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STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONSWITH LEVY NOISE (A FEW ASPECTS)
SZYMON PESZAT, JAGIELLONIAN UNIVERSITY AND POLISH ACADEMYOF SCIENCES
Abstract. The course is concerned with the following topics:• Examples of equations. I will be motivated by the develop-
ment of the theory as well as applications of SPDEs in mod-eling. At this point I will be also concerned with differentconcepts of solutions and their regularity, and their Markovproperty. Finally we will say something about motivation forthe study of SPDEs.
• Stochastic integration in Hilbert spaces (or more general in-finite dimensional spaces), Levy processes in Hilbert spaces.As examples we will consider the so-called cylindrical pro-cesses and an impulsive Levy noise.
• Basic existence result, time regularity.• Long time behaviour of solutions (in particular we will be con-
cerned with the existence and uniqueness of invariant mea-sures).
1. Examples of equations
1.1. Transport equation.
(1)
∂tu(t, x) = ∂xu(t, x) + F (t, x, u(t, x)) +G1(t, x, u(t, x))∂tM(t, x)
+G2(t, x, u(t, x))∂tΠ(t, x), x ∈ R, t > 0,
u(0, x) = u0(x), x ∈ R.
In (1), M = M(t), t ≥ 0, is a Levy (square integrable) martingale andΠ is a compound Poisson process, both taking values in some functionspace. Obviously, as for SDE, (1) should be understood as (symbolic)differential form of some integral (in t) equation. The question is aboutdifferentiability in x.
The first idea, would be to consider (1) as equation on the statespace C1. This would require stochastic integration in the Banachspace C1. Unfortunately it is not possible! Namely, let W be a real-valued standard Brownian motion. It turns out that one can find adeterministic mapping ξ ∈ C([0, 1];C1([0, 1])) such that the random
1
field ∫ t
0
ξ(s)(x)dW (s), x ∈ [0, 1], t ∈ [0, 1],
cannot be modified into a C1-valued process. We can integrate inHilbert spaces (in a similar way as in Euclidean spaces) but only insome Banach spaces (like Lp, W s,p but not C or Ck).
In conclusion, contrary to elementary theory of PDEs, we are forcedto forget about strong solution, and therefore we have to find a way howto treat the linear part ∂xu of the equation. To to this we introduce theconcept of weak solution1 Consider first linear (deterministic) equation
(2)∂tu(t, x) = ∂xu(t, x), t > 0, x ∈ R,u(0, x) = u0(x), x ∈ R.
Clearly, if u0 is differentiable, then the solution is given by the formula
(3) u(t, x) = u0(t+ x).
Obviously, the right hand side of (3) is well defined even if u0 ∈ L2loc(R).
Moreover, it is easy to check that for any test function φ ∈ C10(R),
〈u(t), φ〉 =
∫Ru(t, x)φ(x)dx
= 〈u0, φ〉 −∫ t
0
〈u(s), φ′〉ds.
Thus in the terminology of PDEs, u is a weak solution to (2).Assume that Π = 0, G = G1. Our goal is to invent a reasonable
concept of a weak solution to (1), show its existence and uniquenessand finally study its regularity. Roughly (as we have not define thestochastic integral yet) a weak solution to (1) is a predictable processu taking values in L2
loc such that for any φ ∈ C10(R),
〈u(t), φ〉 = 〈u0, φ〉+
∫ t
0
−〈u(s), φ′〉+ 〈F (s, u(s)), φ〉 ds
+
∫ t
0
〈G(s, u(s))dM(s), φ〉.
Let us give the sense of the stochastic term in the most natural waypossible. Assume that M is of the form
M(t, x) =∑k
Mk(t)ek(x),
1Here weak is in PDEs sense.2
where the sum is finite or infinite, Mk are (possibly correlated) real-valued Levy martingales and ek are given functions of space variablex. Then∫ t
0
〈G(s, u(s))dM(s), φ〉 =∑k
∫ t
0
〈G(s, u(s))ek, φ〉dMk(s).
It is not so obvious however how to show the existence of a solu-tion (in principle compactness method should work (see e.g. Rozovskii(1990), Viot (1974)). This problem is much simpler in the case of theso-called mild solution. We will show later that the motions of weakand mild solutions are equivalent.
The concept of a mild-solution goes back to the well known variationof constants or Duhamel formula. Namely, given a matrix A ∈M(n×n) consider semilinear equation
du
dt(t) = Au(t) + F (t, u(t)), u(0) = u0.
Then, u solves the integral equation
u(t) = eAtu0 +
∫ t
0
eA(t−s)F (s, u(s))ds.
Therefore, there is a hope that the (weak) solution to (1) satisfiesthe integral equation
u(t) = et∂xu0 +
∫ t
0
e(t−s)∂xF (s, u(s))ds+
∫ t
0
e(t−s)∂xG(s, u(s))dM(s),
where for a given function w of x-variable
et∂xw(x) = w(t+ x),
is a solution to the linear equation
(4)∂tu(t, x) = ∂xu(t, x), t > 0, x ∈ R,u(0, x) = w(x), x ∈ R.
For more precise analysis we need to fix a state space E (Hilbertor nice Banach) for our problem. Then our equation will define aMarkov process in E. For the transport equation as the state spacewe can take L2 := L2(R) or weighted space L2
ρ := L2(R, ρ(x)dx). The
linear problem should be well-posed on E. Formally (et∂x) should fora strongly continuous semigroup, that is
• ∀ t ≥ 0, et∂x should be a bounded (i.e. continuous) linear oper-ator on E,• e0∂x = I,
3
• ∀ t, s ≥ 0,
et∂xes∂x = e(t+s)∂x ,
• ∀ψ ∈ E,
limt↓0‖et∂xψ − ψ‖E = 0.
For more information on C0-semigroups see Appendix A, and suchtextbooks as for example Pazy (1983) or Engel and Nagel (2000), Davis(1980), or Lunardi (1995).
One problem should be pointed out: the integrals appearing in thedefinition are of the convolution type. In particular the stochasticcomponent has the form∫ t
0
e(t−s)∂xG(s, u(s))dM(s).
In the case of the transport equation the semigroup is in fact a group
e(t−s)∂x = et∂xe−s∂x .
Hence∫ t
0
e(t−s)∂xσ(s, u(s))dM(s) = et∂x∫ t
0
e−s∂xσ(s, u(s))dM(s).
But in general (see for example the heat equation) a linear part of theequation generates only a semigroup!
1.2. Heat equation. Heat equation can be considered on the wholespace Rd or on a bounded domain O ⊂ Rd. We will consider here theheat equation on the interval [0, 1];
(5)
∂tu(t, x) = ∂2xxu(t, x) + F (t, x, u(t, x)) +G(t, x, u(t, x))∂tM(t, x),
t > 0, x ∈ (0, 1),
u(0, x) = u0(x), x ∈ (0, 1).
Heat equation on a bounded domain should be considered with bound-ary conditions. In our case we should add to (5) either Dirichlet orNeumann or mixed boundary conditions. We will consider the equa-tion with homogeneous boundary conditions. Thus in the first twocases it is assumed that
u(t, x) = 0 for x ∈ ∂O = 0, 1,or
∂nu(t, x) = 0 for x ∈ ∂O = 0, 1.What are the weak and mild formulation of the problem. In the weak
formulation we cannot take the space C20(0, 1) of compactly supported
4
C2-functions as the set of test functions because we will not see theboundary conditions. Taking into account the following calculations∫ 1
0
ψ(x)v′′(x)dx = ψ(1)v′(1)− ψ(0)v′(0) −∫ 1
0
ψ′(x)v′(x)dx
= ψ(1)v′(1)− ψ(0)v′(0)− ψ′(1)v(1) + ψ′(0)v(0)+∫ 1
0
ψ′′(x)v(x)dx,
we can find that the right answers are:
• In case of the Dirichlet boundary conditions the set of test func-tions is
ψ ∈ C2([0, 1]) : ψ(0) = 0 = ψ(1).
• In case of the Neumann boundary conditions the set of testfunctions is
ψ ∈ C2([0, 1]) : ψ′(0) = 0 = ψ′(1).
Then the integral condition in both cases looks like
〈u(t), ψ〉 = 〈u0, ψ〉+
∫ t
0
〈u(s), ψ′′〉+ 〈F (s, u(s)), ψ〉 ds
+
∫ t
0
〈G(s, u(s))dM(s), ψ〉.
In the formula above
〈v, ψ〉 =
∫ 1
0
v(x)ψ(x)dx,
and ∫ t
0
〈G(s, u(s))dM(s), ψ〉 =∑k
∫ t
0
〈G(s, u(s))fk, ψ〉dMk(s),
if M has the representation
M(t) =∑k
Mk(t)fk,
Mk are real-valued martingales, fk functions on [0, 1].We will consider the heat equation on the state space E = L2(0, 1).
To find a mild formulation of the heat equation we need to know thesemigroup generated by the linear part of the equation. To do thislet (AD,Dom(AD)) and (AN ,Dom(AN)) be the Laplace operators withDirichlet and Neumann boundary conditions, respectively. More pre-cisely
ADψ = ψ′′ for any ψ ∈ Dom(AD)5
andANψ = ψ′′ for any ψ ∈ Dom(AN).
HoweverDom(AD) = W 2,2(0, 1) ∩W 1,2
0 (0, 1)
and
Dom(AN) =ψ ∈ W 2,2(0, 1) : ∂nψ(0) = 0 = ∂nψ(0)
.
