ergodicity of stochastic curve shortening flow in the plane

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. MATH. ANAL. c© 2012 Society for Industrial and Applied MathematicsVol. 44, No. 1, pp. 224–244

ERGODICITY OF STOCHASTIC CURVE SHORTENING FLOW INTHE PLANE∗

ABDELHADI ES-SARHIR† AND MAX-K. VON RENESSE†

Abstract. We study models of the motion by mean curvature of an (1+1)-dimensional interfacewith random forcing. For the well-posedness we prove existence and uniqueness for certain degeneratenonlinear stochastic evolution equations in the variational framework of Krylov–Rozovskiı, replacingthe standard coercivity assumption by a Lyapunov-type condition. We also study the long-termbehavior, showing that the homogeneous normal noise model [N. Dirr, S. Luckhaus, and M. Novaga,Calc. Var. Partial Differential Equations, 13 (2001), pp. 405–425], [P. E. Souganidis and N. K. Yip,Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), pp. 1–23] with periodic boundary conditionsconverges to a spatially constant profile whose height behaves like a Brownian motion. For theadditive vertical noise model with Dirichlet boundary conditions we show ergodicity, using the lowerbound technique for Markov semigroups by Komorowski, Peszat and Szarek [Ann. Probab., 38 (2010),pp. 1401–1443].

Key words. degenerate stochastic equations, monotonicity, e-property

AMS subject classifications. 47D07, 60H15, 35R60

DOI. 10.1137/100798235

1. Introduction. Motion by mean curvature of embedded surfaces is a well-studied and rich object of research in geometric PDE theory for which a variety ofmethods have been developed (cf. [24] for a survey). In physics it arises as the sharpinterface limit of the Allen–Cahn equation for the phase field of a binary alloy, describ-ing the motion of an interface between the two phases. Stochastic mean curvature flowwas proposed in [7] as a refined model incorporating the influence of thermal noise.As a result one may think of the following general model, describing the incrementsin the normal direction of the evolving surface (Mt)t≥0 ⊂ R

d:

(1.1) dMnt (x) = −κMt(x)dt+ 〈νMt(x),W (Mt, x, ◦dt)〉, x ∈Mt,

where κMt ∈ R and νMt ∈ Rd denote the mean curvature, respectively, the normal

of Mt and for a given surface M ⊂ Rd, W (M, ., .) : Rd × t �→ R

d is a model specificrandom field with W (Mt, x, ◦dt) being its Stratonovich differential. As an exampleconsider W (M,x, t) = νMt(x)ϕ(x)βt for ϕ ∈ C∞(Rd) and a standard real Brownianmotion β, inducing the dynamics

(1.2) dM(x) = −νMt(x)(κMt(x)dt+ ϕ(x) ◦ dbt

).

As in the deterministic case, one approach to a rigorous definition of (1.1) uses theparametrization of the evolving surfaces (Mt)t≥0 as zero level sets Mt = {f(t, ·) = 0}for a flow of level set functions, interpreting (1.1) as the condition that any R

d-valuedsemimartingale (xt)t∈[0,ε] with xt ∈Mt for all t ∈ [0, ε[ must satisfy(1.3)∫ t

0

〈νMs(xs), ◦dxs〉 = −∫ t

0

κMs(xs)ds+

∫ t

0

〈νMs(xs),W (Ms, xs, ◦ds)〉 ∀t ∈ [0, ε].

∗Received by the editors June 10, 2010; accepted for publication (in revised form) July 14, 2011;published electronically January 19, 2012. This work was supported by the DFG Forschergruppe718 “Analysis and Stochastics in Complex Physical Systems.”

http://www.siam.org/journals/sima/44-1/79823.html†Technische Universitat Berlin, Institut fur Mathematik Straße des 17. Juni 136, D-10623 Berlin,

Germany ([email protected], [email protected]).

224

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


ERGODICITY OF STOCHASTIC MCF IN THE PLANE 225

Combined with Ito’s formula, this leads to an SPDE for f of the form

(1.4) df(x) = |∇f | div(

∇f|∇f |

)(x)dt + 〈∇f(x),W (f(x),∇f(x),∇2f(x);x, ◦dt)〉.

In case of (1.2) this equation reads

(1.5) df(x) = |∇f | div(

∇f|∇f |

)(x) dt + φ(x)|∇f |(x) ◦ dβt,

which in the (d − 1)-dimensional graph case (when f(x, y) = y − u(x) for (x, y) ∈R

d−1 × R) simplifies further to

(1.6) du(x) =√1 + |∇u|2 div

(∇u√

1 + |∇u|2

)(x)dt+

√1 + |∇u(x)|2ϕ(x, u(x))◦dbt.

Due to the degeneracy of the drift operator a rigorous treatment of this familyof SPDE is difficult. In particular, a general well-posedness theory seems to be stillmissing. Motivated by the deterministic counterpart of (1.4) Lions and Souganidisintroduced a notion of stochastic viscosity solutions [12, 13], but certain technicaldetails of this approach are still being investigated [1, 2]. However, some models areincluded such as (1.2) with constant ϕ = ε > 0, which was then studied independentlyin d = 2 by Souganidis and Yip [21], respectively, Dirr, Luckhaus, and Novaga [4],proving a “stochastic selection principle” for ε tending to zero.1

Several approaches for constructing generalized solutions to other versions of (1.1)can be found in the literature; see Yip [23], who selects subsequential limits alongtight approximations, and more recently Roger and Weber [20], who extends therigorous analysis of the sharp interface limit of the one-dimensional stochastic Allen–Cahn equation by Funaki [5]. A detailed analysis of associated formal large deviationfunctionals was started in [8] and remains an active research field.

In this paper we consider variants of (1.1) in the two-dimensional graph case,when the surface M is now just a cord in the two-dimensional (x, y)-plane which canbe parametrized as the graph of a function u. Moreover we impose the cord to bepinned to the x-axis at its end points, resulting in Dirichlet boundary conditions onu. The evolution of M shall be given by

(1.7) dMt = −κMtνMtdt+ eyW (Mt, dt),

where ey =(01

)and W = W ((x, y), t) is a scalar Gaussian field over the (x, y)-plane

and W ((x, y), dt) denotes its Ito differential. In this way (1.7) describes a (mean)curvature flow of the membraneM subject to strictly vertical Gaussian perturbations.Assuming without loss of generality the representation of the noise field as

W ((x, y), t) =

∞∑i=1

φi(x, y)bit

with φi : [0, 1] × R �→ R and an independent sequence {(bit)t≥0} of standard realBrownian motions bi; similarly to the derivation of (1.6), the following SPDE for u is

1This means for ε → 0 the level sets the solutions fεt to (1.5) converge a.s. to some solution of

mean curvature flow even in cases when f0t develops “fattening,” i.e., has zero level sets of positive

Lebesgue measure.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php


226 ABDELHADI ES-SARHIR AND MAX-K. VON RENESSE

obtained:

(1.8) du(x) =∂2xu

1 + (∂xu)2(x) dt+

∞∑i=1

φi(x, u(x)) dbit.

In the deterministic case this equation is also known as the curve shortening flow. Notethat the mild solution approach by da Prato and Zabzcyk [3] is not applicable because(1.8) is not semilinear. Instead we work in the variational SPDE framework pioneeredby Pardoux [15, 16] and Krylov and Rozovskiı [10]. Here we establish an abstractexistence and uniqueness result for a certain class of nonlinear stochastic evolutionequations which are not coercive but satisfy an alternative Lyapunov condition. Thisis then applied to (1.8), which is treated in the Gel’fand triple

H10 ([0, 1]) ⊂ L2([0, 1]) ⊂ H−1([0, 1]),

although the operator A : H10 ([0, 1]) → H−1([0, 1]),

Au =∂2xu

1 + (∂xu)2,

fails to be coercive. By our method we prove well-posedness of (1.8), assuming u0 ∈H1

0 , φi ∈ C1([0, 1]×R), φi(0, .) = φi(1, .) = 0, and, for some finite Λ, z1, z2 ∈ [0, 1]×R,

(1.9)

∞∑i=1

(φi(z1)− φi(z2))2 ≤ Λ2|z1 − z2|2.

