Large deviation principle of occupation measure for stochastic Burgers equation

Large deviation principle of occupation measure for stochastic Burgers equation

Mathieu Gourcy

Laboratoire de Mathématiques, CNRS-UMR 6620, Université Blaise Pascal, 63177 Aubière, France Received 3 March 2006; accepted 4 July 2006

Available online 26 December 2006

Abstract

In this paper we obtain a Large Deviation Principle for the occupation measure of the solution to a stochastic Burgers equation which describes the exact rate of exponential convergence. This Markov process is strongly Feller and has a unique invariant measure. Moreover, the rate function is explicit: it is the level-2 entropy of Donsker–Varadhan.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

On obtient un Principe de Grandes Déviations pour la mesure d’occupation associée à la solution d’une équation de Burgers sto- chastique. Ce résultat décrit convergence exponentielle vers l’unique mesure invariante. La fonction de taux associée est l’entropie de niveau 2 de Donsker–Varadhan.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:60F10; 60J35; 35Q53

Keywords:Stochastic Burgers equation; Large deviations; Occupation measure

1. Introduction and main results

LetH=L²(0,1)equipped with its norm · 2. In this paper we are interested in the large time behavior of the solution to the following stochastic Burgers equation:

dX(t )=

X(t )+1

2DξX²(t )

dt+GdW (t ); X(0, ξ )=x0(ξ )∈H, (1.1)

whereG:H→H is a bounded linear operator,W (t )is a standard cylindrical Wiener process onH, andis the Laplacian on(0,1)with the Dirichlet boundary conditions. Indeed, the problem (1.1) is supplemented by:

X(t,0)=X(t,1)=0, t >0.

It is well known thatis a negative, self-adjoint, non-bounded operator onH with the domain of definition given by D()=

u∈H²(0,1): u(0)=u(1)=0

=H²(0,1)∩H₀¹(0,1),

E-mail address:mathieu.gourcy@math.univ-bpclermont.fr.

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.07.003

where H₀¹=

x: [0,1] →R; x is abs. continuous,x(0)=x(1)=0, and∇x:=D_ξx∈H .

We assume that tr(GG^∗) <∞, i.e. the energy injected by the random force is finite, and that, forQ=GG^∗, Im

(−)^−δ/2

⊂Im Q^1/2

for some 1

2< δ <1, (1.2)

where Im(Q^1/2)is the range of the operatorQ^1/2. The last condition (1.2) means that the noise is not too degenerate.

It is equivalent to say that the domain of definition of(−)^δ/2inHis contained in Im(Q^1/2).

The above equation plays an important role in fluid dynamic for understanding of chaotic behavior. This stochastic model has been intensively studied for 10 years, in particular by Da Prato, Debussche, Dermoune, Weinan, Gatarek, Khanin, Mazel, Sinai and Temam among many others (from a chronological point of view, see [5,4,9,2,3,18]). About large deviations, small noise asymptotic was investigated by Cardon-Weber [1]. More recently, Goldys and Maslowski proved the exponential ergodicity [13].

LetM₁(H )(resp.M_b(H )) be the space of probability measures (resp. signedσ-additive measures of bounded variation) on H equipped with the Borel σ-fieldB(H ). The usual duality relation betweenν∈M_b(H ) andf ∈ bB(H ), the set of bounded and measurable functions onH, will be denoted by

ν(f ):=

H

fdν.

OnM_b(H )(or its subspaceM₁(H )), we will consider the usual weak convergence topologyσ (M_b(H ), C_b(H ))and the so calledτ-topologyσ (M_b(H ), bB(H )), which is much stronger.

Our aim is to establish the large deviation principle (LDP in short) for the occupation measureL_tof the solutionX (or empirical measure of level-2) given by

L_t(A):=1 t t

0

δ_Xs(A)ds, ∀A∈B(H )

δ_a being the Dirac measure ata. Notice thatL_t is an inM₁(H )-valued random variable. This is a traditional subject in probability since the pioneering work of Donsker and Varadhan [11]. The main innovation is that we deal about infinite dimensional diffusions for which their assumptions are not satisfied. For an introduction to large deviations we refer to the books of Deuschel and Stroock [10], and Dembo and Zeitouni [8].