Then the semigroups (eADt) and (eAN t) generated on L2(0, 1) by the op-erators (AD,Dom(AD)) and (AN ,Dom(AN)) are different. Moreover,for ψ ∈ L2(0, 1),
eADtψ =∞∑k=1
e−π2k2t〈ψ, ek〉ek,
where ek(x) =√
2 sin(πkx), and
eAN tψ =∞∑k=0
e−π2k2t〈ψ, fk〉fk,
where fk(x) =√
2 cos(πkx).The mild version of (5) is
u(t) = S(t)u0 +
∫ t
0
S(t− s)F (s, u(s))ds+
∫ t
0
S(t− s)G(s, u(s))dM(s),
where Sis the semigroup generated by linear part of the equation; thatis S(t) = eADt or S(t) = eAN t.
1.3. Heath–Jarrow–Morton. Let P (t, T ) be the price at time t of abond giving 1 at time T ≥ t. Define the forward rate f by the relation
P (t, T ) = e−∫ Tt f(t,θ)dθ.
On can model f in the following way
df(t, θ) = α(t, θ)dt+ 〈σ(t, θ), dL(t)〉U ,where L is a Levy process taking values in a Hilbert space U .
Using the so-called Musiela parametrization
r(t)(x) = f(t, x+t), a(t)(x) = α(t, x+t), b(t)u(x) = 〈σ(t, x+t), u〉U ,we obtain the following expression for the evolution of the curve r(t),t ≥ 0,
r(t) = S(t)r(0) +
∫ t
0
S(t− s)a(s)ds+
∫ t
0
S(t− s)b(s)dL(s),
whereS(t)ψ(x) = ψ(x+ t).
6
Thus formally S is the semigroup generated by the derivative operator,and r is the so-called mild solution to the following stochastic partialdifferential equation
dr =
(d
dxr + a
)dt+ bdL.
In some models b and a depend on r and sometimes also on t, and thusr is a solution to the nonlinear stochastic partial differential equation
dr =
(d
dxr + a(t, r)
)dt+ b(t, r)dL.
One can show that the model is arbitrage-free in the strong sense, thatis the original measure is a martingale measure, if and only if
a(t)(x) =d
dxJ
(∫ x
0
b(t)(y)dy
),
where J is the Laplace exponent of L;
J(v) = −〈a, v〉U + 12〈Qv, v〉U
+
∫U
(e−〈v,u〉U − 1 + χ|u|U<1〈v, u〉U
)ν(dy),
and ν is the Levy measure of L.Note that if L is a real-valued Wiener process, then the non-arbitrage
conditions reads
a(t)(x) =d
dx
1
2
(∫ x
0
b(t)(y)dy
)2
= b(t)(x)
∫ t
0
b(t)(y)dy.
For more details see Bjork, DiMassi, Kabanov and Runggaldier (1997),Bjork and Christiansen (1999), Bjork, Kabanov and Runggaldier (1997),Eberlein and Raible (1999), Filipovic and Tappe (2006), Jakubowskiand Zabczyk (2004), Peszat, Rusinek, and Zabczyk (2007), and Rusinek(2006a), (2006b).
1.4. Lifts of diffusions. Consider an ordinary stochastic differentialequation
(6) dy = f(y)dt+ g(y)dM(s), y(0) = x ∈ Rd,
where M is an Rm-valued martingale, f : Rd 7→ Rd and g : Rd 7→M(d×m).
Let D ⊂ Rd be a closed set and let E be a Hilbert space of (some)functions from D into Rd. Then the following infinite dimensionalequation
dX = F (X)dt+G(X)dM, X(0) = I,7
in which F (X)(x) = f(X(x)), G(X)(x) = g(X(x)) describes the dy-namics of the flow defined by (6). From this observation on can derivebasic properties of the flow, for more details see Brzezniak and Elwor-thy (1996), and Carverhill and Elworthy (1983).
1.5. Other motivations. Other motivations come from the filtering(see e.g. Pardoux (1979), Rozovskii (1990)) theory of delay equa-tions (see e.g. Chojnowska-Michalik (1978)), theory of particle sys-tems, super-processes and cathalytic branching processes (see worksby Dalang, Dowson, Mytnik, Muller, Perkins, Tribe and others).
1.6. Construction of Markov processes. Let (Xn) be a Markovchain (discrete time) on a complete separable metric space E. Then itadmits the representation
Xn+1 = F (Xn, ξn+1),
where (ξn) is a discrete time white noise (can be chosen as a sequenceof independent uniformly distributed random variables on [0, 1]).
In continuous time case we can expect that any (reasonable) Markovprocess X on a Hilbert space E is a solution to the stochastic equation
dX = F (t,X)dt+G(t,X)dL,
where L is a Levy process. This statement is basically true if the statespace of X is finite dimensional, see Ito (1951) and Stroock (2003), andfor the form of the generator Courrege (1965/66).
1.7. Statistical Mechanics. In statistical mechanics often one has aprobability measure on M = RZd or M = L2(E, E , dm)Z
d. In order
to simulate the values (characteristics) of µ one considers a Markovprocess X on M such that µ is invariant and ergodic for X. Typically
dXl =(∑
aklXk + fl(Xl))
dt+ σldLl, l ∈ Zd.
Here A = (akl) is the infinite dimensional matrix of global interactions,and (fl) are local interactions.
2. Form of the equation and equivalence of mild andweak solutions
We will be concerned with equations of the form
dX = (AX + F (t,X)) dt+G(t,X)dL,
where A generates a C0-semigroup S on a Hilbert space E, and L is aLevy process taking values in a Hilbert space U , F : [0,∞)×E0 7→ E,and G : [0,∞)× E0 7→ L(U,E), E0 is a dense subspace of E.
8
Definition 1. A predictable process X taking values in E0 is a mildsolution if it solves the stochastic integral equation
X(t) = S(t)X(0) +
∫ t
0
S(t− s)F (s,X(s))ds
+
∫ t
0
S(t− s)G(s,X(s))dL(s),
the identity for dtdP almost all t and ω.
Definition 2. A predictable process X is a weak solution if for anyψ ∈ Dom (A∗), for any t ≥ 0, P-a.s.
〈X(t), ψ〉E = 〈X(0), ψ〉E +
∫ t
0
〈X(s), A∗ψ〉E + 〈F (s,X(s)), ψ〉E ds
+
∫ t
0
〈G∗(s,X(s))ψ, dL(s)〉E.
We have the following general result.
Theorem 1. The concepts of weak and mild solutions are equivalent;that is a modification of a mild solution is a weak solution, and a mod-ification of a weak solution is a mild solution.
Proof. We give only a sketch of the proof. For more details see e.g.Da Prato and Zabczyk (1992), or Peszat and Zabczyk (2007). First weshow that a weak solution is mild. We will do this in two steps:Step 1 Show that for any z ∈ C1([0,∞); Dom (A∗)),
(7)
〈X(t), z(t)〉E = 〈X(0), z(0)〉E
+
∫ t
0
〈X(s), A∗z(s) + z′(s)〉E + 〈F (s,X(s)), z(s)〉E ds
+
∫ t
0
〈G∗(s,X(s))z(s), dL(s)〉E.
One can do this proving (7) first for z of the form z(t) = ψg(t), g ∈C1([0,∞);R), ψ ∈ Dom (A∗), and then by an approximation argument.Step 2 Let us fix a t > 0. Let z(s) = S∗(t−s)ψ, where ψ ∈ Dom (A∗).Then z′(s) = −A∗z(s). Hence, applying (7) we obtain
〈X(t), z(t)〉E = 〈X(0), z(0)〉E +
∫ t
0
〈F (s,X(s)), z(s)〉Eds
+
∫ t
0
〈G∗(s,X(s))z(s), dL(s)〉E,9
which leads immediately to
〈X(t), ψ〉E = 〈S(t)X(0), ψ〉E +
∫ t
0
〈S(t− s)F (s,X(s)), ψ〉Eds
+
∫ t
0
〈S(t− s)G(s,X(s))dL(s), ψ〉E,
and completes the proof of the implication as Dom (A∗) is dense in E.A proof that any mild solution is weak requires a stochastic Fubini
theorem. Assume for simplicity that F ≡ 0 and that X(0) = 0. Then
X(t) =
∫ t
0
S(t− s)G(s,X(s))dL(s),
and for any ψ ∈ Dom (A∗), we have∫ t
0
〈A∗ψ,X(s)〉Eds =
∫ t
0
∫ t
0
χ[0,s](r)〈A∗ψ, S(s− r)G(r,X(r))dL(r)〉Eds
=
∫ t
0
⟨∫ t
r
S∗(s− r)A∗ψds,G(r,X(r))dL(r)
⟩E
=
∫ t
0
⟨∫ t
r
d
dsS∗(s− r)ψds,G(r,X(r))dL(r)
⟩E
=
∫ t
0
〈(S∗(t− r)− I)ψ,G(r,X(r))dL(r)〉E
=
∫ t
0
〈ψ, S(t− r)G(r,X(r))dL(r)〉E
−∫ t
0
〈ψ,G(r,X(r))dL(r)〉E
= 〈ψ,X(t)〉E −∫ t
0
〈ψ,G(r,X(r))dL(r)〉E .
Hence we obtain the desired identity.
〈ψ,X(t)〉E =
∫ t
0
〈A∗ψ,X(s)〉Eds+
∫ t
0
〈ψ,G(r,X(r))dL(r)〉E .
Provided uniqueness, SPDE defines a Markov familly. Usually theexistence follows from the Banach fixed point argument. Then we getthe continuous (in L2) dependence of the solution on the initial data.Hence we have Feller, and consequently strong Markov property!
