997888rarr sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) (185)
is continuous
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 21
Proof By (178) and (179) for every ℎ isin 119867 we have
sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817
le sum
119895isinN
10038171003817100381710038171003817119863120590
119895
(ℎ)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ)10038171003817100381710038171003817
le (1 + ℎ) sum
119895isinN
1205812
119895lt infin
(186)
showing (184) Moreover for every 119895 isin N the mapping
119867 997891997888rarr 119867 119863120590119895
(ℎ) 120590119895
(ℎ) (187)
is continuous because for all ℎ1 ℎ
2isin 119867 we have
10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ1) minus 119863120590
119895
(ℎ1) 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) 120590
119895
(ℎ2) minus 119863120590
119895
(ℎ2) 120590
119895
(ℎ2)10038171003817100381710038171003817
le10038171003817100381710038171003817119863120590
119895
(ℎ1)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ1) minus 120590
119895
(ℎ2)10038171003817100381710038171003817
+10038171003817100381710038171003817119863120590
119895
(ℎ1) minus 119863120590
119895
(ℎ2)10038171003817100381710038171003817
10038171003817100381710038171003817120590119895
(ℎ2)10038171003817100381710038171003817
(188)
Let ] be the counting measure on (NP(N)) which is givenby ](119895) = 1 for all 119895 isin N Then we have
sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) = intN
119863120590119895
(ℎ) 120590119895
(ℎ) ] (119889119895) (189)
Hence because of the estimate10038171003817100381710038171003817119863120590
119895
(ℎ) 120590119895
(ℎ)10038171003817100381710038171003817le (1 + ℎ) 120581
2
119895 ℎ isin 119867 119895 isin N (190)
the continuity of the mapping (185) is a consequence ofLebesguersquos dominated convergence theorem
For a mapping 120601 isin 1198622
119887(R119898
119867) and elements 1205771 120577
119898isin
D(119860lowast
) we define the mappings 120572120601120577
R119898
rarr R119898 and 120590119895
120601120577
R119898
rarr R119898 119895 isin N as
120572120601120577
(119910) = ⟨119860lowast
120577 120601 (119910)⟩ + ⟨120577 120572 (120601 (119910))⟩
120590119895
120601120577(119910) = ⟨120577 120590
119895
(120601 (119910))⟩
(191)
Proposition 67 Let 120601 isin 1198622
119887(R119898
119867) and 1205771 120577
119898isin D(119860
lowast
)
be arbitrary Then for every 1199100
isin R119898 there exists a (up toindistinguishability) unique strong solution to the SDE
119889119884119905= 120572
120601120577(119884
119905) 119889119905 + sum
119895isinN
120590119895
120601120577(119884
119905) 119889120573
119895
119905
1198840= 119910
0
(192)
Proof By virtue of the assumption 120601 isin 1198622
119887(R119898
119867) and (177)ndash(179) there exist a constant ge 0 such that
10038171003817100381710038171003817120572120601120577
(1199101) minus 120572
120601120577(119910
2)10038171003817100381710038171003817R119898
le 10038171003817100381710038171199101 minus 119910
2
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
(193)
and a sequence (120581119895)119895isinN sub R
+with sum
119895isinN 1205812
119895lt infin such that for
every 119895 isin N we have100381710038171003817100381710038171003817120590119895
120601120577(119910
1) minus 120590
119895
120601120577(119910
2)100381710038171003817100381710038171003817R119898
le 120581119895
10038171003817100381710038171199101 minus 1199102
1003817100381710038171003817R119898 119910
1 119910
2isin R
119898
100381710038171003817100381710038171003817120590119895
120601120577(119910)
100381710038171003817100381710038171003817R119898le 120581
119895(1 +
10038171003817100381710038171199101003817100381710038171003817R119898
) 119910 isin R119898
(194)
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩)119889119904
= 0 119905 ge 0
int
119905and120591
0
sum
119895isinN
100381610038161003816100381610038161003816⟨120585 120590
119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩
100381610038161003816100381610038161003816
2
119889119904 = 0
119905 ge 0
(210)
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 23
By the continuity of the processes 119883 and 119884 we obtain for all120585 isin D(119860
lowast
) the following identities
⟨119860lowast
120585 ℎ⟩ + ⟨120585 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
⟨120585 120590119895
(ℎ) minus 119863120601 (119910) 120590119895
120601120577(119910)⟩ = 0 119895 isin N
(211)