Under (1.2), it is known thatXt is a Markov process with a unique invariant measureμ(cf. [7]). So the ergodic theorem says that, almost surely underPμ,Lt converges weakly toμ. We establish in this note a much more stronger result:

Theorem 1.1.Assume thattr(GG^∗) <+∞and(1.2) (throughout this paper). Let0< λ0<₂^π_Q², whereQis the norm ofQas an operator inH and

Φ(x)=e^λ0x22, Mλ0,L:= ν∈M₁(H )

H

Φ(x)ν(dx)L

. (1.3)

The familyPν(L_T ∈ ·)asT → +∞satisfies the large deviation principle(LDP)with respect to(w.r.t. in short)the topologyτ, with speedT and the rate functionJ, uniformly for any initial measure inMλ₀,LwhereL >1 is any fixed number. HereJ:M₁(H )→ [0,+∞]is the level-2entropy of Donsker–Varadhan defined by(3.2)below.

More precisely we have:

(i) J is a good rate function on M₁(H )equipped with the topology τ of the convergence against bounded and Borelian functions, i.e.,[Ja]isτ-compact for everya∈R⁺;

(ii) for all open setGinM1(H )with respect to the topologyτ, lim inf

T→∞

1

T log inf

ν∈Mλ0,L

Pν(L_T ∈G)−inf

G J; (1.4)

(iii) for all closed setF inM1(H )with respect to the topologyτ, lim sup

T→∞

1

T log sup

ν∈Mλ0,L

Pν(L_T ∈F )−inf

F J. (1.5)

Furthermore we have

J (ν) <+∞ ⇒ νμ, ν(H₀¹)=1and

H₀¹

∇x²₂dν <+∞, (1.6)

whereμis the unique invariant probability measure of(Xt).

The LDP w.r.t. the topologyτ is much stronger than that w.r.t. the usual weak convergence topology as in Donsker and Varadhan [11]. Sometimes considered as a technical detail, the topologyτis crucial here: interesting consequences of this LDP can be deduced for many physical quantities of the system such asxH¹= ∇x2, or more generally xH^α:= (−)^α/2x2for 0α1, which are not continuous onH. In fact, we establish

Corollary 1.2. Let B a separable Banach space, and f:H₀¹→B a measurable function, bounded on the balls {x; ∇x2R}, and satisfying

∇xlim2→∞

f (x)_B

∇x²₂ =0. (1.7)

Then,Pν(LT(f )∈ ·)satisfies the LDP onB, with speedT and the rate functionIf given by I_f(z)=inf J (ν); J (ν) <+∞,

H₀¹

f (x)dν(x)=z

, ∀z∈B,

uniformly over initial distributionsνinMλ₀,Lfor any fixedL >1.

For instance,f:H₀¹→B:=H^α withf (x)=x for anyα∈ [0,1)is allowed, so that the LDP inH^α holds for Pν(1/T_T

0 X_tdt∈ ·). An other particular case of the above corollary is the following: for everyp∈(0,2), Pν

1 T

T

0

∇X(t )^p

2dt∈ ·

satisfies the LDP onRwith speedT and the rate functionI defined by I (z)=inf J (ν); J (ν) <+∞,

H

∇x^p₂dν(x)=z

, ∀z∈R (1.8)

uniformly over initial distributionsνinMλ₀,L(for anyL >1).

Finally, we introduce(e_k)_kthe complete orthonormal system inL²(0,1)which diagonalizeson its domain, and by−λ_kthe corresponding eigenvalues. We have

e_k(x)= 2

πsinkπ x, λ_k=π²k², k∈N^∗= {1,2, . . .}. Remarks 1.3.

(i) Let us see the meaning of our assumptions: tr(Q) <+∞and (1.2). Assume thatGek=σkek for everyk1.

Then

GW (t )= ∞ k=1

σkβk(t )ek, (1.9)

where(βk)k∈N^∗ is a family of independent real valued standard Brownian motions. Then tr(Q) <+∞and condition (1.2) is satisfied if

c

k |σ_k| C k^1/2+ε

for two positive constantscandCand some smallε >0.

A more general example of noise for which our assumptions hold is G:=(−)^−βB, 1

4< β <1 2,

whereBis any linear bounded and invertible operator onH. Indeed tr(GG^∗)B²_H→Htr(^−2β) <+∞for 2β >

1/2. Since Im(G)=Im(^−β)and by the polar decomposition, Im(G)⊂Im(√

GG^∗), the condition (1.2) is then verified withδ=2β.

(ii) Our approach here is well adapted to the case of a multiplicative (or correlated) random forcing term, that is, the noiseGW (t )can be replaced by

g X(t, ξ )

GW (t ),

whereg:H→ [α, β]is Lipschitz continuous, 0< α < β <∞,Gsatisfies (1.2) and tr(GG^∗) <+∞. Indeed, fol- lowing [4], the strong Feller property and the topological irreducibility hold. All estimates necessary for the LDP in Theorem 1.1 still hold in the actual case, and then all previous results remain valid.