10
3. Levy processes
Let (Ω,F ,P) be a probability space.
Definition 3. A process L : Ω× [0,∞) 7→ U is Levy if and only if
(i) L(0) = 0,(ii) L is stochastically continuous,(iii) L has independent stationary increments.
Remark 1. Any Levy process admits a cadlag modification. Recallthat “cadlag” means right continuous and having left limits.
Example 1. Any Wiener process W in U is Levy. Moreover,
E ei〈W (t),u〉U = e−t2〈Qu,u〉U , u ∈ U, t ≥ 0,
where Q is the covariance of W .
Example 2. Any compound Poison process on U is Levy. Let ν be afinite Borel measure on U . Recall that compound Poisson process withjump or equivalently Levy measure ν is given by
L(t) :=
Π(t)∑j=1
Xj,
where Π is a Poisson process with intensity λ = ν(U) < ∞2, and Xj
are independent identically distributed random variable (briefly i.i.d)with the distribution
P (Xj ∈ Γ) =ν(Γ)
ν(U), Γ ∈ B(U).
It is easy to show that
E ei〈L(t),u〉U = e−tΨ(u),
where
Ψ(u) :=
∫U
(1− ei〈u,v〉U
)ν(dv).
Moreover, L is integrable if and only if∫U
|u|Uν(du) <∞,
and if this is a case, then
EL(t) = t
∫U
uν(du).
2P(Π(t) = k) = e−λt (λt)k
k! , it is a Levy process with jump measure λδ1.11
L is square integrable if and only if∫U
|v|2Uν(dv) <∞,
and if this is a case, then
E 〈L(t)− EL(t), u〉U 〈L(t)− EL(t), v〉U =
∫U
〈z, u〉U〈z, v〉Uν(dz).
Remark 2. The concept of a compound Poisson process is crucialfor understanding the characterization of an arbitrary Levy process.The Levy–Khinchin theorem says that an arbitrary Levy process isa sum of a Wiener process, an uniform movement and a “compoundPoisson process L with infinite jump measure”. In order to construct aprocess L on U with infinite (but σ-finite) jump measure we can divideU into a countable sum U =
⋃Un of measurable sets Un such that
ν(Un) = 1 and Un ∩ Um = ∅ for n 6= m. Then one may try to writeL =
∑Ln, here Ln are independent compound Poisson processes each
with Levy measure νn being restriction of ν to Un. The problem is withconvergence. Usually the series does not converges in any reasonablesense!. The idea is to write
U = U0 ∪ U c0 ,
where U0 =⋃n∈I Un is such that
∫U0
|u|2Uν(du) <∞,∫Un
|u|Uν(du) <∞, ∀n ∈ I,
and Ic is finite. We can define then the Levy process with intensity νas ∑
n∈I
(Ln(t)− ELn(t)) +∑n6∈I
Ln(t).
The first sum is a sum of square integrable martingales with
∑n∈I
E |Ln(t)− ELn(t)|2U =
∫U0
|u|2Uν(du) <∞.
12
Therefore it converges in probability (and P-a.s) uniformly in t from anybounded interval, due to the Doob submartingale inequality 3 yielding
P
(supt∈[0,T ]
∑n≤N,n∈I
|Ln(t)− ELn(t)|2U ≥ r
)≤
∫⋃n≤N Un
|u|2Uν(du)
r.
4. Levy–Khinchin formula and Levy–Khinchindecomposition
In the theory of Levy processes the Levy–Khinchin formula and theLevy–Khinchin decomposition are crucial.
Theorem 2. (Levy–Khinchin formula) Given a ∈ U , Q ∈ L+1 (U)
and a non-negative Borel measure ν on U satisfying µ(0) = 0 and
(8)
∫U
|u|2U ∧ 1ν(du) <∞,
there is a Levy process L such that
(9) E ei〈L(t),u〉U = e−tΨ(u),
where
(10)
Ψ(u) :=− i〈a, u〉U + 12〈Qu, u〉U
+
∫U
(1− ei〈u,v〉U + χ|v|U≤1(v)i〈u, v〉U
)ν(dv).
Conversely, for each Levy process L there are a ∈ U , Q ∈ L+1 (U), and
a Borel measure ν, ν(0) = 0, satisfying (8) such that (9) holds withΨ given by (10).
Remark 3. Measure ν is called Levy measure or jump measure of L.One can show that for Γ ∈ U \ 0,
ν(Γ) = E π(1,Γ) = E∑t≤1
χΓ (∆L(t)) ,
where ∆L(t) := L(t)−L(t−). Therefore, ν(Γ) is the expected numberof jumps of L up to time 1 from the set Γ.
3If X is a right continuous submartingale (that is E(X(t)|Fs) ≥ X(S)), then
r P
(supt∈[0,T ]
X(t) ≥ r
)≤ EX+(T ).
13
Theorem 3. (Levy–Khinchin decomposition) Let R > 0. AnyLevy process L taking values in U can be written in the form
(11) L(t) = aRt+W (t) +MR(t) + LR(t),
where ar ∈ U , W is a Wiener process in U , MR is Levy process being asquare integrable martingale (with respect to filtration generated by L),and LR is a compound poisson process. Moreover,
(a) MR, LR and W are independent,(b) W does not depend on R,(c) if ν is the Levy measure of L, then MR has the Levy measure
χ|u‖U<R(u)ν(du), and LR has the Levy measure χ|u|U≥R(u)ν(du),(d) the covariance operator 4 of MR is given by
〈QRu, v〉U =
∫|z|U≤R
〈u, z〉U〈v, z〉Uν(dz).
For details see Parthasarathy (1967), Linde (1986), Peszat and Zabczyk(2007), and for finite dimensional case Sato (1999), Applebaum (2004).
5. Stochastic integration with respect to Levy processes
Our goal is to develop a theory of integration of operator-valuedprocesses
ψ : [0,∞)× Ω 7→ L(U,H)
with respect to a Levy process L taking values in U . By Levy–Khinchindecomposition theorem
L(t) = aRt+W (t) +MR(t) + LR(t).
where aR ∈ U , W is a Wiener process in U , MR is a square integrableLevy martingale, and LR is a compound Poisson process. Since LR hastrajectories of bounded variation (in fact there are piecewise constant)we can integrate with respect to LR pathwise. Thus
dLR(t) =∑j
Yjδτj(dt),
where τj are moments of jumps and Yj are the sizes of jumps.Clearly, integration with respect to aRt is the deterministic integra-
tion with respect to Lebesgue measure dt. Therefore, what is left isthe theory of integration with respect to square integrable martingalesW (with continuous paths) and MR (with discontinuous paths).
4As MR(0) = 0, MR is mean zero.14
Assume that M is a square integrable mean zero martingale withrespect to filtration (Ft) taking values in U having stationary inde-pendent increments. Let Q be its covariance. Then it is easy to seethat
|M(t)|2U − TrQt
and
M(t)⊗M(t)−Qtare martingales, respectively real- and L+
1 (U)-valued. In the generaltheory of stochastic integration (see e.g. Metivier (1982)) this meanthat the angle bracket 〈M,M〉 and operator angle bracket 〈〈M,M〉〉 ofM equal TrQt and Qt, respectively.
Let
ψ =∑k
αkχ(tk,tk+1],
where αj are L(U,H)-valued random variables, be a simple function.We assume that ψ is predictable in the sense, that for any j and anyu ∈ U , αju is Ftj -measurable H-valued random variable. Write∫ t
0
ψ(s)dM(s) :=∑j
αj (M(tj+1 ∧ t)−M(tj ∧ t)) .
Then, by simple calculation, one can show that∫ t
0
ψ(s)dM(s), t ≥ 0,
is a square integrable H-valued martingale and that
(12) E∣∣∣∣∫ t
0
ψ(s)dM(s)
∣∣∣∣2H
=
∫ t
0
E ‖ψ(s)Q1/2‖2L(HS)(U,H)ds, t ≥ 0.
This isometry gives us clue how to extend the integral. First of allnote that ψ(s) does not have to be bounded operator. What is reallynecessarily is the fact that ψ(s)Q1/2 is Hilbert–Schmidt for almost alls.
Let H := Q1/2(U). We equip H with the scalar product inducedfrom U by Q1/2, that is
〈u, v〉H :=⟨Q−1/2u,Q−1/2v
⟩U,
where Q−1/2 is the pseudo-inverse od Q1/2,
Q−1/2u = z ⇔ Q1/2z = u, and |z|U = inf|y|U : Q1/2y = u
.
We will call H the Reproducing Hilbert Kernel Space of M , (shortlyRKHS), and M a cylindrical martingale in H.
15
Remark 4. Generally M does not take values inH unless dimH <∞.It does, however, in any Hilbert space V such that the embeddingH → V is Hilbert–Schmidt.
Remark 5. RKHS and Levy measure are intrinsic characteristics ofLevy processes, that is if L takes values in U and V is a Hilbert spacecontaining U then the RHS and the Levy measure do not depend nthe choice of U or V .
It follows from (12), that the class of admissible integrants equals
L2M := L2(Ω× [0,∞),P , dPdt;L(HS)(H, H)),
where P is the σ-field of predictable sets. Moreover, for any ψ ∈ L2M ,∫ t
0
ψ(s)dM(s), t ≥ 0,
is a square integrable H-valued martingale and that
(13) E∣∣∣∣∫ t
0
ψ(s)dM(s)
∣∣∣∣2H
=
∫ t
0
E ‖ψ(s)‖2L(HS)(H,H)ds, t ≥ 0.