Condition (1.9) is well known, for example, in the theory of isotropic flows, whereit guarantees the existence of a forward stochastic flow dΦ = F (Φ, dt) of homeomor-phisms of [0, 1] × R+ driven by the martingale field F (z, t) =

∑∞i=1 φi(z)b

it; cf. [11,

Theorem 4.5.1].In section 3.2 we apply this general framework also to the above-mentioned model,

(1.10) du(x) =∂2xu

1 + (∂xu)2(x) dt+ ε

√1 + (∂xu)2 ◦ dbt,

in case ε ≤√2 with periodic boundary conditions and which was treated in [21, 4] by

completely different (comparison) arguments.Moreover, we show below that both (1.8) and (1.10) admit uniquely defined

generalized solutions for initial conditions u0 ∈ L2(0, 1) which is a Markov process(uxt ;x ∈ L2([0, 1]); t ≥ 0) on L2([0, 1]), inducing a Feller semigroup on the space ofbounded continuous functions on L2([0, 1]). However, in view of the poor regularityof the operator A, a more explicit characterization of the L2([0, 1])-valued process(uxt )t≥0 by some SPDE or even just an associated Kolmogorov operator on smoothfinitely based test functions does not seem to be available. This is very similar tothe generalized solutions for abstract SPDE with only m-accretive drift operators ob-tained in [19] by means of nonlinear semigroup theory. The advantage in the presentcase is, however, that the variational approach is embedded such that we know thesolution (uxt )t≥0 is strong if (1.9) holds and the initial condition x = u0 has a weakderivative in L2([0, 1]).

Finally we study the long-term behavior of these models. For (1.10) we showthat the fluctuations around the spatial average vanish asymptotically and that the

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



mean value process converges weakly to a scaled standard Brownian motion. Thelong-term behavior of (1.8) is more complex, and here we prove the ergodicity of thegeneralized solution (uxt ) in the case of additive noise by verifying the conditions ofa recent abstract result by Komorowski, Peszat, and Szarek for Markov semigroupswith the so-called e-property [9, Theorem 1]. We point out that [9, Theorem 3] doesnot apply directly in our situation because the deterministic flow does not convergeto equilibrium locally uniformly with respect to the initial condition. However, forthe verification of the critical lower bound in our case we exploit the fact that thestochastic flow admits a Lyapunov function with compact sublevel sets.

2. Well-posedness of certain noncoercive variational SPDEs.

2.1. Strong solutions for a class of noncoercive SPDEs with regularinitial condition. Although we are mainly interested in the example (1.8) we shallformulate here a general existence and uniqueness result in the abstract variationalframework of [10] for stochastic evolution equations, following with only a few changesthe excellent presentation in [18]. Let

V ⊂ H

be a continuous and dense embedding of two separable Hilbert spaces with corre-sponding inner products 〈., .〉V and 〈., .〉H . Via the Riesz isomorphism on H , thisinduces the Gel’fand triple

V ⊂ H ⊂ V ∗

such that in particular

V ∗〈u, v〉V = 〈u, v〉H ∀ u ∈ H, v ∈ V.

In addition we shall also assume that the inner product 〈., .〉V induces a closedquadratic form on H . This implies the existence of a densely defined self-adjointoperator L : H ⊃ D(L) → H on H such that V = D(

√L), 〈u, v〉V = 〈u, Lv〉H for

u ∈ V, v ∈ D(L) and such that the closure of L : V ⊃ D(L) → V ∗, still denoted byL, defines an isometry. Moreover we assume that L has discrete spectrum with cor-responding eigenbasis (ei)i≥n, which will be the case if, for example, the embeddingV ⊂ H is compact.

Let (W (t))t≥0 be a cylindrical white noise on some separable Hilbert space (U,〈., .〉U ) defined on some probability space (Ω,P,F), and let Ft = σ(Ws, s ≤ t) be theassociated filtration. For X = H , respectively, X = V , we denote by L2(U,X) theclass of Hilbert–Schmidt mappings from U to X , equipped with the Hilbert–Schmidtnorm ‖T ‖2L2(U,X) =

∑i≥1〈Tui, T ui〉X , where (ui)i≥1 is some orthonormal basis of U .

Let

A : V → V ∗, σ : V → L2(U, V )

be measurable maps. Then the existence and uniqueness result below applies to H-valued Ito-type stochastic differential equations of the form

(2.1)

{du(t) = Au(t)dt+ σ(u(t))dWt,u(0) = u0 ∈ H.

Below we shall work under the following set of assumptions on the coefficients Aand B.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



(H1) (Hemicontinuity.) For all u, v, x ∈ V the map

R � λ→V ∗ 〈A(u + λv), x〉V

is continuous.(H2) (Weak monotonicity.) There exists c1 ∈ R such that for all u, v ∈ V

2 V ∗〈Au−Av, u − v〉V + ‖σ(u)− σ(v)‖2L2(U,H) ≤ c1‖u− v‖2H .

(H3) (Lyapunov condition.) For n ∈ N, the operatorAmapsHn := span{e1, . . . , en}⊂ V into V and there exists a constant c2 ∈ R such that

2 〈Au, u〉V + ‖σ(u)‖2L2(U,V ) ≤ c2(1 + ‖u‖2V ) ∀u ∈ Hn, n ∈ N.

(H4) (Boundedness.) There exists a constant c3 ∈ R such that

‖A(u)‖V ∗ ≤ c3(1 + ‖u‖V ).

Remark 2.1. Note that (H3) replaces the standard coercivity assumption in [10]:

(A) 2 V ∗〈Au, u〉V + ‖σ(u)‖2L2(U,H) ≤ c2‖u‖2H − c4‖u‖αV ∀v ∈ V

for some positive constant c4 and α > 1. Both conditions (H3) and (A) yield thecompactness of the Galerkin approximation in V . Condition (A) is used indirectly byapplying the finite-dimensional Ito formula to the square of the H-norm. In our casewe use condition (H3) directly by application of the finite-dimensional Ito formula tothe squared V -norm functional.

Basically, a solution to (2.1) is a V -valued process such that the equation holdsin V ∗ in integral form; cf. [10]. The following precise definition is taken from [18].

Definition 2.2. A continuous H-valued (Ft)-adapted process (u(t))t∈[0,T ] iscalled a solution of (2.1) if for its dt⊗P-equivalence class [u] we have [u] ∈ L2([0, T ]×Ω, dt⊗ P, V ) and P-a.s.

u(t) = u(0) +

∫ t

0

A(u(s)) ds+

∫ t

0

σ(u(s)) dWs, t ∈ [0, T ],

where u is any V -valued progressively measurable dt⊗ P-version of [u].Now we can state the main result of this section as follows.Theorem 2.3. Assume that conditions (H1)–(H4) hold. Then for any initial

data u0 ∈ V , there exists a unique solution u to (2.1) in the sense of Definition 2.2.Moreover,

E

(sup

t∈[0,T ]

‖u(t)‖2H

)<∞.