(iii) The class (1.3) of allowed initial distributions for the uniform LDP is sufficiently rich. For example, choosing Llarge enough, it includes all the Dirac probability measuresδxwithxin any ball ofH.

(iv) Our LDP is more precise than the exponential convergence ofPtto the invariant measureμestablished in [13].

Indeed the LDP furnishes the exact rate of the exponential convergence in probability of the empirical measuresLT

toμ. Moreover by Theorem 6.4 in [21], under the strong Feller and topological irreducibility assumption for(Pt), the LDP in Theorem 1.1 is equivalent to saying that the essential spectral radius in some weighted functions spacesbuB is zero.

(v) The assumption (1.2) plays a crucial role for Theorem 1.1: if the noise acts only on a finite number of modes (i.e.,σ_k=0 for allk > Nin (1.9)) as in Kolmogorov’s turbulence theory, we believe that the LDP w.r.t. theτ-topology is false. It is a challenging open question for establishing the LDP ofL_T w.r.t. the weak convergence topology in the last degenerate noise case.

(vi) For the 2D-stochastic Navier–Stokes equation, we can prove, under suitable conditions, a LDP on someD(A^α), for ¹₄< α <¹₂. HereAis also the Laplacian, but regarded as an operator on the subspace of theL²-vector fields with free divergence. That will be carried out in a future work.

This paper is organized as follows. In Section 2, we recall known results on existence and uniqueness of solution, and existence of an invariant probability measure for Eq. (1.1). In Section 3 we give some general facts about large deviations for strong Feller and irreducible Markov processes and we obtain the uniform lower bound (1.4). Then we prove the convergence of the Galerkin approximations for the considered equation in Section 4. The exponential tightness is investigated in Section 5, and the uniform upper bound (1.5) for the strongτ-topology in Section 6. Finally, the extension to non-bounded functionals onHis discussed in the last Section 7.

2. Solutions of the equation and their properties

Let us specify what we understand by solution. Generally, we are concerned with two ways of giving a rigorous meaning to solutions of stochastic differential equations in infinite dimensional spaces, that is, the variational one [17,15] and the semigroup one [6]. Correspondingly, as in the case of deterministic evolution equations, we have two notions of strong, and “mild” solution. In most situations, one finds that the concept of strong solution is too limited to include important examples. The weaker concept of mild solution seems to be more appropriate. In the sequel, we are working with this concept, that we define more precisely now.

We denote byS(t )the semigroup generated byonL²(0,1), or from a formal point of view,S(t )=e^t .

Definition 2.1.We say thatX∈C([0, T], L²(0,1))is a “mild” solution of problem (1.1) ifX(t )is adapted toFt, the σ-algebra of the cylindrical Wiener process until timetand for arbitrary 0t, we have

X(t )=S(t )x₀+ t

0

S(t−s)1

2D_ξX²(s)ds+ t

0

S(t−s)GdW (s) (2.1)

for anyx0∈L²(0,1),Palmost surely.

Note that all the terms in (2.1) take sense since the mapping F:u∈C

[0, T], L¹(0,1)

→ t

0

S(t−s)1

2D_ξu(s)ds∈C

[0, T], L²(0,1)

is well defined (see [7] p. 260) and the stochastic convolution W:=_t

0S(t−s)GdW (s)also (see (4.4) below).

Da Prato, Debussche, Temam established in [5] for the first time existence and uniqueness for a stochastic Burgers equation cylindrically perturbed, that is whenGis the identity operator. The method they used to obtain local existence in time of a solution consists in considering a fixed path of the noise, to get into a deterministic setting and use a fixed point argument. Then the time of explosion is shown to be infinite, by means of a priori bounds on the solution. The same proof gives in our setting:

Theorem 2.2.Stochastic Burgers equation(1.1)admits a unique mild solution and for allT >0, X∈C

[0, T], L²(0,1)

∩L²

[0, T], C[0,1]

.

The solution satisfies Markov and strong Markov properties (see [6]). We can also consider the transition semigroup associated to the dynamics given by

P_tΦ(x):=EΦ X(t, x)

=E^xΦ X(t )

, ∀Φ∈bB(H ).

As in [5], this semigroup admits an invariant measure. Moreover, under our condition (1.2) on the noise, the following interesting properties hold.

Lemma 2.3.

(i) The transition semigroup (P_t) corresponding to the forced Burgers equation (1.1)satisfies the strong Feller property. That is, for any bounded Borelian functionΦ onH and anyt >0, the functionPtΦ(·)is continuous onH.