For more information on stochastic integration in Hilbert spaces seeMetivier (1982), Peszat and Zabczyk (2007).
Theorem 4. Let L be a Levy process with the exponent Ψ;
E ei〈L(t),u〉U = e−tΨ(u).
Let F : [0, T ] 7→ U = U∗. Then
Eei∫ T0 F (s)dL(s) = e−
∫ T0 Ψ(F (s))ds,
provided that the integrals are well defined in the Riemman sense.
Proof. Let (tnj ) be the partition of [0, T ]. We have
E ei∫ T0 F (s)dL(s) = lim
n→∞E ei
∑j〈F (snj ),L(snj+1)−L(snj )〉U
= limn
∏j
E ei〈F (snj ),L(snj+1−snj )〉U
= limn
∏j
e−(snj+1−snj )Ψ(F (snj ))
= limn
e−∑j Ψ(F (snj ))(snj+1−snj )
= e−∫ T0 Ψ(F (s))ds.
16
Example 3. Let
Z(t) =
∫ t
0
e−sdπ(s),
where π is a Poisson process with intensity 1. Then
E eixZ(t) = e−∫ t0 Ψ(xe−s)ds,
where
Ψ(x) = 1− eix =
∫R
(1− eixy
)δ1(dy).
Therefore, since
e−sδ1(Γ) = δ1(esΓ) = δe−s(Γ),
we have ∫ t
0
Ψ(xe−s)ds =
∫R
(1− eixy
) ∫ t
0
e−sδ1ds(dy)
=
∫R
(1− eixy
) ∫ t
0
δe−s(dy)ds.
Note that ∫Rf(y)
∫ t
0
δe−sdsdy =
∫ t
0
f(e−s)ds.
Changing the variables r = e−s, dr = −rds, we obtain∫ t
0
f(e−s)ds =
∫ 1
e−tf(r)
dr
r.
Thus the corresponding jump measure equals
χ(e−t,1](r)dr
r.
For the complete exposition of the theory of integration with respectto infinite-dimensional martingale see Metivier (1982), or Peszat andZabczyk (2007).
6. In general integrants have to be predictable
It is know that if we integrate with respect to a Wiener process, thenit is enough to assume that the integrant is measurable, adapted andwith probability 1, locally square integrable with respect to time. Thefollowing examples shows that in general the integrant should be alsopredictable.
17
Example 4. Let Π be a Poisson process with intensity λ. Let τ be themoment of the first jump of Π. Then χ[0,τ) is a measurable adaptedprocess. We note that χ[0,τ) is not predictable. Clearly a predictableprocess is χ[0,τ ]. Note that χ[0,τ ] is a modifcation of χ[0,τ).
Let Π be the compensated process. Then
X(t) :=
∫ t
0
χ[0,τ)(s)dΠ(s) = −λt ∧ τ +
∫ t
0
χ[0,τ)(s)dΠ(s) = −λt ∧ τ.
Clearly, X is not a martingale, nor a local martingale. It has decreasingtrajectories. On the other hand, the process
Y (t) :=
∫ t
0
χ[0,τ ](s)dΠ(s) = −λt∧τ+
∫ t
0
χ[0,τ ](s)dΠ(s) = −λt∧τ+χt≥τ
is a martingale.
Obviously if X is cadlag and adapted, then X(t−), t ≥ 0, is pre-dictable. Unfortunately, in important cases X does not have a cadlagmodification (see Section 11.1). It can be mean square continuous, thatis
lims↑t
E |X(t)−X(s)|2E = 0, ∀ t ≥ 0.
Then there is its predictable modification due to the following generalresult (see Gikhmann and Skorokhod (1980) or Peszat and Zabczyk(2007), Prop. 3.21).
Theorem 5. Any measurable stochastically continuous adapted processhas a predictable modification.
7. Poisson random measures
Let (E, E) be a measurable space, and let PZ+([0,∞) × E) be the
space of all measures on [0,∞) × E with values in Z+ := 0, 1, . . . ∪∞. We consider on PZ+
([0,∞) × E) the σ-field generated by map-pings
PZ+([0,∞)× E) 3 ρ 7→ ρ(Γ) ∈ Z+, Γ ∈ B([0,∞))× E .
Let µ be a σ-finite measure on (E, E).
Definition 4. A Poisson random measure on [0∞)×E with intensitymeasure λ(dtdξ) := dtµ(dξ) is a random element π in PZ+
([0,∞)×E)such that
(i) for every Γ ∈ B([0,∞))× E , the random variable π(Γ) has thePoisson distribution
P (π(Γ) = k) = e−λ(Γ) (λ(Γ)t)k
k!,
18
(ii) for any disjoin Γi, . . .Γk ∈ B([0,∞))× E , the random variablesπ(Γ1), . . . , π(Γk) are independent.
Definition 5. Let π be a Poisson random measure with intensity mea-sure dtµ(dξ). We call
π(dtdξ) := π(dtdξ)− dtµ(dξ)
the compensated Poisson random measure.
7.1. Construction of a Poisson random measure. Assume thatµ is finite. Then the Poisson random measure with intensity measuredtµ(dξ) can be written as follows
π(dtdξ) =∑j
δτj ,ξj ,
where τj are the moments of jumps of a Poisson process with intensityµ(E) and ξj are independent random variables with distribution
P (ξj ∈ Γ) =µ(Γ)
µ(E).
In general E =⋃En, where En are disjoint and of finite measure,
and π =∑πn, where πn are independent Poisson random measures
with intensity measures dtχEn(ξ)µ(dξ).
7.2. Stochastic integration (deterministic integrands). We willintroduce here the stochastic integral with respect to a Poisson randommeasure of deterministic mappings. Namely, let f : [0,∞)×E 7→ U besimple, that is
f =∑k
ukχ(tj ,tj+1]χEj ,
where uj ∈ U , 0 ≤ tj ≤ tj+1 and Ej ∈ E . Define∫ t
0
∫E
f(s, ξ)π(dtdξ) =∑k
ukπ((tj ∧ t, tj+1 ∧ t]× Ej).
Then by simple calculation one can show that
(14) E∣∣∣∣∫ t
0
∫E
f(s, ξ)π(dsdξ)
∣∣∣∣U
=
∫ t
0
∫E
|f(s, ξ)|Udtµ(dξ),
and that
(15) E∣∣∣∣∫ t
0
∫E
f(s, ξ)π(dsdξ)
∣∣∣∣2U
=
∫ t
0
∫E
|f(s, ξ)|2Udtµ(dξ).
Therefore we can extend the stochastic integral with respect to π or πto the class of measurable integrable (square integrable functions).
19
8. Poisson random measure corresponding to Levyprocess
A Levy process L on U with Levy measure ν, defines Poisson randommeasure
π((s, t]× Γ) :=∑r∈(s,t]
χΓ(∆L(r)),
where ∆L(r) := L(r)−L(r−). The measure (defined first on cylinders(s, t] × Γ has an extension to Poisson random measure on [0,∞) × Uwith the intensity measure dtν(du).
The Levy–Khinchin decomposition can be written in the followingform:
Theorem 6. (Levy–Khinchin decomposition) Let R > 0. AnyLevy process L taking values in U can be written in the form(16)
L(t) = aRt+W (t) +
∫ t
0
∫|v|U<R
vπ(dsdv) +
∫ t
0
∫|v|U≥R
vπ(dsdv).
9. Processes on l2
Assume that
U = l2 :=
(xk) ∈ RN :
∑k
x2k <∞
.
Let (Zk) be a sequence of real-valued Levy processes each with Levymeasure νk, and let (λk) be a sequence of non-negative real numbers.
9.1. Square integrable case. Assume that each Zk is square inte-grable, mean zero, normalized EZ2
k(1) = 1, and that Zk are uncor-related. Since Zk are mean zero, λkZk are martingales. Note thatZ := (λkZk) is a square integrable in l2 if and only if∑
k
λ2k <∞.
Let ek be the canonical basis of l2. The covariance operator Q isgiven by
〈Qej, ek〉l2 = λjλk EZj(1)Zk(1) = λ2kδk,j.
Thus Qek = λkek. The RKHS of Z is equal to
H = Q1/2(U) =
(xk) :
∑k
x2k
λ2k
<∞
20
with the scalar product
〈(xk), (yk)〉H =∑k
xkykλ2k
,
where we adopt the convention that 0/0 = 0.
9.2. Jump measure. If Zj are independent, then the jump measureν of Z is concentrated on axes, and
ν (xj) : xj ∈ Γ, xk = 0, j 6= k = ν(λkΓ)
for any k ∈ N and any Γ ∈ B(R). Moreover, Z takes values in l2 if andonly if ∫
l2|x|2l2ν(dx) <∞,
or equivalently if ∑k
∫R|λkxk|2 ∧ 1νk(dxk) <∞,
which can be written equivalently as follows∑k
∫(−λ−1
k ,λ−1k )
x2kνk(dxk) + νkR \ (−λ−1
k , λ−1k )
<∞.
10. Impulsive white noise
Let O be an open not necessarily bounded domain in Rd (possiblyO = Rd). Let π be a Poisson random measure on [0,∞)×O×R withintensity of jump measure dtdxν(dσ). Assume that∫
Rσ2 ∧ 1ν(dσ) <∞.
Consider the distributions-valued process
Z(t) =
∫ t
0
∫|σ|<R
σπ(dsdxdσ) +
∫ t
0
∫|σ|≥R
σπ(dsdxdσ).