Proof. The proof follows the standard path of spectral Galerkin approximation;the only difference between this proof and those of [10, 18] is the compactness argu-ment; cf. Lemma 2.4 below. To this aim let (en)n≥1 be an orthonormal basis in H ofeigenfunctions for the operator L : H ⊃ D(L) → H . Clearly (en)n≥1 ⊂ V and the setspan{en, n ≥ 1} is dense in V . Let Hn := span{e1, . . . , en} and define Pn : V ∗ → Hn

by

Pny :=

n∑i=1

V ∗〈y, ei〉V ei, y ∈ V ∗.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Then we have that Pn|H is just the orthogonal projection onto Hn in H . We shalldefine the family of n-dimensional Brownian motions by setting

Wnt :=

n∑i=1

〈Wt, fi〉Ufi =n∑

i=1

Bi(t)fi,

where (fi)i≥1 is an orthonormal basis of the Hilbert space U . We now consider then-dimensional SDE

(2.2)

{dun(t) = PnAu

n(t)dt+ Pnσ(un(t))dWn

t ,un(0, x) = Pnu0(x),

}

which is identified with a corresponding SDE dx(t) = bn(x(t))dt+σn(x(t))dBnt in R

n

via the isometric map Rn → Hn, x →

∑ni=1 xiei. By [18, Remark 4.1.2] conditions

(H1) and (H2) imply the continuity of the fields x → bn(x) ∈ Rn and x → σn(x) ∈

Rn×n. Moreover, assumption (H2) implies

2〈bn(x) − bn(y), x− y〉Rn + |σn(x) − σn(y)|2L2(Rn,Rn) ≤ c1|x− y|2 ∀x, y ∈ Rn,

and, by the equivalence of norms on Rn, (H3) gives the bound

2〈bn(x), x〉 + |σn(x)|2L2(Rn,Rn) ≤ c5(1 + |x|2)

for some c5 ∈ R. Hence, (2.2) is a weakly monotone and coercive equation in Rn

which has a unique globally defined solution; cf. [18, Chapter 3].Lemma 2.4. Let un be the solution to (2.2). Then there is a constant c2 ≥ 0

independent of n such that for T > 0 we have

sup0≤t≤T

E‖un(t)‖2V ≤ (c2T + E(‖u0‖2V

))ec2T .

Proof. Due to the definition of Pn we may write

〈un(t), ei〉 = 〈un(0), ei〉+∫ t

0

⟨n∑

k=1

V ∗〈A(un(s)), ek〉V ek ds, ei

⟩

+

⟨∫ t

0

Pnσ(un(s)) dWn

s , ei

⟩.

Hence, the Ito formula in Rn yields

‖un(t)‖2V = ‖un0‖2V + 2

∫ t

0

〈PnA(un(s)), un(s)〉V ds+

∫ t

0

‖Pnσ(un(s))‖2L2(Un,V ) ds

+Mn(t), t ∈ [0, T ],

P-a.s, where Un := span {f1, f2, . . . , fn} ⊂ U and

Mn(t) := 2

∫ t

0

〈un(s), Pnσ(un(s)) dWn

s 〉V , t ∈ [0, T ],

is a local martingale. We consider a sequence of Ft- stopping times τj with τj ↑ +∞as j → +∞ and such that ‖un(t∧τj)(ω)‖V is bounded uniformly in (t, ω) ∈ [0, T ]×Ω,Mn(t ∧ τj), t ∈ [0, T ], is a martingale for each j ∈ N. Then we have

E‖un(t ∧ τj)‖2V =E‖un0‖2V + 2

∫ t

0

E1[0,τj]〈PnA(un(s)), un(s)〉V ds

+

∫ t

0

E1[0,τj]‖Pnσ(un(s))‖2L2(Un,V ) ds.

(2.3)Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Now using the definition of the operators A and Pn we can write

〈PnA(un(s)), un(s)〉V =

⟨n∑

i=1

V ∗〈A(un(s)), ei〉V ei, un(s)⟩

V

=

n∑i=1

V ∗〈A(un(s)), ei〉V 〈ei, un(s)〉V .

Since un(t) ∈ Hn for t ∈ [0, T ] and (en)n≥1 ⊂ V by assumption (H3) we can write

V ∗〈A(un(s)), ei〉V = 〈A(un(s)), ei〉H ;

this yields

〈PnA(un(s)), un(s)〉V =

n∑i=1

〈A(un(s)), ei〉H〈ei, un(s)〉V

=

n∑i=1

〈A(un(s)), ei〉Hλi〈ei, un(s)〉H ,

where {λi ≥ 0} are the eigenvalues of the operator L.Therefore we have

〈PnA(un(s)), un(s)〉V = 〈A(un(s)), un(s)〉V .

Hence, the operator Pn may be dropped in the first integral on the right-handside term of (2.3) such that by the second part of assumption (H3)

E‖un(t ∧ τj)‖2V ≤ E‖un0‖2V + c2

∫ t

0

(1 + E‖un‖2V )ds.

Hence letting j → +∞ and using Fatou’s lemma we obtain

E‖un(t)‖2V ≤ E‖un0‖2V + c2

∫ t

0

(1 + E‖(un(s))‖2V ) ds.

Now Gronwall’s lemma yields

(2.4) E‖un(t)‖2V ≤ (c2T + E‖un0‖2V )ec2T .

For the estimate of E‖un0‖2V , we use the definition of Pn and write

‖un0‖2V = ‖Pnu0‖2V = 〈Pnu0, Pnu0〉V =n∑

i=1

n∑j=1

V ∗〈u0, ei〉V 〈ei, ej〉V V ∗〈u0, ei〉V

=n∑

i=1

λi〈u0, ei〉2H ≤∞∑i=1

λi〈u0, ei〉2H = ‖u0‖2V .

From this point all remaining arguments from [18, Chapter 4] carry over withoutchange in order to complete the proof of Theorem 2.3.

According to [10] the following Ito formula for ‖ut‖H holds.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Theorem 2.5. Let u0 ∈ L2(Ω,F0,P, H) and v ∈ L2([0, T ]× Ω, dt ⊗ P, V ∗), θ ∈L2([0, T ]×Ω, dt⊗ P, L2(U,H)), both progressively measurable. Define the continuousV ∗-valued process

u(t) := u0 +

∫ t

0

v(s) ds+

∫ t

0

θ(s) dWs, t ∈ [0, T ].

If for its dt⊗ P-equivalence class [u] we have [u] ∈ L2([0, T ]×Ω, dt⊗ P, V ), thenu is an H-valued continuous Ft-adapted process,

E

(sup

t∈[0,T ]

‖u(t)‖2H

)<∞,

and the following Ito formula holds for the square of its H-norm P-a.s.:(2.5)

‖u(t)‖2H = ‖u0‖2H + 2

∫ t

0

(V ∗

〈v(s), u(s)〉V + ‖θ(s)‖2L2(U,H)

)ds+ 2

∫ t

0

〈u(s), θ(s) dWs〉

for all t ∈ [0, T ], where u is any V -valued progressively measurable dt ⊗ P-version of[u].

Remark 2.6. Consider two solutions u(1) and u(2) of (2.1) with initial condition

u(1)0 ∈ V and u

(2)0 ∈ V , respectively. Applying Theorem 2.5 to u = u(1)−u(2) together

with condition (H2) and Gronwall’s lemma, we obtain

(2.6) E‖u(1)(t)− u(2)(t)‖2H ≤ ‖u(1)0 − u(2)0 ‖2He2c1t.

2.2. Generalized solutions for initial condition in H. By means of (2.6)it is possible to construct a unique generalized solution to (2.1) for initial conditionin u0 ∈ H . In particular this yields a unique Feller process on H which extends theregular strong solutions of (2.1).

Proposition 2.7. Assume (H1)–(H4). Then there exists a unique time homoge-neous H-valued Markov process (uxt , t ≥ 0, x ∈ H) such that t → uxt solves the SPDE(2.1) in the sense of Definition 2.2 whenever x = u0 ∈ V . Moreover, (uxt ) induces aFeller semigroup on H; i.e., the space Cb(H) of bounded continuous function on His invariant under the the operation ϕ → Ptϕ, where Ptϕ(x) = E(ϕ(uxt )), x ∈ H, forany t ≥ 0.