Consequently the mapping 120585 997891rarr ⟨119860lowast
120585 ℎ⟩ is continuous onD(119860
lowast
) and hence we have ℎ isin D(119860lowastlowast
) by the definitionof the domain provided in (4) By Proposition 7 we have119860 = 119860
lowastlowast and thus we obtain ℎ isin D(119860) proving (195) ByProposition 7 the domainD(119860
lowast
) is dense in 119867 and thus
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(119910) isin 119879
ℎM 119895 isin N (212)
showing (196) Moreover for all 120585 isin D(119860lowast
) we have
⟨120585119860ℎ + 120572 (ℎ) minus 119863120601 (119910) 120572120601120577
(119910)
minus1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))⟩ = 0
(213)
Since the domain D(119860lowast
) is dense in 119867 together withProposition 63 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
(119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
+1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910)))
= 119863120601 (119910) 120572120601120577
(119910) minus1
2sum
119895isinN
119863120601 (119910) (⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910)(120572120601120577
(119910) minus1
2sum
119895isinN
⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩) isin 119879ℎM
(214)
which proves (197)(2)rArr(1) Let ℎ
0isin M be arbitrary By Proposition 61
and Remark 60 there exist 1205771 120577
119898isin D(119860
lowast
) and aparametrization 120601 119881 rarr 119880 cap M around ℎ
0such that the
inverse 120601minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and120601 has an extension 120601 isin 119862
2
119887(R119898
119867) Let ℎ isin 119880 cap M be
arbitrary and set 119910 = ⟨120577 ℎ⟩ isin 119881 By relations (195) (197)and Proposition 62 we obtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
(215)
and thus
119860ℎ = 119863120601 (119910)( ⟨119860lowast
120577 ℎ⟩
+⟨120577 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
minus 120572 (ℎ) +1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
(216)
Together with Lemma 66 this proves the continuity of 119860 on119880capM Since ℎ
0isin Mwas arbitrary this proves that119860 is con-
tinuous onMFurthermore by (196) and Proposition 62 we have
120590119895
(ℎ) = 119863120601 (119910) 120590119895
120601120577(ℎ) for every 119895 isin N (217)
Moreover by (195) (197) and Propositions 62 and 63 weobtain
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)
= 119863120601 (119910)(⟨120577119860ℎ + 120572 (ℎ)
minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ)⟩)
= 119863120601 (119910) (⟨119860lowast
120577 ℎ⟩ + ⟨120577 120572 (ℎ)⟩)
minus1
2sum
119895isinN
119863120601 (119910) ⟨120577119863120590119895
(ℎ) 120590119895
(ℎ)⟩
= 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
(1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
minus119863120590119895
(ℎ) 120590119895
(ℎ))
(218)
This gives us
119860ℎ + 120572 (ℎ) = 119863120601 (119910) 120572120601120577
(119910)
+1
2sum
119895isinN
1198632
120601 (119910) (120590119895
120601120577(119910) 120590
119895
120601120577(119910))
(219)
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
24 International Journal of Stochastic Analysis
Now let 119884 be the strong solution to (192) with initialcondition119910
0= ⟨120577 ℎ
0⟩ isin 119881 Since119881 is open there exists 120598 gt 0
such that 119861120598(119910
0) sub 119881 We define the stopping time
120591 = inf 119905 ge 0 119884119905notin 119861
120598(119910
0) (220)
Since the process 119884 has continuous sample paths and satisfies1198840= 119910
0 we have 120591 gt 0 and P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (221)
Therefore defining the119867-valued process119883 = 120601(119884) we haveP-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (222)
Moreover using Itorsquos formula (Theorem 26) and incorporat-ing (217) (219) we obtain P-almost surely
119883119905and120591
= 120601 (1199100)
+ int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) 120601 (119884
119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904
= 120601 (1199100) + int
119905and120591
0
(119860120601 (119884119904) + 120572 (120601 (119884
119904))) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(120601 (119884119904)) 119889120573
119895
119904
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119883
119904)) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
(119883119904) 119889120573
119895
119904 119905 ge 0
(223)
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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