Taking into account the representation
π(dsdxdσ) =∑
δτk,xk,σk ,
we obtain the following a bit formal expression for Z
Z(t) =
∑|σk|<R,τk≤t
σkδτk,xk − t∫|σ|<R
σdxν(dσ)
+∑
|σk|≥R,Tk≤t
σkδτk,xk .
Intuitively, at random points (τk, xk) at time and space Z gives randomimpulses of random size σk.
21
Remark 6. One can show that
M(t) =
∫ t
0
∫|σ|<R
σπ(dsdxdσ)
is a square integrable martingale in a sufficiently large space, and thatits RKHS equals
H = L2(O,B(O), aRdx), aR :=
∫|σ|<R
σ2ν(dσ).
Thus, in particular, M takes values in any Hilbert space V such thatthe embedding H → V is Hilbert–Schmidt.
Remark 7. The jump measure µ of Z is the image of the measuredxν(dσ) under the transformation
O × R 3 (x, σ) 7→ σδx ∈ D(O),
where D(O) is the space of distribution on O.
Therefore our definition is the following.
Definition 6. Impulsive cylindrical (or white ) noise with intensity ofjumps measure dxλ(dσ) is the Levy process on the space of distribu-tions with the Levy measure ν being the image of dxλ(dσ) under thetransformation (x, σ) 7→ σδx.
Remark 8. Impulsive cylindrical process L takes values in a Hilbertspace U provided that ∫
U
|u|2U ∧ 1ν(du) <∞.
Let U = H−α be the Sobolev space of order −α with α > d/2. Then,by Sobolev embedding,
C := supx∈O|δx|H−α <∞.
Therefore ∫H−α|u|2H−α ∧ 1λ(du) =
∫O
∫Rσ2|δx|H−αdxν(dσ)
≤ C
∫Rσ2 ∧ 1λ(dσ) <∞.
Consequently, L takes values in H−α.22
11. Regularity of stochastic convolution
We start this section with results on the lack of a cadlag modificationfor SPDEs driven by a process whose jump measure is not supportedon the state space. Then we present different tools useful for studyregularity of stochastic convolutions.
11.1. Lack of cadlag modification. As the following example showsin some cases the solution to linear stochastic evolution equation doesnot have a cadlag modification.
Example 5. Let U and H be Hilbert spaces such that
(i) H is densely embedded into U .(ii) One has∫ T
0
‖S(s)‖2L(HS)(H,H) ds <∞. ∀T > 0.
(iii) For any t > 0, S(t) has a continuous extension to an operatorS(t) ∈ L(U,H).
(iv) For any u ∈ U \H,
limt↑0|S(t)u|H =∞.
Let Z be a square integrable mean zero random variable in U withRKHS H, and let L be a compound Poisson process with Levy measureν which is the distribution of Z. Then
X(t) =
∫ t
0
S(t− s)dL(s) =∑τn<t
S(t− τn)Zn,
where τn are the jump times of L and Zj are independent copies of Z.Then, by (ii),
supt≤T
E |X(t)|2H <∞
butlimt↑τn|X(t)|H = lim
t↑τn|S(t− τn)Zn|H =∞,
since Zn take values in U \H.Explicitly, take H = L2(0, 1), U = W−1,2
0 (0, 1), S the heat semigroupgenerated by the Laplace operator with Dirichlet boundary conditions,and Z = ηδξ, where ξ ∈ (0, 1), and η is a mean zero random variable.
We have the following have ben proven by Brzezniak and Zabczyk(2009), and Peszat and Zabczyk (2007).
Theorem 7. If the jump measure of the noise is not supported on Ethen the stochastic convolution does not have cadlag trajectories in E.
23
11.2. Factorization. Stochastic integral with respect to the squareintegrable martingale as a square integrable martingale has a cadlagmodification. This is not always true for stochastic convolution pro-cesses
X(t) :=
∫ t
0
S(t− s)Ψ(s)dM(s), t ≥ 0.
One way to show the continuity of trajectories is to use the so-calledfactorization
X(t) = Γ(1)Iα(Xα)(t),
where
Xα(t) =1
Γ(1− α)
∫ t
0
(t− s)−αS(t− s)ψ(s)dM(s), t ≥ 0,
and Iα is the fractional derivative operator given by
Iαψ(t) =1
Γ(α)
∫ t
0
(t− s)α−1S(t− s)ψ(s)ds,
and Γ is the Euler Γ-function. It is easy to show that
Iα ∈ L(Lp(0, T ;H), C([0, T ];H))
provided that 1/q < α < 1.For the Wiener integral it is usually not hard to show that Xα has
trajectories in Lq(0, T ;H) with some 1/q < α < 1. Therefore thecontinuity of trajectories of X follows. For the Levy process this isusually not true.
11.3. Kotelenez. Kotelenez (1982) proved the regularity of stochasticconvolution ∫ t
0
S(t− s)dM(s), t ≥ 0,
driven an arbitrary square integrable martingale in H for contractionsemigroups S, that is under assumption that
‖S(t)‖L(H,H) ≤ eωt, t ≥ 0.
We outline here some different proof due to Hausenblas and Seidler(2001) and (2006). Their method is based on the Nagy dilation theorem(see Davies (1980) or Nagy and Foias (1970)).
Theorem 8. (Nagy) If S is a C0-semigroup of contractions on H,then there is a Hilbert space H containing H and a unitary group R onH such that S = PR, where P ∈ L(H,H) is a projection.
24
If S(t) = PR(t), t ≥ 0, where P ∈ L(H,H) and R is a group, then∫ t
0
S(t− s)dM(s) = P
∫ t
0
R(t− s)dM(s) = PR(t)
∫ t
0
R(−s)dM(s).
Since
Y (t) :=
∫ t
0
R(−s)dM(s), t ≥ 0,
is a square integrable martingale, Y has cadlag trajectories in H andconsequently X has cadlag trajectories in H as R is strongly continu-ous.
11.4. General criterion for the absence of discontinuities ofthe second kind. We recall here some basic facts from Gikhman andSkorokhod (1980).
Let ξ = (ξ(t), t ∈ [0, T ]) be a separable process taking values in ametric space (U, ρ). We extent ξ in R putting ξ(t) = ξ(0) for t < 0 andξ(t) = ξ(T ) for t ≥ T . The following result holds (see Gikhman andSkorokhod (1980), Lemma 3 and Theorem 1 of Chapter 3).
Theorem 9. Assume that there are an increasing function g : (0,∞) 7→(0,∞) and a function q : (0,∞) × (0,∞) 7→ (0,∞) such that for allC, h > 0,
(17)P [ρ(ξ(t), ξ(t− h)) > Cg(h)] ∩ [ρ(ξ(t), ξ(t+ h)) > Cg(h)]
≤ q(C, h),
and
(18) G :=∑n
g(T2−n) <∞, Q(C) :=∑n
2nq(C, T2−n) <∞.
Then with probability 1, ξ has no discontinuities of the second kind,and for all N > 0,
P
sup
t,s∈[0,T ]
ρ(ξ(t), ξ(s)) > N
≤ P
ρ(ξ(0), ξ(T )) >
N
2G
+Q
(N
2G
).
Remark 9. Assume that there are p, r,K > 0 such that for all t ∈[0, T ] and h > 0,
(19) E [ρ(ξ(t), ξ(t− h))ρ(ξ(t), ξ(t+ h)]p ≤ Kh1+r.
Let 0 < r′ < r. Then (17) and (18) hold with
g(h) := hr′/(2p) and q(C, h) :=
K
C2ph1+r−r′ .
25
For, by Chebyshev’s inequality
P [ρ(ξ(t), ξ(t− h)) > Cg(h)] ∩ [ρ(ξ(t), ξ(t+ h)) > Cg(h)]≤ P
ρ(ξ(t), ξ(t− h))ρ(ξ(t), ξ(t+ h)) > C2g2(h)
≤ Kh1+r
C2pg2p(h)=Kh1+r−r′
C2p.
Note that in this case G =∑
n(T2−n)r′/(2p) <∞, and
(20)
Q
(N
2G
)=∑n
2nK(2G)2p
N2p(T2−n)1+r−r′
=K(2G)2p
N2pT 1+r−r′
∑n
2−n(r−r′)
=K(2G)2pT 1+r−r′
1− 2r′−rN−2p.
Remark 10. Let q ≥ 1. Assume (17) and (18). Since
E supt,s∈[0,T ]
(ρ(ξ(t), ξ(s)))q = q
∫ ∞0
P
sup
t,s∈[0,T ]
ρ(ξ(t), ξ(s)) ≥ N
N q−1dN,
Theorem 9 yields
E supt,s∈[0,T ]
(ρ(ξ(t), ξ(s)))q
≤ (2G)qE (ρ(ξ(T ), ξ(0))q + 1 + q
∫ ∞1
Q
(N
2G
)N q−1dN.
Combining Theorem 9 with Remarks 9 and 10, and identity (20), weobtain the following result.
Corollary 1. Assume (19). Then with probability 1, ξ has no discon-tinuities of the second kind and for any 1 ≤ q < 2p,
E supt,s∈[0,T ]
(ρ(ξ(t), ξ(s)))q
≤ (2G)qE (ρ(ξ(T ), ξ(0))q + 1 +q
2p− qK(2G)2pT 1+r−r′
1− 2r′−r.
12. Stationary solutions to linear problem
12.1. Wiener case. We are concerned with stationary solutions to thefollowing Ornstein–Uhlenbeck equation
(21) dX = AXdt+BdW,26
where A generates a C0 semigroup S on H, B ∈ L(U,H) and W is acylindrical Wiener process on U . Let
Qt :=
∫ t
0
S(t− s)BB∗S∗(t− s)ds.