Proof. For x ∈ V ⊂ H define t → uxt ∈ H as the unique solution to (2.1) withinitial condition u0 = x. For arbitrary x ∈ H , choose a sequence (xk)k in V such that‖xk − x‖H → 0. Then by (2.6) the sequence of processes (t → uxk

t )k∈N is Cauchy inC([0,∞);L2(Ω, H)) with respect to the topology of locally uniform convergence, anddefine (t → uxt ) as the unique limit. For ϕ ∈ Cb(H) define Ptϕ(x) as above. Then(2.6) obviously yields

(2.7) E‖uxt − uyt ‖2H ≤ e2c1t‖x− y‖2H , t ≥ 0,

which implies that Ptϕ ∈ Cb(H) for ϕ ∈ Cb(H). To prove that (uxt )x∈Ht≥0 is Markov,

by the monotone class argument it suffices to show that for all x ∈ H

(2.8) E(ψ(uxt ) · ϕ1(u

xs1) · · ·ϕn(u

xsn))= E

(Pt−snψ(u

xsn) · ϕ1(u

xs1) · · ·ϕn(u

xsn))

for any 0 ≤ s1 ≤ s2 · · · ≤ sn < t and ϕ1, . . . , ϕn, ψ ∈ Cb(H) ∩ Lip(H). By (2.7) wehave

|Pt−sϕ(x)− Pt−sϕ(y)| ≤ ec2(t−s) Lip(ϕ)‖x − y‖H ∀ϕ ∈ Lip(H);

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



hence it will be enough to show (2.8) for x ∈ V , where it follows by standard argumentsfrom the uniqueness of solutions of (2.1) their adaptedness to the filtration Fs, s ≥ 0,which for s ≤ t is independent of the sigma algebra of increments Gs,t = σ(Wσ −Ws; s ≤ σ ≤ t); cf. [18, Proposition 4.3.5]. This proves the existence of (uxt ; t ≥ 0;x ∈H) as in the claim of the theorem. Trivially, uniqueness of (uxt ) follows from (2.7),which holds for any H-valued closure of solutions to (2.1).

3. Application: Stochastic curve shortening flows in (1+1) dimension.We shall now apply the previous abstract result to the two models (1.8) and (1.10)with Dirichlet, respectively, periodic boundary conditions. In the deterministic casethis model is also known as the curve shortening flow. Below we refer to (1.10) as thehomogeneous normal noise model and to (1.8) as the vertical (inhomogeneous) noisemodel. In both cases our treatment is based on the simple but essential observation(also previously used in, e.g., [14]) that for d = 1 the drift operator in the SPDE (1.6)above may be written as

(3.1) Au =∂2xu

1 + (∂xu)2= ∂x(arctan(∂xu)),

which fits into our slightly modified Krylov and Rozovskiı framework as will be shownbelow.

3.1. The (pinned) vertical noise model. To this aim let

H10 ([0, 1]) ⊂ L2([0, 1]) ⊂ H−1([0, 1])

be the Gel’fand triple, which is induced from the Dirichlet Laplacian L = Δ onL2([0, 1]). For u ∈ H1

0 ([0, 1]), let Au ∈ H−1([0, 1]) be defined by

(3.2) H−10

〈Au, v〉H10= −

∫[0,1]

arctan(∂xu)∂xvdx ∀v ∈ H10 ([0, 1]),

which is clearly hemicontinuous in the sense of condition (H1), due to the continuityand uniform boundedness of ζ → arctan ζ. Trivially A is also bounded in the sense of(H4) because

(3.3) ‖Au‖H−1([0,1]) = supv∈H1

0 ([0,1]),‖v‖H10≤1

∫[0,1]

arctan(∂xu)∂xvdx ≤(π2

)1/2.

Moreover, by the monotonicity of arctan

(3.4) H−1 〈Au−Av, u−v〉H10= −

∫[0,1]

(arctan(∂xu)−arctan(∂xv))(∂xu−∂xv)dx ≤ 0.

The eigenvectors of L = Δ are ei = (x → sin(i2πx)), i ∈ N; hence Au = ∂2xu/(1 +(∂xu)

2) ∈ H10 ([0, 1]) for any u ∈ Hn = span{e1, . . . , en} ⊂ H1

0 ([0, 1]). Moreover,

(3.5) 〈Au, u〉H10= −

∫[0,1]

∂2xu

1 + (∂xu)2∂2xu(x)dx ≤ 0 ∀u ∈ Hn.

Let (φi)i∈N denote a sequence of linear independent C1-functions on [0, 1]×R suchthat φ(0, y) = φ(1, y) = 0 for all y ∈ R and such that the assumption (1.9) holds for thenoise field, and furthermore let U denote the Hilbert space obtained from the closure of

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



the span of {φi, i ∈ N} with respect to the inner product 〈∑n

i=1 λiφi,∑m

j=1 ηjφj〉U :=∑n∧mi=1 λiηi.Define the diffusion operator B : H1

0 ([0, 1]) → L(U,L2([0, 1]) by

B(u)[φ](x) = φ(x, u(x)) ∈ L2([0, 1]).

Note that B(u) is in fact in L2(U,L2([0, 1])) since

‖B(u)(φi)‖2L2([0,1]) =

∫[0,1]

φi(x, u(x))2dx =

∫[0,1]

|φi(x, u(x)) − φi(0, u(0))|2dx

≤ (Lip(φi))2

∫[0,1]

(x2 + u2(x))dx = (Lip(φi))2

(1

3+ ‖u‖2L2([0,1])

),

such that

(3.6) ‖B(u)‖2L2(U,L2([0,1])) =∑i

‖B(u)(φi)‖2L2([0,1]) ≤(1

3+ ‖u‖2L2([0,1])

)· Λ2.

Moreover,

‖B(u)−B(v)‖2L2(U,L2([0,1])) =∑i

‖B(u)[φi]−B(v)[φi]‖2L2([0,1])

=∑i

∫[0,1]

(φi(x, u(x)) − φi(x, v(x)))2dx

≤ Λ2‖u− v‖2L2([0,1]).(3.7)

Similarly, B(u)[φ] ∈ H10 ([0, 1]) for u ∈ H1

0 ([0, 1]), and by the chain rule for weaklydifferentiable functions,

‖B(u)(φi)‖2H10 ([0,1])

=

∫[0,1]

(∂xφi(x, u(x)))2dx =

∫[0,1]

〈∇φi(x, u(x)), (1, ∂xu(x))〉2 dx

=

∫[0,1]

⟨∇φi(x, u(x)),

(1, ∂xu(x))√1 + |∂xu(x)|2

⟩2

(1 + |∂xu(x)|2) dx.

We remark that since for i ≥ 1, φi ∈ C1([0, 1]× R), condition (1.9) is equivalentto claiming that for all z ∈ [0, 1]× R, v ∈ R

2 we have∑i≥1

〈∇φi(z), v〉2 ≤ Λ2.

Thus we have

(3.8) ‖B(u)‖2L2(U,H10 ([0,1]))

=∑i

‖B(u)(φi)‖2H10 ([0,1])

≤ (1 + ‖u‖2H10([0,1])

) · Λ2

In view of (3.3)–(3.8) we conclude that conditions (H1)–(H4) are satisfied in thegiven case with constants c1 = c2 = Λ2 and c3 =

√π/2. Hence, by Theorem 2.3 we

arrive at the following result.Theorem 3.1. Assume condition (1.9) holds for the noise field and u0 ∈ H1

0 ([0, 1]).Then for any T > 0 there is an (up to dt⊗P-equivalence in [0, T ]×Ω) unique H1

0 ([0, 1])-valued process (ut)t∈[0,T ] solving the SPDE (1.8) in the sense of Definition 2.2.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Proposition 2.7 readily yields generalized solutions for the initial condition inL2([0, 1]) as follows.