(ii) For everyt >0,P_t(x, O) >0for allx∈H and all non-empty open subsetOofH. Hence,(P_t)is also topolog- ically irreducible.

(iii) In particular, the transition semigroup(P_t), corresponding to the forced Burgers equation(1.1)admits a unique invariant measureμ, which charges all non-empty open subsets ofH.

Part (i) is well known when the cylindrical noise is considered (see [7]). In our case of a finite trace class noise, the non-degeneracy condition (1.2) is essential. More precisely,δ <1 allows to obtain a bound on the derivative of the semigroup by using the Bismut–Elworthy formula as in [2] or [12]. The conditionδ >¹₂ is borrowed from the finite trace assumption, crucial in the application of Itô’s formula for the exponential tightness.

The point (ii) was proved by Goldys and Maslowski in [13] for our class of noise. We recall that(P_t)is topologically irreducible if, for all non-empty open setΓ inH, and allx∈H, we haveP_t(x, Γ ) >0 for somet >0.

According to the general theory [7], we obtain (iii) as first corollary, sometimes called Doob’s theorem, of the two preceding points together with the existence of invariant measure. In fact this result gives also the convergence of the transition probabilities to the invariant measure.

Our aim is to complete the study of Eq. (1.1) by giving information on the rare events and the exact rate of exponential convergence by means of a large deviation principle, one of the strongest ergodic behaviors of Markov processes.

3. General results about large deviations

In this section, we introduce some necessary notations and definitions and give general results (essentially follow- ing [19]) on large deviations for Markov processes.

3.1. Notations and entropy of Donsker–Varadhan

We first compare the “topological irreducibility” defined above (often called irreducibility in the literature on SPDE) with the probabilistic irreducibility for a Markov process which is the more general assumption under which the large deviations result we use (as Lemma 3.2 below) holds true (see [14,19] for details).

Letνbe a probability measure onH; a transition kernel operatorP onHis saidν-irreducible (resp.ν-essentially irreducible) if for allAinH such thatν(A) >0, and for allxinH (resp. forνalmost allxinH), we can findn∈N such thatPⁿ(x, A) >0. Whenνcharges all non-empty open subsets ofH, theν-irreducibility implies the topological irreducibility. But for the strong FellerP, the topological irreducibility implies theν-irreducibility for allνsuch that ννP (see [19]).

Thus by Lemma 2.3, for the unique invariant measureμof our model,P_t isμ-irreducible for everyt >0. In reality for our model, we have the much stronger property that all the probability measures in the family

P_t(x,·), x∈H, t >0

are equivalent, and they are also equivalent toμ(see [7, p. 41]).

Consider theH:=L²(0,1)-valued continuous Markov process Ω, (Ft)t0,F,

Xt(ω)

t0, (Px)x∈H

whose semigroup of Markov transitions kernels is denoted by(Pt(x,dy))t0, where

Ω =C(R⁺, H ) is the space of continuous functions fromR⁺ toH equipped with the compact convergence topology;

Ft=σ (X_s,0st )for anyt0 is the natural filtration;

F=σ (X_s,0s)andPx(X₀=x)=1.

Hence,Px is the law of the Markov process with initial statex inH. For any initial measureνonH, letPν(dω):=

HPx(dω)ν(dx).

The empirical measure of level-3 (or process level) is R_t:=1

t t

0

δ_θsXds,

where(θ_sX)_t =X_s+t for allt, s0 are the shifts on Ω. HenceR_t is a random element ofM₁(Ω), the space of probability measures onΩ.

The level-3 entropy functional of Donsker and VaradhanH:M1(Ω)→ [0,+∞]is defined by, H (Q):= E^Q¯h_F0

1(Q¯_ω(−∞,0];Pw(0)), if Q∈M₁^s(Ω),

+∞, otherwise, (3.1)

where

M₁^s(Ω)is the space of those elements inM1(Ω)which are moreover stationary;

Q¯is the unique stationary extension ofQ∈M₁^s(Ω)toΩ¯ :=C(R, H );Ft^s=σ (X(u);sut )onΩ¯,∀s, t∈R, st;

Q¯_ω(−∞,t]is the regular conditional distribution ofQ¯ knowingFt^−∞;

h_G(ν, μ)is the usual relative entropy or Kullback information ofνwith respect toμrestricted on theσ-fieldG, given by

h_G(ν, μ):=

_dν

dμ|_Glog(_dμ^dν|_G)dμ, ifνμonG,

+∞, otherwise.

The level-2 entropy functionalJ:M1(H )→ [0,∞]is defined by J (β)=inf

H (Q); Q∈M₁^s(Ω)andQ₀=β

, ∀β∈M₁(H ), (3.2)

whereQ₀(·)=Q(X(0)∈ ·)is the marginal law at timet=0.