Assume that X(0) is a random element in H, Note that
X(t) = S(t)X(0) +
∫ t
0
S(t− s)BdW (s)
takes values in H if and only if∫ t
0
‖S(t− s)N‖2L(HS)(U,H)ds <∞.
or equivalently if and only if Qt ∈ L+1 (H) for any or all t > 0.
Note that∫ t
0S(t− s)BdW (S) has the distribution N(0, Qt).
Definition 7. We call probability measure m on H an invariant orstationary measure (solution) to (21) if and only if
m = S(t)m ∗N(0, Qt), ∀ t ≥ 0.
Theorem 10. (Zabczyk (1985)) The following conditions are equiv-alent
(i) there is a stationary solution to (21),(ii) one has
supt≥0
TrQt <∞.
Moreover, if (i) or (ii) holds, then any invariant measure m is of theform
m = β ∗N(0, Q∞),
where Q∞ = limt→∞Qt, and β is a probability measure on H which isinvariant for the semigroup S, that is S(t)β = β for t ≥ 0.
For the proof we need the Bochner theorem.
Theorem 11. (Bochner) Let φ : U 7→ C. Then the following condi-tions are equivalent:
(i) φ = η for some probability measure η on U ,(ii) φ(0) = 0, φ is positive-definite, that is∑
j,k
φ(uj − ui)λjλj ≥ 0.
and S-continuous, that is for any ε > 0 there is Sε ∈ L+1 (U)
such that
Reφ(u) ≥ 1− ε, ∀u ∈ U : 〈u, Sεu〉U ≤ 1.27
12.2. Proof of Theorem 10. Let m be an invariant measure. Then
m = S(t)m ∗ γt,where γ + t = N(0, Qt). Thus
m(u) = m(S∗(t)u) e−12〈Qtu,u〉U .
Since
Re e12〈Qtu,u〉U m(u) ≤ Re m(u) ≤ 1,
we have
e12〈Qtu,u〉U ≤ 1
Re m(u),
and consequently
〈Qtu, u〉U ≤ 2 log1
Re m(u), ∀ t ≥ 0.
By S-continuity, there is an S ∈ L+1 (U) such that
Re µ(u) ≥ 1
2, ∀u : 〈Su, u〉U ≤ 1.
Thus〈Qtu, u〉U ≤ 2 log 2, ∀u : 〈Su, u〉U ≤ 1.
Consequently,0 ≤ Qt ≤ 2 log 2S,
and hencesupt≥0
TrQt ≤ 2 log 2TrS <∞.
Hence (i)⇒ (ii). To see that (ii)⇒ (i) note that
TrQt =
∫ t
0
‖S(s)B‖2L(HS)(U,H)ds.
For, let en be an othonormal basis on H. We have∑n
〈Qten, en〉U =∑n
⟨∫ t
0
S(s)BB∗S∗(s)dsen, en
⟩U
=∑n
∫ t
0
〈B∗S∗(s)en, B∗S∗(s)en〉U
=
∫ t
0
‖B∗S∗(s)‖2L(HS)(H,U)ds.
Thus, as ‖T‖L(HS)(U,H) = ‖T ∗‖L(HS)(H,U) we obtain that
Q∞ =
∫ ∞0
S(s)BB∗S∗(s)ds
28
is nuclear. Therefore N(0, Q∞) is invariant measure as the limit ofdistributions of the solution X(t) starting from 0.
We are showing now that if m is an invariant measure, then
m = β ∗N(0, Q∞)
where β is probability measure invariant for S. To see this note that
m(u) = m(S∗(t)u)e−12〈Qtu,u〉U .
By (ii),
φ(u) := limt→∞
m(S∗(t)u) = m(u)e12〈Q∞u,u〉U .
It is a characteristic functional of a probability measure β by theBochner theorem. The fact that β is invariant with respect to S isobvious.
12.3. Examples of non-uniqueness of invariant measure for lin-ear equation.
Example 6. Let U = L2(0,∞), let S(t)u(x) = etu(t + x), u ∈ U ,x, t ≥ 0. It is easy to check that S is a C0-semigroup on U . Let
φ(x) = e−kx for x ∈ [k, k + 1).
Then S(1)φ = φ. Thus if q has the uniform distribution on [0, 1], andβ is the distribution of S(q)ψ. Then β is invariant for S. Let A be thegenerator of S. Consider the equation
dX = AXdt+ fdW,
where W is a real-valued Wiener process and
f(x) = e−x2
.
Then, with Ba = fa, B ∈ L(R, U),
supt≥0
Tr
∫ t
0
S(s)BB∗S∗(s)ds <∞.
There are at least two (in fact infinitely many) stationary solutions.
Example 7. Consider the system
dx = 0dt,
dy = −1
2ydt+ dW,
Let
µ = N
((00
),
(0 00 1
)),
29
and let κ be a probability measure on R×0. Then κ∗µ is an invariantmeasure.
12.4. Levy case. Let L be a Levy process in U with Levy measureν, and let A be a generator of a C0-semigroup S on U . Consider thefollowing Ornstein–Uhlenbeck equation
(22) dX = AXdt+ dL.
Recall that for any independent of L random variable X(0) in U , thesolution starting from X(0) is given by
X(t) = S(t)X(0) +
∫ t
0
S(t− s)dL(s).
Assume that
L(t) = at+W (t) +M(t) + L0(t),
where a ∈ U , W is a Wiener process in U with the (nuclear) covarianceQW , and M + L0 is a pure jump process with Levy measure ν.
Recall that if m is an invariant (or stationary) measure for the equa-tion, then for any t > 0 the distribution of X(t) is equal to m, providedthat the distribution of X(0) is m.
Proposition 1. If there is a stationary solution then
supt≥0
Tr
∫ t
0
S(s)QWS∗(s)ds <∞.
Proof. Let
Qt :=
∫ t
0
S(s)QWS∗(s)ds, t ≥ 0.
We have
m = S(t)m ∗N(0, Qt) ∗ γt, ∀ t ≥ 0,
where γt is the distribution of∫ t
0
S(t− s)d(as+M(s) + L0(s)).
Passing to the characteristic functionals we obtain
m(u) = m(S∗(t)u)e−12〈Qtu,u〉U γt(u), u ∈ U.
Thus
〈Qtu, u〉U ≤ 2 log1
Re m(u)as
Re m(S∗(t)u)γt(u) ≤ 1.30
By Bochner theorem, there is a trace class operator S ∈ L+1 (U) such
that
Re m(x) ≥ 1
2, ∀u : 〈Su, u〉U ≤ 1.
Then〈Qtu, u〉U ≤ 2 log 2, ∀u : 〈Su, u〉U ≤ 1.
ConsequentlyQt ≤ 2 log 2S, ∀ t ≥ 0.
From now on assume that W = 0 and a = 0. Thus we are concernedwith the pure jump case.
Proposition 2. (Chojnowska-Michalik (1987)) If γt convergesweakly to a probability measure γ, then there is an invariant measurem for (22). Moreover, any invariant measure m is of the form
(23) m = β ∗ γ,where β is any invariant measure for S.
Proof. If γ is the weak limit of γt, then by the Krylov–Bogolyubovtheorem γ is invariant. What is left is to show that any invariantmeasure has the form (23). To do this assume that m is an invariantmeasure. Then
m = S(t)m ∗ γt, ∀ t ≥ 0,
and consequently
m(u) = m(S(t)u)γt(u), ∀ t ≥ 0, ∀u ∈ U.Since γt(u)→ γ(u), there is a limit
ψ(u) := limt→∞
m(S∗(t)u) =m(u)
γ(u).
We have to show that ψ is a characteristic function of a probabilitymeasure β. Then the invariance of β with respect to S is obvious since
ψ(S∗(t)u) = ψ(u), ∀ t ≥ 0, ∀u ∈ U.To to this we can use the following general result whose proof can beshown in Parthasarathy (1967), Th 2.1 , p. 58.
Theorem 12. Let (λn), (µn) and (νn) be sequences of measures on aPolish space X . Assume that
λn = µn ∗ νn, ∀n.Then, if (λn) and (νn) are relatively weakly compact, then (µn) is rel-atively weakly compact.
31
We apply the theorem to the case of X = U , λn = m, µn = S(tn)mand νn = γtn where tn ↑ ∞. By the theorem there is a sequence tn ↑ ∞such that S(tn)m converges weakly as n ↑ ∞ to a certain measure β.
Therefore ψ = β s required.
Assume that the semigroup S generated by A is exponentially stable,that is there are C and ω > 0 such that
‖S(t)‖L(U,U) ≤ Ce−ωt, ∀ t ≥ 0.
Theorem 13. (Chojnowska-Michalik (1987) If S is exponentiallystable then the following conditions are equivalent:
(i) there is an invariant measure for (22).(ii) the distributions γt of∫ t
0
S(t− s)dL(s)
converge weakly as t ↑ ∞.(iii) The jump measure ν of L satisfies∫
U
log+ |u|Uν(du) <∞.
Moreover, the measure if exists must me unique and is equal γ =limt→∞ γt.
Proof. The implication (ii) ⇒ (i) follows from Proposition 2. In or-der to show the uniqueness note that by Proposition 2, any invariantmeasure m must by of the form m = β ∗ γ, where β is invariant forS. Therefore it s enough to note that any invariant for S probabilitymeasure must by equal to δ0. To see this note that
β(u) = limt→∞
m(S∗(t)u).