Proposition 3.2. Under condition (1.9) there is a unique L2([0, 1])-valuedMarkov process (uxt , t ≥ 0, x ∈ L2([0, 1])) such that t → uxt is a strong solution to

(1.8) when x = u0 ∈ H1,20 ([0, 1]). Moreover, (uxt )t≥0 induces a Feller semigroup on

Cb(L2([0, 1])).

3.2. The (periodic) homogeneous normal noise model. In order to applyTheorem 2.3 to the model (1.2) from [21, 4] with φ(x) = ε and periodic boundaryconditions on [0, 1] we convert the Stratonovich SPDE

(1.10) du(x) =∂2xu

1 + (∂xu)2(x) dt + ε

√1 + (∂xu)2 ◦ dbt

into Ito form. Assuming C3-regularity of u, then for the bracket process of(√1 + (∂xut(x))2)t≥0 and (bt)t≥0 one computes

d⟨√

1 + (∂xu·(x))2, b·⟩t=

∂xu(x)√1 + (∂xu(x))2

d〈∂xu·(x), b·〉

=∂xut(x)√

1 + (∂xut(x))2ε∂x

(√1 + (∂xut(x))2

)dt

= ε(∂xut(x))

2∂2xut(x)

1 + (∂xut(x))2dt

such that the Ito equivalent of (1.10) is

(3.9) du(x) =

(ε2

2∂2xu+

(1− ε2

2

)∂x arctan(∂xu)

)(x)dt + ε

√1 + (∂xu(x))2dbt.

Introducing the subspace L20([0, 1]) = L2([0, 1]) ∩ {f |

∫[0,1]

f(x)dx = 0} of zero

mean functions and assuming periodic boundary conditions for u, then (3.9) yieldsfor the projections

(3.10) u = u− [u], [u] =

∫[0,1]

udx

the system of equations

du(x) =

(ε2

2∂2xu+

(1− ε2

2

)∂x arctan(∂xu)

)(x)dt

+ε(√

1 + (∂xu(x))2 −[√

1 + (∂xu(x))2])dbt,(3.11)

d[u] = ε[√

1 + (∂xu(x))2]dbt.(3.12)

Equation (3.11) is independent of (3.12) and will be treated in the frameworkof section 2. To this aim let H1([0, 1]) = {f ∈ H1([0, 1]) | f(0) = f(1)} denote theperiodic Sobolev functions and let H1

0 := H10 ∩ L2

0([0, 1]), which is a Hilbert spacewith the norm ‖f‖2

H10

=∫[0,1]

(∂xf)2dx. Due to the Poincare inequality for zero mean

functions, H10 ([0, 1]) embeds into L2

0([0, 1]) continuously and densely. Hence, choosing

V := H10 ⊂ H := L2

0([0, 1]) ⊂ H−1

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



gives an admissible Gel’fand triple in the sense of section 2. Define the operatorAε : V �→ V ∗,

Aε =ε2

2Δ +

(1− ε2

2

)A,

where Δ is the periodic Laplacian and A is defined analogously as in (3.2).Moreover, choosing U = R as a one-dimensional Hilbert space, we have that

(dbt)t≥0 is a cylindrical white noise on U , and in this terminology,

σ : Hn ⊂ C∞([0, 1]) ∩ H10 �→ L2(U, H

10 ),

where for z ∈ U , ζ ∈ Hn

σ(ζ)(z) = ε(√

1 + (∂xζ(·))2 −[√

1 + (∂xζ(·))2])

· z ∈ H10 ([0, 1]).

Clearly, Aε is hemicontinuous in the sense of (H1). For (H2), due to (√1 + x2 −√

1 + y2)2 ≤ (x− y)2 for x, y ∈ R, for fixed u, v ∈ Hn it holds that

‖σ(u)− σ(v)‖2L2(U,H) ≤ ε2∫[0,1]

(√1 + (∂xu)2 −

√1 + (∂xv)2

)2(x)dx

≤ ε2∫[0,1]

(∂xu− ∂xv)2dx = −ε2 V ∗〈Δu −Δv, u− v〉V ,(3.13)

whence

2 V ∗〈Aεu−Aεv, u− v〉V + ‖σ(u)− σ(v)‖2L2(U,H) ≤ 2

(1− ε2

2

)V ∗

〈Au −Av, u− v〉V ,

(3.14)

verifying (H2) with c1 = 0 provided ε ≤√2. Similarly, for smooth u ∈ H1

0 ,

〈Aεu, u〉V = −(ε2

2‖∂2xu‖2L2 +

(1− ε2

2

)⟨∂2xu

1 + (∂xu)2, ∂2xu

⟩L2

)

and

‖σ(u)‖2H10= ε2

∥∥∥∂x√1 + (∂xu)2∥∥∥2L2

= ε2

∥∥∥∥∥ ∂xu · ∂2xu√1 + (∂xu)2

∥∥∥∥∥2

L2

such that in this case

(3.15) 2 〈Aεu, u〉V + ‖σ(u)‖2L2(U,V ) = −2

∫[0,1]

(∂2xu)2

1 + (∂xu)2dx ≤ 0,

showing that (H3) is satisfied for any choice of ε, just as (H4), since Δ : H10 �→ H−1

is an isometry. Hence, by Theorem 2.3 for u0 ∈ H10 ([0, 1]) the SPDE (3.11) admits a

unique strong solution with E(∫ T

0 ‖us‖2H10

ds) <∞, from which we obtain [u] via (3.12)

by stochastic integration. Inserting this back into (3.10) then gives a solution u to(3.9) in the sense of Definition 2.2. Finally, applying the Ito formula to ‖u− v‖2L2([0,1]

and noting that (3.13) remains true, we find that for ε ≤√2 and t ≥ 0

E(‖ut − vt‖2L2([0,1])

)≤ ‖u0 − v0‖2L2([0,1]).

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Hence, by the same arguments as in Proposition 2.7 the strong solutions inducea unique Feller process allowing for weak solutions if the initial datum is only inL2([0, 1]). We summarize our result as follows.

Theorem 3.3. For ε ≤√2 and initial datum u ∈ H1

0 ([0, 1]) in the periodiccase, model (1.10) is well defined as a unique strong variational solution to (3.9)with periodic boundary conditions. Moreover, (3.9) defines a unique Feller process onL2([0, 1]) through the family of solutions ((ut)t≥0, u0 ∈ H1

0 ([0, 1])).Remark 3.4. This result should also hold for (1.6) in (1+1) dimensions with more

than one Brownian and suitably regular φ’s. However, in Ito formulation the equationbecomes messy.

4. Long-term behavior.

4.1. The homogeneous normal noise model.Proposition 4.1. Let ε <

√2, let (ut)t≥0 be a strong solution to (3.9) with

initial condition u0 ∈ H1([0, 1]), and let u = u − [u] be its zero-mean component.Then limt→∞ E‖u‖2

H10([0,1])

= 0.

Proof. Using that Ito’s formula holds for the process

‖ut‖2L2([0,1]) =: zt

we see (zt)t is a continuous local semimartingale with Doob–Meyer decomposition

1

2dzt = −Ψ(ut) dt+

1

2s(us) dbt,(4.1)

with drift

Ψ(u) = −〈Aεu, u〉L2 − 1

2‖σ(u)‖2L2([0,1])

,

and with diffusion coefficient

s(u) = 〈u, σ(u)〉L2([0,1]).

Since Ψ(0) = 0 it holds that Ψ(us) ≥ 0 for ε ≤√2 due to (3.13) and (3.14). For the

bracket processes (〈M〉t)t≥0 of the local martingale part dM = s(us)dbt in (4.1) wefind

E(〈M〉T ) =∫ T

0

E[s(us)2]ds

≤ 2

(∫ T

0

E(‖us‖2L2([0,1])) ds

)1/2(∫ T

0

E(1 + ‖∂xus‖2L2([0,1])) ds

)1/2

,

which is finite (cf. Lemma 2.4 or (4.5) below). As a result the process ‖ut‖2L2([0,1]) is anonnegative supermartingale. Hence there exists some nonnegative random variableZ such that

(4.2) limt→∞ ‖ut‖2L2([0,1]) = Z almost surely.