Lastly introduced in [19], we define the restriction of the Donsker Varadhan entropy to theμcomponent, by H_μ(Q):= H (Q), ifQ0μ,

+∞, otherwise and for the level-2 entropy functional

Jμ(β):= J (β), ifβμ, +∞, otherwise.

For our model, let us first establish the

Lemma 3.1.We haveJ (ν) <+∞ ⇒νμ. Moreover,J=J_μonM₁(H )and[J=0] = {μ}.

Proof. Considerνsuch thatJ (ν) <∞. We recall the expression (3.1) of the Level-3 entropy. ForQ∈M₁^s(Ω)such thatQ₀=ν,H (Q) <∞, and for everyt >0, noting that the entropy of marginal measure is not larger than the global entropy, we have by Jensen inequality,

H (Q)=E^Q¯h_F1Q¯_ω(−∞,0];Pw(0)

=1 tE^Q¯h_F0

t

Q¯_ω(−∞,0];Pw(0)

1

th_F0

t(Q;Pν)1

thσ (w(t ))(Q;Pν)

1

th_{B(H )}(ν;νP_t).

Taking infinimum over suchQ, we get J (ν)1

th_{B(H )}(ν;νP_t). (3.3)

So the Kullback information ofνwith respect toνP_t is finite, which implies by definition thatννP_t. Since all P_t(x,dy), t >0, x∈Hare equivalent toμ([7]), we have

νPt(·)=

H

P (t, x,·)ν(dx)μ.

ThusννPtμ, as desired.

By definition, we haveJJ_μand they are equal on ν∈M1(H )such thatνμ

.

Since any probability measureνonHsuch thatJ (ν) <∞is absolutely continuous with respect toμ, we haveJ=J_μ onM₁(H ).

At the end, if the probability measureβ is such thatJ (β)=0 thenβμandβ=βP_t for everyt >0 by (3.3).

By the uniqueness in Lemma 2.3, we haveβ=μand the proof is finished. 2 3.2. The lower bound

Let us first recall the definition of the projective limitτ_pof the strongτ-topology, τ_p:=σ

M₁(Ω),

t0

bF⁰_t

,

wherebF⁰t is the set of functions onΩ, that are bounded and measurable forF⁰t.

The following level-3 lower bound of Large Deviations forτ_p was established by Wu (see [19, Theorem B.1]) under more general conditions.

Lemma 3.2.([19])For any open setOin(M1(Ω), τp), lim inf

t→∞

1

t logPx(R_t∈O)−inf

O H_μ, μ-a.e. initial statex∈H.

Recall thatHμ=H by Lemma 3.1 and that a good rate function admits compact level sets (by definition). Our goal here, is to prove the

Proposition 3.3.IfJ is a good rate function on(M1(H ), τ )and the uniform upper bound(1.5)is satisfied, then the level-3uniform lower bound holds true:for any measurable open subsetOin(M₁(Ω), τ_p),

lim inf

t→∞

1

t log inf

ν∈Mλ0,L

Pν(R_t∈O)−inf

O H.

In particular, the desired Level-2lower bound(1.4)holds(by the contraction principle).

Proof. For anyQ∈Ofixed, we can take aτ_pneighborhood ofQinM₁(Ω)of form N (Q, δ):= Q∈M₁(Ω)such that

F_idQ−

F_idQ

< δ, ∀i=1, . . . , d

contained inO, whereδ >0, 1d∈NandFi∈bF⁰nfor somen∈N. It is sufficient to establish that for everyQin Osuch thatH (Q) <∞

lim inf

t→∞

1

t log inf

ν∈Mλ0,L

Pν

R_t ∈N (Q, δ)

−H_μ(Q). (3.4)

But by Egorov’s lemma, Lemma 3.2 implies the existence of a Borelian subsetKinH withμ(K) >0 such that for anyε >0

xinf∈K

1 t logPx

R_t∈N

Q,δ

2

−H_μ(Q)−ε (3.5)

for alltlarge enough. Let us fixa >0. For any 0ba, we have

F_id(Rt◦θ_b−R_t)

2(a+1) t F_i∞

and then for all 0ba and for alltlarge enough (depending onaandδ), Pν

R_t∈N (Q, δ) Pν

X_b∈K; R_t◦θ_b∈N

Q,δ 2

Pν(X_b∈K) inf

x∈KPx

R_t∈N

Q,δ

2

. Integrating for 0ba, and dividing byayields

Pν

Rt∈N (Q, δ)

E^νLa(K) inf

x∈KPx

Rt∈N

Q,δ

2

. (3.6)

Hence, for proving (3.4), by (3.5) and (3.6), it is enough to establish that for any Borelian subsetKwithμ(K) >0, we can finda >0 such that

inf

ν∈Mλ0,L

E^νL_a(K) >0.