Since S∗(t)u → 0 as t → ∞, we have β(u) = m(0) = 1, and henceβ = δ0.
In order to show (i) ⇒ (ii), note that if m is an invariant measurethen
m = S(t)m ∗ γt.Since S(t)m converges weakly to δ0 we conclude by Theorem 12, thatγt converges.
The proof of the equivalence of (ii) and (iii) is the most difficult part.We will present a sketch of the proof of (iii) ⇒ (ii). The strategy isthe following. Let
Z(t) =
∫ t
0
S(t− s)dL(s).
32
We will show that for each t, Z(t) has the same distribution as a certainLevy process Lt(1), with the Levy exponent
Ψt(u) = −iatu+
∫U
(1− ei〈u,v〉U + χ|v|U<1(v)i〈u, v〉U
)νt(dv).
where at ∈ U and νt is a Levy measure. Using stability of S we willshow that at converges to certain a. We will show that νt converges toa certain ν satisfying ∫
U
|u|2U ∧ γ(du) <∞,
and hence Ψt(u) converges to
Ψ(u) = −iau+
∫U
(1− ei〈u,v〉U + χ|v|U<1(v)i〈u, v〉U
)ν(dv).
To calculate (at, νt) we need to calculate the characteristic functionalof Z(t). To this end we use Theorem 4. Let B = v ∈ U : |v|U < 1.We have
E ei〈Z(t),u〉U = e−Ψt(u),
where
Ψt(u) =
∫ t
0
Ψ(S∗(t− s)u)ds =
∫ t
0
Ψ(S∗(s)u)ds
= −i
∫ t
0
〈a, S∗(s)u〉Uds
+
∫ t
0
∫U
(1− ei〈S∗(s)u,v〉U − χB(v)i〈S∗(s)u, v〉U
)ν(dv)
= −i
⟨u,
∫ t
0
S(s)uds
⟩U
+ i
⟨a,
∫ t
0
∫U
[χB(v)S(s)v − χB(S(s)v)] ν(dv)ds
⟩U
+
∫ t
0
∫U
(1− ei〈S∗(s)u,v〉U − χB(s(s)v)i〈u, S(s)v〉U
)ν(dv).
Therefore,
νt =
∫ t
0
S(s)νds,
and we have to show that∫U
|u|2U ∧ 1
∫ ∞0
S(s)µ(du)ds <∞.33
Additionally have to show that
It :=
∫ t
0
∫U
[χB(v)S(s)v − χB(S(s)v)] ν(dv)ds
converges. Using more detailed calculations one can shows that theabove claims are equivalent to∫
U
log+ |u|Uν(du).
We will do some calculations for the convergence of It. We have It =I1t − I2
t , where
I1t =
∫ t
0
∫|u|U≤1,|S(s)u|U>1
S(s)uν(du)ds
and
I1t =
∫ t
0
∫|u|U>1,|S(s)u|U≤1
S(s)uν(du)ds.
We have
|I1t |U ≤
∫ t
0
∫|u|U≤1
|S(s)u|2Uν(du)ds
≤∫ ∞
0
∫|u|U≤1
|u|2Uν(du)C2e−2ωsds <∞.
Clearly
|I2t |U ≤
∫ ∞0
∫|u|U>1,|S(s)u|U≤1
|S(s)u|Uν(du)ds
≤∫ ∞
0
∫|u|U∈(1,eωt/2)
|S(s)u|Uν(du)ds+
∫ ∞0
∫|u|U≥eωs/2
ν(du)ds.
We have ∫ ∞0
∫|u|U≥eωs/2
ν(du)ds =
∫ ∞0
∫ 2ω
log |u|U≥tν(du)ds
=2
ω
∫U
log+ |u|Uν(du).
34
Finally, ∫ ∞0
∫1<|u|U<eωs/2
|S(s)u|Uν(du)ds
≤ C
∫ ∞0
∫1<|u|U<eωs/2
|u|Ue−ωsν(du)ds
≤ C
∫ ∞0
∫1<|u|U<eωs/2
e−ωs2 ν(du)ds
≤ C
∫ ∞0
e−ωs2 dsν|u|U > 1 <∞.
For more details see Chojnowska-Michalik (1987) and Applebaum(2006). Finite dimensional case is treated also in Sato (1999).
Appendix A. Hilbert–Schmidt and nuclear operators
In what follows U and H are real separable Hilbert spaces. The spaceof all bounded (i.e. continuous) linear operators from U to H is denotedby L(U,H). A bounded linear operator T : U 7→ H is Hilbert–Schmidtif and only if ∑
k
|Tek|2H <∞
for every or equivalently for some orthonormal basis ek of U . Thespace of Hilbert–Schmidt operators from U toH is denoted by L(HS)(U,H).It is a Hilbert space with the norm
‖T‖L(HS)(U,H) :=
(∑k
|Tek|2H
)1/2
.
Given u ∈ U and h ∈ H we denote by u⊗ h a linear operator from Uto H given by
u⊗ h(v) = 〈u, v〉U h, v ∈ U.A bounded linear operator T : U 7→ H is trace class or nuclear if andonly if it can be written in the form
T =∑k
uk ⊗ hk,
where uk ⊂ U , hk ⊂ H and∑k
|uk|U |hk|H <∞.
35
The space L1(U,H) of all nuclear operators is equipped with the norm
‖T‖L1(U,H) := inf
∑k
|uk|U |hk|H : T =∑k
uk ⊗ hk
.
Given T ∈ L1(U) := L1(U,U) we denote by TrT the trace of T ,
TrT :=∑k
〈Tek, ek〉U ,
where ek is an orthonormal basis of U . It is known that TrT doesnot depend on the choice of the orthonormal basis ek of U .
Let L+1 (U) be the class of all nuclear operators T ∈ L1(U) such that
T = T ∗ ≥ 0.
Remark 11. Let X be a square integrable (i.e. E |X|2U <∞) randomvector in U . Then its covariance operator Q defined by
〈Qu, v〉U := E 〈X − EX, u〉U〈X − EX, v〉U , u, v ∈ U,belongs to L+
1 (U). Moreover,
TrQ = E |X − EX|2U .
Example 8. Let U = H = L2(E, E , γ). Then the operator
Tψ(x) =
∫E
G(x, y)ψ(y)γ(dy)
is Hilbert–Schmidt if and only if∫E
∫E
G2(x, y)γ(dx)γ(dy) <∞.
Moreover,
‖T‖2L(HS)(U,U) =
∫E
∫E
G2(x, y)γ(dx)γ(dy) <∞.
Appendix B. C0-semigroups
Definition 8. Let (E, ‖ · ‖E) be a Banach space. A family S(t), t ≥ 0,of bounded linear operators from E into E is a strongly continuous (C0
in short) semigroup if
• S(0) = I, (I stands for the identity operator on E),• S(t+ s) = S(t)S(s) for all t, s ≥ 0,• ‖S(t)ψ − ψ‖E → 0 as t ↓ 0, for every ψ ∈ E.
The second property listed above is the semigroup property, and thelast one is the strong continuity property. For the proof of the followingtheorems see e.g. Pazy (1983)
36
Theorem 14. Let S = (S(t)) be a C0-emigroup on E. Then there areconstants γ ≥ 0 and M ≥ 1 such that
‖S(t)‖L(E,E) ≤Meγt, ∀ t ≥ 0.
Definition 9. If S is a C0-semigroup such that there is a γ ≥ 0 suchthat ‖S(t)‖L(E,E) ≤ eγt for t ≥ 0, then S is called C0-semigroup of(generalized) contractions.
Definition 10. Let S be a C0-semigroup. A linear operator A definedby
Dom (A) =
ψ ∈ E : lim
t↓0
1
t(S(t)ψ − ψ) exists
and
Aψ = limt↓0
1
t(S(t)ψ − ψ) for ψ ∈ Dom (A)
is the infinitesimal generator of the semigroup.
Theorem 15. The generator of a C0-semigroup is densely defined andclosed.
Appendix C. Ito–Nisio theorem
The following result due to Ito and Nisio is concerned with differenttypes of convergence of a series of independent random variables takingvalues in b separable Banach space E. For its proof we refer the readerto Kwapien and Woyczynski (1992) or the original paper of Ito andNisio (1968). In its formulation Sn := X1 + . . .+Xn and L(Sn) denotesthe distribution of Sn.
Theorem 16. Let (Xk) be a sequence of independent random variablestaking values in E. Then the following conditions are equivalent:
(i) the series∑
kXk converges a.s.,(ii) the series
∑kXk converges in probability,
(iii) distributions L(Sn) converge weakly.
It additionally (Xk) are symmetric, then conditions (i)−(iii) are equiv-alent to each of the following condition:
(iv) the sequence (L(Sn)) is relatively weakly compact,(v) there exists a random variable S with values in E, and a familly
D ⊂ E∗, separating points of E, such that for each x∗ ∈ D,series
∑k x∗(Xk) converges a.s. to x∗(S),
(vi) there exists a probability measure µ on E, and a familly D ⊂E∗, separating points of E, such that for each x∗ ∈ D, series∑
k x∗(Xk) converges in distribution to x∗(µ).
37
The following result whose proof can be found in (?), Corollary 2.2.1is of particular interest.
Theorem 17. Let p > 0, and let Xk are independent random elementsin E. Then the series
∑kXk converges a.s. if and only if for some
a > 0, or equivalently for any a > 0, the following conditions aresatisfied:
(i) the series∑
k P (‖Xk‖E > a) <∞,(ii) the series
∑kXkχ‖Xk‖E≤a converges in p-th mean.