For the identification of the limit Z we integrate (4.1) and take expectations to obtainthe identity

(4.3) E‖ut‖2L2([0,1]) +

∫ t

s

EΨ(uσ) dσ = E‖us‖2L2([0,1]) ∀ 0 ≤ s ≤ t.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



From (3.14) (in case when u = u and v = 0 ∈ H10 ([0, 1])) we see that

(4.4) Ψ(u) ≥(1− ε2

2

)〈Au, u〉 =

(1− ε2

2

)∫[0,1]

α(∂xu)dx,

where the function

α : R → R≥0, α(s) = arctan(s) · s,

is even, convex, continuous and increasing on R≥0. Combining this with

‖u‖L2([0,1]) ≤ ‖u‖L∞([0,1] ≤ ‖∂xu‖L1([0,1]) ∀u ∈ H10 ([0, 1])

and Jensen’s inequality, from (4.3) we deduce

E‖ut‖2L2([0,1]) + 2

(1− ε2

2

)∫ t

s

α(E‖us‖L2([0,1])

)ds ≤ E‖us‖2L2([0,1]) ∀ 0 ≤ s ≤ t.

In particular, ∫ ∞

0

α(E‖us‖L2([0,1])

)ds <∞,

which implies that

lim inft→∞ α

(E‖ut‖L2([0,1])

)= 0.

Thus we may find a sequence (tk) tending to infinity such that

limk

E(‖utk‖L2([0,1])

)= 0,

and hence also for some subsequence tk′

limk′

‖utk′‖L2([0,1]) = 0 almost surely,

which proves that Z = 0. From (4.3) it also follows that the family {ut, t ≥ 0} isbounded in the Hilbert space L2

(Ω;L2([0, 1])

), hence weakly precompact and thus

also in fact weakly convergent to zero, due to the uniqueness of the limit Z = 0in (4.2). From (4.3) we also see that the norms of u in L2

(Ω;L2([0, 1])

)are non-

decreasing, hence convergent, such that the convergence of u to zero is actually strongin L2

(Ω;L2([0, 1])

).

Finally, for the strong H10 convergence recall that (3.15) implies for the sequence

um of smooth spectral approximations of u that

E‖umt ‖2H1

0≤ ‖um‖2

H10

∀ t ≥ 0.

Since um converges to u in L2loc

(R≥0;L

2(Ω;L2[0, 1]))), selecting subsequences um

′,

this passes on to u first for dt-almost all t ≥ 0 but then also for all t ≥ 0, due to thelower semicontinuity of the H1

0 -norm on L2([0, 1]), using the L2([0, 1])-sample pathcontinuity of the process u. Finally, due to the Markov property and uniqueness ofum, via conditioning this property propagates and we arrive at

(4.5) E‖ut‖2H10≤ E‖us‖2H1

0∀ t ≥ s ≥ 0.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Hence by the same arguments as above for the L2([0, 1])-case we obtain that u con-verges strongly to zero in L2(Ω; H1

0 ([0, 1])), which is the claim.We can thus describe the long-term behavior of the model (3.9) in the periodic

case as follows.Theorem 4.2. For initial condition u ∈ H1([0, 1]) and ε ≤

√2, let u be

the solution to (3.9). Then for T → ∞, the family of L2([0, 1])-valued processes(uT )t≥0 := (uT+t−uT )t≥0 converges weakly to the L2([0, 1])-valued process ε(1 ·βt)t≥0

on the path space C(R≥0, L2([0, 1])), where β is a standard real Brownian motion.

Proof. Thanks to Proposition 4.1 and the decomposition (3.10), (3.11), (3.12) thetheorem is shown once we have established the precompactness of the laws. For thiswe use the following version of Aldous’s tightness criterion; cf. [6, Theorem 2.2.2].

Lemma 4.3. Let {(Xn)t≥0}n be a sequence of adapted continuous processesdefined on some stochastic bases (Ωn, (Fn

t )t≥0, Pn), n ∈ N, assuming values in aseparable metric space (X , d). Then the corresponding sequence of laws is tight onC(R≥0, X), provided the following two conditions hold:

(i) For each t from a dense subset T ⊂ [0,∞) the sequence of laws of (Xnt )n∈N

is tight on (X , d).(ii) For every uniformly bounded sequence (τn)n of Fn-stopping times and δ > 0

it holds that

limρ→0

lim supn→∞

supθ∈[0,ρ]

Pn(d(Xn

τn , Xnτn+θ) ≥ δ

)= 0.

Checking such kinds of conditions is standard, hence we give only a sketch. Con-dition (i) is implied by (4.5) and the compactness of H1

0 in L2([0, 1]). Due to mono-tonicity of the drift part in (4.1) it suffices to prove (ii) for the martingale part of‖u‖2L2([0,1]). Since

|σ(u)|2 ≤ 2 + 2E(‖ut‖2H1

0

)≤ 2 + 2‖u0‖2H1

0

its expected quadratic variation grows at most linearly in time, from which (ii) isobtained via Chebyshev inequality and Ito isometry.

4.2. The vertical additive noise model. In this final section we show exis-tence and uniqueness of an invariant measure for the L2([0, 1])-valued Markov process(uxt ; t ≥ 0, x ∈ L2([0, 1])) of generalized solutions obtained in Proposition 3.2 for theSPDE (1.8), now in the additive noise case when

(4.6) du =∂2xu

1 + (∂xu)2dt+QdWt, u(0) = u0 ∈ H1,2

0 ([0, 1]),

where W is cylindrical white noise on some abstract Hilbert space U and Q ∈L2(U,H

1,20 ([0, 1])). As an example consider the case of U = L2([0, 1]) and Q =

(−Δ)−β for β > 3/4, with Δ being the Dirichlet Laplacian on [0, 1], i.e.,

du(x) =∂2xu

1 + (∂xu)2(x) dt +

∑k

(k2π)−2β sin(k2πx) db(k)t .

Note also that for additive noise condition (H2) is satisfied with c1 = 0. Asa consequence of (2.6), the Feller semigroup on L2 induced from the generalizedsolutions u of (1.8) by Ptϕ(x) = E(ϕ(uxt )) has the so-called e-property [9]; i.e., for allbounded Lipschitz continuous functions ϕ : L2 �→ R

(4.7) |Ptϕ(x) − Ptϕ(y)| ≤ Lip(ϕ)‖x− y‖ ∀x, y ∈ L2.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Theorem 4.4. Let (Pt)t≥0 denote the Feller semigroup on L2([0, 1]) correspond-ing to the generalized solution to (4.6). Then (Pt) is ergodic; i.e., there is a unique

(Pt)-invariant probability measure μ on L2([0, 1]). In particular, limt→∞ 1t

∫ t

0 〈Ptϕ, ν〉 =〈ϕ, μ〉 for any Borel probability measure ν ∈ M1(L

2([0, 1])) and any bounded contin-uous ϕ : L2([0, 1]) �→ R.