Notice that

E^νL_a(K)μ(K) 2

1−Pν

L_a(K)−μ(K)μ(K) 2

and by the assumed level 2 upper bound, lim sup

a→+∞

1

alog sup

ν∈Mλ0,L

Pν

L_a(K)−μ(K)μ(K) 2

−inf

F J (ν),

where

F = β∈M1(H ): β(K)−μ(K)μ(K) 2

is closed for theτ-topology. So once infFJ >0, we shall obtain for allalarge enough

ν∈Minfλ0,L

E^νL_a(K)μ(K) 2

1−exp

−ainfFJ 2

>0.

It remains to prove that infFJ >0. To this end we may assume that infFJ <+∞. In that case, sinceJ is a good rate function (our condition), infFJ is attained by someβ0∈F. ButJ (β)=0⇔β=μ(Lemma 3.1) andμ /∈F, so infFJ=J (β₀) >0 as desired. 2

3.3. Cramer functionals and weak upper bound

Let us introduce the uniform upper Cramer functional over a non-empty family of initial measuresAinM₁(H ), Λ(V|A):=lim sup

t→∞

1 t log sup

ν∈AE^νexp t L_t(V ) and several other Cramer functionals,

Λ(V|x):=Λ V|{δx}

=lim sup

t→∞

1

t logE^xexp t Lt(V )

, Λ^∞(V ):=Λ

V|{δ_x;x∈H}

=lim sup

t→∞

1 t log sup

x∈H

E^xexp t L_t(V )

, Λ⁰(V ):=sup

x∈H

Λ(V|x)=sup

x∈H

lim sup

t→∞

1

t logE^xexp t L_t(V )

, (3.7)

whereV is a bounded and Borelian function onH.

The functionals Λ⁰(V ) andΛ^∞(V ) are respectively the pointwise and uniform Cramer functionals introduced already in [10]. ForΛ:bB(H )→Rany one of the above functionals, define its Legendre transformation:

Λ^∗_w(ν)= sup

V∈C_b(H ) H

Vdν−Λ(V )

, ∀ν∈M_b(H ),

Λ^∗(ν)= sup

V∈bB(H ) H

Vdν−Λ(V )

, ∀ν∈M_b(H ), (3.8)

whereM_b(H )is the space of all signedσ-additive measures of bounded variation on(H,B).

Remark that{δx}x∈H⊂

L>0Mλ₀,L, we have for any bounded and measurable functionV, Λ⁰(V )sup

L>0

Λ(V|Mλ₀,L)Λ^∞(V ).

Since(Pt)is Feller, we have by [19, Proposition B.13]

Λ⁰_∗ (ν)=

Λ⁰_∗

w(ν)= Λ^∞_∗

(ν)= Λ^∞_∗

w(ν)=J (ν), ∀ν∈M₁(H ) which implies the l.s.c. forJ and the fact that

sup

V∈bB(H ) H

Vdν−sup

L>0

Λ(V|Mλ0,L)

= sup

V∈C_b(H ) H

Vdν−sup

L>0

Λ(V|Mλ0,L)

=J (ν), ∀ν∈M₁(H ).

So by Gärtner and Ellis theorem (see [8]), we have always the following general weak* upper bound

Lemma 3.4.LetM₁(H )be equipped with the weak convergence topology. For any compact subsetKinM₁(H )w.r.t.

the weak convergence topology, and for anyε >0, there is a neighborhoodN (K, ε)ofKinM₁(H )such that

lim sup

t→∞

1

t log sup

ν∈Mλ0,L

Pν

L_t ∈N (K, ε)

−infν∈KJ (ν)+ε, if infν∈KJ (ν) <∞,

−¹_ε, otherwise.

Now to obtain the upper bound in Theorem 1.1 w.r.t. the weak convergence topology, we need to prove the expo- nential tightness ofL_t.

4. Convergence of a Galerkin method

Let us introduce the approximation system associated with Eq. (1.1):

dXn(t )=

X_n(t )+1

2Π_nD_ξ(X_n)²(t )

dt+G_ndW (t ); X_n(0)=Π_nx, (4.1)

where Π_n is the orthogonal projection on H_n, the finite dimensional space spanned by the first n eigenvectors (e₁, . . . , e_n), andG_n:=Π_nG.