In the case of Hilbert space-valued random elements the theoremabove yields the so-called Three Series Theorem
Theorem 18. Let Xk are independent random elements in a Hilbertspace E. Then the series
∑kXk converges a.s. if and only if for some
a > 0, or equivalently for any a > 0, the following three series converge:
(i)∑
k P (‖Xk‖E > a),(ii)
∑k EXkχ‖Xk‖E≤a,
(iii)∑
k Var(Xkχ‖Xk‖E≤a
).
References
[1] Albeverio, S. and Rudiger, B. (2005). Stochastic integrals and the Levy-Itodecomposition on separable Banach spaces, Stoch. Anal. Appl. 23, 217–253.
[2] Applebaum, D. (2004). Levy Processes and Stochastic Calculus (Cambridge,Cambridge University Press,).
[3] Applebaum, D. (2006). Martingale-valued measures, Ornstein–Uhlenbeckprocesses with jumps and operator self-decomposability in Hilbert space,Seminaire de Probabilite, Lecture Notes in Mathematics, Vol. 1897, (Berlin,Springer), pp. 173–198.
[4] Bjork, T. and Christensen, B.J. (1999). Interest rate dynamics and consistentforward rate curves, Math. Finance 9, 323–348.
[5] Bjork, T., DiMassi, G., Kabanov, Y. and Runggaldier, W. (1997). Towardsa general theory of bound markets, Finance Stoch. 1, 141–174.
[6] Bjork, T., Kabanov, Y. and Runggaldier, W. (1997). Bond market structurein the presence of marked point process, Math. Finance 7, 211–239.
[7] Brzezniak, Z. and Elworthy, K. D. (1996). Stochastic flows of diffeomor-phisms, Stochastic analysis and applications (Powys, 1995), pp.107–138,World Sci. Publ., River Edge, NJ.
[8] Carverhill, A.P. and Elworthy, D. (1983). Flow of stochastic dynamical sys-tems: the Functional Analysis approach, Z. Wahrsch. Verw. Gebiete, 65,245–267.
[9] Chojnowska-Michalik, A. (1978). Representation theorem for general stochas-tic delay equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. 26, 634–641.
[10] Chojnowska-Michalik, A. (1987). On processes of Ornstein–Uhlenbeck typein Hilbert spaces, Stochastics 21, 251–286.
38
[11] Courrege, Ph. (1965/66). Sur la forme integro-differentielle des operateursde C∞K dans C0 satisfaisant au principe du maximum, in Sem. Theorie duPotentiel, Expose 2.
[12] Da Prato, G., Kwapien, S. and Zabczyk, J. (1987). Regularity of solutions oflinear stochastic equations in Hilbert spaces, Stochastics 23, 1–23.
[13] Da Prato, G. and Zabczyk, J. (1992a). Stochastic Equations in Infinite Di-mensions (Cambridge, Cambridge University Press).
[14] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite DimensionalSystems (Cambridge, Cambridge University Press).
[15] Eberlein, E. and Raible, S. (1999). Term structure models driven by generalLevy processes, Math. Finance 9, 31–53.
[16] Engel, K-J., and Nagel, R. (2000). One-Parameter Semigroups for Liner Evo-lution Equations (Springer).
[17] Filipovic, D. (2001). Consistency problems for Heath-Jarrow-Morton interestrate models, Lecture Notes in Mathematics, Vol. 1760 (Berlin, Springer).
[18] Filipovic, D. and Tappe, S. (2006). Existence of Levy term structure models,submitted.
[19] Gikhman, I.I. and Skorokhod, A.V. (1980). The Theory of Stochastic Pro-cesses I, (Springer).
[20] Hausenblas, E. and Seidler, J. (2001). A note on maximal inequality for sto-chastic convolutions, Czechoslovak Math. J. 51, 785–790.
[21] Hausenblas, E. and Seidler, J. (2006). Maximal inequalities and exponentialintegrability for stochastic convolutions driven by martingales, preprint.
[22] Heath, D., Jarrow, R. and Morton, A. (1992). Bond pricing and the termstructure of interest rates: A new methodology, Econometrica 61, 77–105.
[23] Ichikawa, A. (1986). Some inequalities for martingales and stochastic convo-lutions, Stochastic Anal. Appl. 4, 329–339.
[24] Ito, K. (1951). On stochastic differential equations, Mem. Amer. Math. Soc.4, 1–51.
[25] Ito, K. (1984). Foundations of Stochastic Differential Equations in InfiniteDimensional Spaces, (Philadelphia, SIAM,).
[26] Ito, K. and Nisio, M. (1968). On the convergence of sums of independentBanach space valued random variables, Osaka Math. J. 5, 35–48.
[27] Jakubowski, J. and Zabczyk, J. (2004). HJM condition for models with Levynoise, IM PAN Preprint 651, (Warsaw).
[28] Jakubowski, J., M. and Zabczyk, J. (2007). Exponential moments for HJMmodels with jumps, Finance Stochastics, to appear.
[29] Knoche, C. (2004). SPDEs in infinite dimension with Poisson noise, C. R.Acad. Sci. Paris, Ser. I 339, 647–652.
[30] Kotelenez, P. (1982). A submartingale type inequality with applications tostochastic evolution equations, Stochastics 8, 139–151.
[31] Kotelenez, P. (1984). A stopped Doob inequality for stochastic convolutionintegrals and stochastic evolution equations, Stochastic Anal. Appl. 2, 245–265.
[32] Kotelenez, P. (1987). A maximal inequality for stochastic convolution inte-grals on Hilbert space and space-time regularity of linear stochastic partialdifferential equations, Stochastics 21, 345–458.
39
[33] Kwapien, S. and Woyczynski, W.A. (1992). Random Series and StochasticIntegrals: Single and Multiple, (Birkhauser, Boston, Basel, Berlin).
[34] Lasota, A. and Szarek, T. (2006). Lower bound technique in the theory of astochastric differential equation, J. Differential Equations 231, 513–533.
[35] Linde, W. (1986). Probability in Banach Spaces - Stable and Infinitely Divis-ible Distributions, (Wiley, New York).
[36] Lunardi, A. (1995). Analytic Semigroup and Optimal Regularity of ParabolicProblems, (Birkhauser).
[37] Metivier, M. (1982). Semimartingales: A Course on Stochastic Processes, (DeGruyer, Berlin).
[38] Mueller, C. (1998). The heat equation with Levy noise, Stochastic Process.Appl. 74, 67–82.
[39] Mytnik, L. (2002). Stochastic partial differential equation driven by stablenoise, Probab. Theory Relat. Fields 123, 157–201.
[40] Nagy, S. and Foias, C. (1970). Harmonic Analysis of Operators on HilbertSpace, (North-Holland, Amsterdam).
[41] Pardoux, E. (1979). Stochastic partial differential equations and filtering ofdiffusion processes, Stochastics 3, 127–67.
[42] Parthasarathy, K.R. (1967). Probability Measures on Metric Spaces, (Aca-demic Press, New York).
[43] Pazy, A. (1983). Semigroups of Linear Operators and Applications to PartialDifferential Equations, (Springer, Berlin).
[44] Peszat, S., Rusinek, A., and Zabczyk, J. (2007). On a stochastic partialdifferential equation of bond market, submitted
[45] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equationswith Levy Noise, (Cambridge Univ. Press, Cambridge).
[46] Peszat, S. and Zabczyk, J. (2006). Stochastic heat and wave equationsdriven by an impulsive noise, in Stochastic Partial Differential Equationsand Applications- VII, eds. G. Da Prato and L. Tubaro, Lect. Notes PureAppl. Math., 245, (Chapman & Hall/CRC, Boca Raton), 229–242.
[47] Rozovskii B.L. (1990). Stochastic Evolution Equations. Linear Theory andApplications to Non-linear Filtering, (Kluwer).
[48] Rusinek, A. (2006a). Invariant measures for a class of stochastic evolutionequations, Preprint IMPAN 667, Warszawa.
[49] Rusinek, A. (2006b). Invariant measures for forward rate HJM model withLevy noise, Preprint IMPAN 669.
[50] Saint Loubert Bie, E. (1998). Etude d’une EDPS conduite par un bruit pois-sonien, Probab. Theory Related Fields 111, 287–321.
[51] Sato, K.I. (1999). Levy Processes and Infinite Divisible Distributions, (Cam-bridge University Press, Cambridge).
[52] Stroock, D.W. (2003). Markov Processes from K. Ito’s Perspective, (PrincetonUniversity Press, Princeton).
[53] Tehranchi, M. (2005). A note on invariant measures for HJM models, FinanceStochastics 9, 387–398.
[54] Viot M. (1974). Solution en lois d’une equation aux derivees partielles nonlineaires, These, Universite Pierre et Marie Curie, Paris.
40
[55] Walsh, J.B. (1986). An introduction to stochastic partial differential equa-
tions, in Ecole d’ete de probabilites de Saint-Flour XIV - 1984, Lecture Notesin Math., 1180, Springer, Berlin New York, 265–439.
[56] Yosida, K. (1965). Functional Analysis, Grundlehren Math. Wiss., Vol. 123,(Springer, Berlin).
[57] Zabczyk, J. (1983). Stationary distributions for linear equations driven bygeneral noise, Bull. Polish Acad. Sci. 31, 197–209.
[58] Zabczyk, J. (1889). Symmetric solutions of semi-linear stochastic equations,in Stochastic Partial Differential Equations and Applications, II (Trento,1988), Lecture Notes in Math., 1390, (Springer, Berlin), 237–256.
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