Let QT (x, ·) := 1T

∫ T

0μut dt, where μut denotes the distribution at time t of the

generalized solution uxt to (4.6) with initial condition u0 = x ∈ L2.Proposition 4.5. For any x ∈ L2 the family of measures {QT (x, ·), T ≥ 1} is

tight on L2([0, 1]).Proof. Assume first that ∈ H1,2

0 ([0, 1]). In view of

|ξ| − α ≤ arctg ξ · ξ ≤ β + |ξ|, ξ ∈ R, α, β > 0,

it holds that

H−1〈Av, v〉H1 = −∫ 1

0

arctg(∂xv) · ∂xv dx ≤ −∫ 1

0

|∂xv| dx+ α

≤ −c‖v‖W 1,1(0,1) + α(4.8)

for some c > 0 by the Poincare inequality.Now let t → ut be the solution to (4.6) with regular initial condition x = u0 ∈

H1,20 ([0, 1]). Then Theorem 2.5 holds. Hence by the Ito formula for ‖ut‖2L2([0,1] and

(4.8) we have

E‖u(t)‖2 = E‖u(0)‖2 + 2E

∫ t

0V ∗〈A(u(s)), u(s)〉V ds+ E

∫ t

0

‖Q‖2LHS(U,H) ds

≤ E‖u(0)‖2 − c E

∫ t

0

‖u(s)‖W 1,1(0,1) +Dt,

(4.9)

where D := α+ ‖Q‖2LHS(U,H). In particular,

(4.10) E

(1

t

∫ t

0

‖u(s)‖W 1,1(0,1) ds

)≤ 1

c

(E‖x‖2 +D

)∀t ≥ 1.

Since the functional L2([0, 1]) � u → ‖u‖W 1,1(0,1) ∈ R ∪ {∞} has compact sublevel

sets in L2([0, 1]), the claim follows for regular initial condition x = u0 ∈ H1,20 ([0, 1]).

For the tightness of QT (x, .) with general x ∈ L2, recall (e.g., [17, remark onp. 49]) that it is sufficient (and necessary) to find for arbitrary ε > 0, δ > 0 a finiteunion of δ-balls Sδ =

⋃i=1,...,kBδ(xi) ⊂ L2 such that

QT (x, Sδ) > 1− ε ∀T > 1.

To this aim choose z ∈ Bδε/4(x) ∩ H1,20 (0, 1) and a finite union of δ/2-balls Sδ/2 =⋃

i=1,...,kBδ/2(xi) such that QT (z, Sδ/2) ≥ 1− ε2 . Let Sδ =

⋃i=1,...,k Bδ(xi) and choose

a bounded Lipschitz function ϕ on L2 with χSδ/2≤ ϕ ≤ χSδ

and Lip(ϕ) ≤ 2δ . Hence,

using (4.7), for all T > 1

QT (x, Sδ) ≥1

T

∫ T

0

Psϕ(x)ds ≥1

T

∫ T

0

Psϕ(z)ds−2

δ‖x− z‖

≥ QT (z, S δ2)− 2‖x− z‖

δ> 1− ε.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Lemma 4.6. For x ∈ L2(0, 1), let (vx(t))t≥0 be the (generalized) solution of (4.6)corresponding to Q = 0. Then it holds that

limt→+∞ ‖vx(t)‖ = 0.

Proof. Adapting arguments from [14], consider the case with initial data v0 ∈C∞

0 (0, 1) (space of C∞-differentiable function compactly supported in [0, 1]). We setM := ‖v′0‖∞ and define a function h(t) in which the following hold:

(i) h is of class C∞(R) and satisfies

h(t) = arctan t for |t| ≤M,

|h(t)| ≤ |t|, t ∈ R.

(ii) h′ is a bounded function on R that satisfies infx∈R h′(x) ≥ μ > 0 for a positive

constant μ.(iii) h′′ is a bounded function on R.For T > 0 fixed, consider the equation

(4.11)

{dv(t) = (h(vx(t)))x dt,

v(0) = v0.

Following an argument similar to that in [14] and a maximum principle for uniformlyparabolic equation, we can prove that the classical solution v of (4.11) satisfies

sup0≤t≤T

‖vx‖∞ ≤M.

Hence from the construction of h we deduce that this solution is also the solution of(4.6) with Q = 0 corresponding to the initial data v0 ∈ C∞(0, 1). Now we remarkthat for the function z �→ arctan z we can write

arctan z = k(z) · z for all z ∈ R

for some positive decreasing function k on R. Therefore by using the energy estimatefor the function v(t) we can write

1

2

d

dt‖v(t)‖2 = −〈arctan vx(t), vx(t)〉L2(0,1)

≤ − infz∈B(0,M)

k(z) ‖vx(t)‖2

≤ − infz∈B(0,M)

k(z) ‖v(t)‖2.

(4.12)

Thus we obtain

‖v(t)‖2 ≤ e−2t inf

z∈B(0,M)k(z)

‖v0‖2.

This implies the statement of the lemma for regular initial datum v0. For generalv0 ∈ L2(0, 1) we proceed by approximation and let vn0 be a sequence of functions inC∞

0 (0, 1) which converges to v0 in L2(0, 1) for n → +∞. For n ≥ 0 we denote byvn(t) the solution corresponding to the initial condition vn0 . By using the fact that

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



vn(t) → 0 as t → 0 and a triangle inequality argument, we deduce the statement ofthe lemma for general initial datum v0 ∈ L2(0, 1).

Lemma 4.7. For x ∈ L2(0, 1), let (vx(t))t≥0 be the (generalized) solution of (4.6)corresponding to Q = 0. Then for every x ∈ L2, T > 0 and ε > 0, it holds that

(4.13) P(‖uxT − vxT ‖ < ε) > 0.

Proof. First we suppose that x ∈ V and denote by (vxt )t≥0 the solution corre-sponding to (4.6) with Q = 0. We write

(4.14) z(t) = u(t)− v(t), t ≥ 0.

Then the process z(t)t≥0 solves the equation{dz(t) = (Au(t)−Av(t))dt +QdWt,z(0) = 0.

We set

z(t) = y(t) +QWt.

Then we have

dy(t) = (Au(t)−Av(t)) dt.

Therefore,

1

2

d

dt‖y(t)‖2 = V ∗〈Au(t)−Av(t), y(t)〉V dt

= V ∗〈Au(t)−Av(t), z(t)〉V dt−V ∗ 〈Au(t)−Av(t), QWt〉V

≤ 2(π2

) 12 ‖QWt‖V ,

where we used the monotonicity of A and (3.3) to obtain the estimate in the last line.Thus we deduce for 0 ≤ t ≤ T

‖y(t)‖ ≤ c T sup0≤t≤T

‖QWt‖V

for some positive constant c. We now use the splitting of z(·) and the Poincareinequality to obtain for 0 ≤ t ≤ T

(4.15) ‖z(t)‖ ≤(c T +

1

2

)sup

0≤t≤T‖QWt‖V .

For the case where x ∈ H we proceed by approximation and use the uniform bound(3.3) to obtain the same estimate as in (4.15) for the process z(t) = ux(t) − vx(t),x ∈ H . Since Q is a Hilbert–Schmidt operator from U to V , (QWt)t≥0 is a continuousGaussian random process with values in V . Hence, for all δ > 0

P

(sup

0≤t≤T‖QWt‖V < δ

)> 0.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Now let ε > 0 and take δ > 0 such that (cT + 1/2)δ < ε. Then

P

(‖z(t)‖ < ε

)> P

(sup

0≤t≤T‖QWt‖V < δ

)> 0.

Remark 4.8. The property (4.13) is sometimes called the stability of the model(4.6). Our proof of Lemma 4.7 is rather standard but will not work in case of multi-plicative noise.

Proposition 4.9. For every δ > 0 and every x ∈ L2([0, 1]) it holds that

(4.16) lim infT→∞

QT (x,Bδ(0)) > 0.

Proof. We proceed in three steps. Let δ > 0 and x ∈ L2([0, 1]) be given.Step 1. For R > 0 let CR = {u ∈ L2|u ∈ W 1,1

0 (0, 1), ‖u‖1,1 ≤ R}, which is acompact subset of L2([0, 1]). From (4.10) and Chebyshev’s inequality we deduce

QT (0, L2([0, 1]) \ CR) ≤c

R∀ T > 1.