The convergence of a similar approximation but with a non-linearity truncated by the function f_n(x)= nx²

n+x²

was investigated by Da Prato and Debussche [3]. The aim of this section is to establish some a priori estimates onX_n, and the convergence of the approximation method (4.1).

Theorem 4.1.The solutionsXnof (4.1)converge to the solutionX of (1.1)inC([0, T];H )and in L²([0, T];H₀¹) almost surely.

4.1. A priori estimate for the finite dimensional approximations

From now on, we denote by·,·the inner product in H. Let us apply Itô’s formula to the finite dimensional diffusionX_n. SinceX_n(t )∈H_n, remark that

X_n(t ), Π_nD_ξX²_n(t )

=

Π_nX_n(t ), D_ξX_n²(t )

=

X_n(t ), D_ξX²_n(t )

= 1 0

X_n(t, ξ )D_ξX²_n(t, ξ )dξ=

X_n³(t, ξ ) 3

ξ=1 ξ=0

=0 because of no-slip boundary conditions. So, we obtain:

dX_n(t )²

2=2

X_n(t ),dXn(t )

+tr(Qn)dt

=

−2∇X_n(t )²

2+tr(Qn) dt+2

X_n(t ), G_ndW (t ) .

In the same spirit, denoting byd[Y, Y]t the quadratic variation process of a semi-martingaleY, we can also compute with the Itô formula

de^{λ0Xn(t )22}=e^{λ0Xn(t )22}

λ₀dX_n(t )²

2+λ²₀ 2 d

X_n²2,X_n²2

t

=e^{λ0Xn(t )22}

−2λ0∇X_n(t )²

2+λ₀tr(Qn)+2λ²₀G^∗_nX_n(t )²

2

dt +2λ0e^{λ0Xn(t )22}

X_n(t ), G_ndW (t ) .

For any smooth functionf onH_n=Π_n(L²), we defineg:=Lnf if f

Xn(t )

−f Xn(0)

− t

0

g Xn(s)

ds

is a local martingale. The following lemma, being a consequence of Itô’s formula, is well known to probabilists and it is crucial.

Lemma 4.2.([16])If f is smooth onHn, andf 1, then Mt:=e⁻

_t

0 Lnf

f (Xn(s))ds

f Xn(t ) is a local martingale.

In view of the above definition we have forf (x)=e^λ0x22 (x∈H_n), Lnf (x):=f (x)

−2λ0∇x²₂+λ₀tr(Qn)+2λ²₀G^∗_nx²

2

and

−Lnf (x)

f (x) =2λ0∇x²₂−λ₀tr(Qn)−2λ²₀G^∗_nx²

2

2λ0∇x²₂−λ₀tr(Q)−2λ²₀Qx²₂. Moreover, by the Poincaré inequality

x2∇x2

π , ∀x∈H

we obtain for 0< λ₀₂^π_Q², since 1−^λ0_π^Q2 ¹₂

−Lnf (x) f (x) 2λ0

∇x²₂

1−λ₀Q π²

−tr(Q) 2

λ₀∇x²₂−λ₀tr(Q).

So we conclude by Lemma 4.2 that N_tⁿ:=exp

λ₀

t

0

∇X_n(s)²

2ds−λ₀tr(Q)t

e^{λ0Xn(t )22} (4.2)

is a supermartingale. This proves the following crucial exponential estimate:

Lemma 4.3.Let0< λ0<₂^π_Q². For anyxinH, we have E^xexp

λ0

t

0

∇Xn(s)²

2ds

e^{λ0Xn(t )22}e^λ0tr(Q)te^λ0x22, ∀t >0. (4.3) In particular, we have

sup

n∈NE^xe^{λ00t∇Xn(s)22ds}<∞

so(X_n)_nis uniformly bounded inL²(Ω× [0, T], H₀¹).

This kind of estimates was also investigated by Da Prato and Debussche [2] for proving some properties on deriv- atives of the transition semigroup.

4.2. Proof of Theorem 4.1

Let us introduce the stochastic convolution, or Ornstein–Uhlenbeck process W(t )=

t

0

S(t−s)GdW (s)

which is the mild solution of the linear equation with additive noise

dW(t )=W(t )dt+GdW (t ); W(0)=0. (4.4)

SinceQ=GG^∗has finite trace, it is known (see [6, p. 99 and p. 148]), that the stochastic integralWis the limit in L²(Ω, C([0, T];H ))and inL²(Ω, L²([0, T];H₀¹))of its finite dimensional approximation defined by

Wⁿ(t )= t

0

S(t−s)GndW (s)=ΠnW(t ).