Hence we may pick some R > 0 such that QT (0, CR) >34 for all T > 1. From now on

we omit the subscript R, i.e., C = CR.Step 2. Claim: There is some ε1 > 0, a γ1 > 0, and a finite sequence T1, . . . , Tk,

Ti > 0 such that

1

k

∑i=1,...,k

PTi(x,Bδ(0)) > γ1 ∀ x ∈ Cε1 ,

where Cε1 = {u ∈ L2([0, 1]) | dL2(u,C) < ε1} and PT (x, ·) is the transition probabilitycorresponding to (ux(t))t≥0 at time T . In fact, by Lemma 4.6 for each x ∈ L2([0, 1])there exists a Tx <∞ such that vxTx

∈ Bδ/4(0). For T > 0 and δ > 0 let

D(x, T, δ) := P{‖vxT − uxT ‖L2([0,1]) ≤ δ},

which is strictly positive by Lemma 4.7. Hence it follows that PTx(x,B δ2(0)) ≥

D(x, Tx, δ/4) =: γx > 0. Similarly as in the second part of Proposition 4.5 wemay use (4.7) to deduce that for each x ∈ L2([0, 1]) there exists rx > 0 such thatPTx(y,Bδ(0)) > γx/2 for all y ∈ Brx(x). Since C is compact we may select a finitesequence (xi, ri), i = 1, . . . , k, such that C ⊂

⋃i=1,...,k B(xi, ri). Setting Ti := Txi the

claim follows with ε1 := mini=1,...,k ri and γ1 := mini=1,...,k γi/2k.Step 3. Choose ρ > 0 such that

QT (x,Cε1) >1

2∀ x ∈ Bρ(0).

This is possible by a similar argument to that in the second part Proposition 4.5.Finally, by reasons analogous to those in step 2, we may find some T0 > 0 and γ2 > 0such that PT0(x,Bρ(0)) > γ2.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



Hence,

lim infT

QT (x,Bδ(0)) = lim infT

1

T

∫ T

0

Ps(x,Bδ(0))ds

= lim infT

1

k

∑i=1,...,k

1

T

∫ T

0

Ps+Ti+T0(x,Bδ(0))ds

= lim infT

1

k

∑i=1,...,k

1

T

∫ T

0

∫L2([0,1])

∫L2([0,1])

× PTi(z,Bδ(0))Ps(y, dz)PT0(x, dy)ds

≥ lim infT

1

T

∫ T

0

∫Bρ(0)

∫Cε1

1

k

∑i=1,...,k

× PTi(z,Bδ(0))Ps(y, dz)PT0(x, dy)ds

≥ γ1 lim infT

1

T

∫ T

0

∫Bρ(0)

Ps(y, Cε1)PT0(x, dy)ds,

which, by Fatou’s lemma, is bounded from below by

≥ γ1

∫Bρ(0)

lim infT

1

T

∫ T

0

Ps(y, Cε1)PT0(x, dy)ds

= γ1

∫Bρ(0)

lim infT

QT (y, Cε1)PT0(x, dy)ds

>1

2γ1PT0(x,Bρ(0)) >

1

2γ1γ2 > 0.

Proof of Theorem 4.4. In view of (4.7), the claim is now a consequence of [9, The-orem 1] (with T = L2([0, 1]) thanks to Proposition 4.5), asserting that a stochasticprocess on a Polish metric space whose semigroup on functions preserves Lipschitz con-stants and whose family of time averaged distributions are tight for any starting pointadmits a unique invariant probability measure, provided it has at least one stronglyrecurrent state such as here the state 0 ∈ L2([0, 1]) in (4.16) of Proposition 4.9.

Acknowledgment. We are indebted to Martin Hairer for directing us towardsthe critical reference [9] at an early stage of this project.

REFERENCES

[1] R. Buckdahn and J. Ma, Pathwise stochastic Taylor expansions and stochastic viscositysolutions for fully nonlinear stochastic PDEs, Ann. Probab., 30 (2002), pp. 1131–1171.

[2] M. Caruana, P. Friz, and H. Oberhauser, A (Rough) Pathwise Approach to Fully Non-linear Stochastic Partial Differential Equations, preprint, arXiv.org:0902.3352, 2009.

[3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, EncyclopediaMath. Appl. 44, Cambridge University Press, Cambridge, UK, 1992.

[4] N. Dirr, S. Luckhaus, and M. Novaga, A stochastic selection principle in case of fatteningfor curvature flow, Calc. Var. Partial Differential Equations, 13 (2001), pp. 405–425.

[5] T. Funaki, The scaling limit for a stochastic PDE and the separation of phases, Probab.Theory Related Fields, 102 (1995), pp. 221–288.

[6] A. Joffe and M. Metivier, Weak convergence of sequences of semimartingales with applica-tions to multitype branching processes, Adv. in Appl. Probab., 18 (1986), pp. 20–65.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



[7] K. Kawasaki and T. Ohta, Kinetic drumhead model of interface. I, Progr. Theoret. Phys., 67(1982), pp. 147–163.

[8] R. Kohn, F. Otto, M. G. Reznikoff, and E. Vanden-Eijnden, Action minimization andsharp-interface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math.,60 (2007), pp. 393–438.

[9] T. Komorowski, S. Peszat, and T. Szarek, On ergodicity of some Markov processes, Ann.Probab., 38 (2010), pp. 1401–1443.

[10] N. V. Krylov and B. L. Rozovskiı, Stochastic evolution equations, in Current Problems inMathematics, Vol. 14, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii,Moscow, 1979, pp. 71–147 (in Russian).

[11] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Stud. Adv.Math. 24, Cambridge University Press, Cambridge, UK, 1990.

[12] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations:Non-smooth equations and applications, C. R. Acad. Sci. Paris Ser. I Math., 327 (1998),pp. 735–741.

[13] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic pde with semilinear stochasticdependence, C. R. Acad. Sci. Paris Ser. I Math., 331 (2000), pp. 617–624.

[14] M. Nara and M. Taniguchi, The condition on the stability of stationary lines in a curvatureflow in the whole plane, J. Differential Equations, 237 (2007), pp. 61–76.

[15] E. Pardoux, Equations aux derivees partielles stochastiques de type monotone, in Seminaire

sur les Equations aux Derivees Partielles (1974–1975), III, College de France, Paris, 1975,Exp. 2, p. 10.

[16] E. Pardoux, Sur des equations aux derivees partielles stochastiques monotones, C. R. Acad.Sci. Paris Ser. A-B, 275 (1972), pp. A101–A103.

[17] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probab. Math. Statist. 3.Academic Press, New York, 1967.

[18] C. Prevot and M. Rockner, A Concise Course on Stochastic Partial Differential Equations,Lecture Notes in Math. 1905, Springer, Berlin, 2007.

[19] A. Rascanu, Deterministic and stochastic differential equations in Hilbert spaces involvingmultivalued maximal monotone operators, Panamer. Math. J., 6 (1996), pp. 83–119.

[20] M. Roger and H. Weber, Tightness for a Stochastic Allen–Cahn Equation, ArXiv e-prints,2009.

[21] P. E. Souganidis and N. K. Yip, Uniqueness of motion by mean curvature perturbed bystochastic noise, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), pp. 1–23.

[22] H. Weber, Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation,Comm. Pure Appl. Math., 63 (2010), pp. 1071–1109.

[23] N. K. Yip, Stochastic motion by mean curvature, Arch. Ration. Mech. Anal., 144 (1998),pp. 313–355.

[24] X.-P. Zhu, Lectures on Mean Curvature Flows, AMS/IP Stud. Adv. Math. 32, AMS, Provi-dence, RI, 2002.

Dow

nloa

ded

11/1

8/14

to 1

29.4

9.23

.145

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

ergodicity of stochastic curve shortening flow in the plane

Documents