Notice thatWⁿ is the mild solution of the finite dimensional linear equation with additive noise

dWⁿ(t )=Wⁿ(t )dt+GndW (t ); W(0)=0. (4.5)

Let us prove that the convergences above hold in fact a.s. inC([0, T];H )andL²([0, T];H₀¹). Indeed the a.s.

convergence of Wⁿ toW in L²([0, T];H₀¹) is obvious. For the convergence in C([0, T], H ), since for a.e. ω, t→W(t, ω)is continuous from [0, T] to H, then K := {W(t, ω);t ∈ [0, T]} is compact inH. Notice that if h∈H,Πnh→hinH and that the mappingsh→Πnh,n1 are equi-continuous onH forΠnH→H=1. So the above pointwise convergence is uniform over the compact subsetKby Arzela–Ascoli’s theorem: asn→ ∞,

sup

t∈[0,T]

W(t, ω)−Π_nW(t, ω)

H→0.

Our proof below, as in [7], will be completely deterministic. Fix anyω∈Ω such thatWⁿ(ω)→W(ω)both in C([0, T], H )andL²([0, T], H₀¹)and we shall remove “ω” in the proof below.

Let us define

y:=X−W=S(t )x+1 2 t

0

S(t−s)D_ξ(X)²(s)ds,

y_n:=X_n−Wⁿ=S(t )Π_nx+1 2 t

0

S(t−s)Π_nD_ξ(X_n)²(s)ds and

z_n:=y−y_n=X−X_n−

W−Wⁿ .

Recall thatXis bounded inL^∞([0, T], H )and also inL²([0, T], H₀¹)almost surely. Indeed we have the following a-priori estimates (see [7, p. 264]):

y(t )²

2e⁸

t

0W(s)²_∞dsx²2+2 t

0

e⁸

t

rW(s)²_∞dsW(r)⁴

∞dr and

T

0

Dξy(t )²

2dt8 T

0

W(t )²

∞y(t )⁴

2dt+ T

0

W(t )⁴

∞dt

and the fact thatH₀¹⊂C[0,1]is a compact continuous embedding. The same proof as in [7, p. 264] yields the same estimates forynwithWreplaced byWⁿ, so the sequence(Xn)nis bounded inL^∞([0, T], H )andL²([0, T], H₀¹) almost surely (see also [3]). We can assume without loss of generality that the preceding bounds hold for our “ω”.

It remains to show the convergence ofz_nto 0 in the desired spaces. Notice thatz_nis solution of dzn

dt =z_n+1

2D_ξX²−1

2Π_nD_ξX_n² from which we can deduce the a priori estimate:

1 2

d

dtz_n²₂+ D_ξz_n²₂= 1

2D_ξ(X)²−1

2Π_nD_ξ(X_n)², z_n

= 1

2D_ξ

X²−X_n² , z_n

+

1

2(I−Π_n)D_ξ X²

, z_n

:=I₁+I₂.

Noting thatX−X_n=z_n+(I−Π_n)W, we have I1 = −1

2

(Xn+X)(X−Xn), Dξzn

= −1 2

(X_n+X)z_n, D_ξz_n

−1 2

(X_n+X)(I−Π_n)W, D_ξz_n :=I₁₁+I₁₂.

We can boundI₁₁as follows

|I11|1

2X_n+X_∞z_n2D_ξz_n2

1

4D_ξz_n²₂+ X_n+X²_∞z_n²₂ and for the second

|I12|1

2Xn+X_∞(I−Πn)W

2Dξzn2

1

4D_ξz_n²₂+ X_n+X²_∞(I−Π_n)W²

2. Similarly, for the remaining term, we have

|I2|(I−Π_n)Dξ(X)²

2+ X²_∞zn²2. Hence we obtain the inequality

d

dtz_n²₂+ D_ξz_n²₂2

X²_∞+ X_n+X²_∞

z_n²₂+2(I−Π_n)D_ξ(X)²

2

+2Xn+X²_∞(I−Πn)W²

2. (4.6)

By Gronwall’s inequality we get z_n(t )²

2exp ^t

0

2X(s)²

∞+2X_n(s)+X(s)²

∞ds(I−Π_n)x²

2

+2 t

0

exp t

s

2X(r)²

∞+2X_n(r)+X(r)²

∞dr

X_n(s)+X(s)²

∞(I−Π_n)W(s)²

2ds

+2 t

0

exp t

s

2X(r)²

∞+2X_n(r)+X(r)²

∞dr

(I−Π_n)D_ξX(s)²

2ds.

In the sequel we denote the norm in the corresponding spaces respectively by

|u|²_L2(0,T ,H ):=

T

0

u(t )²

2dt,

|u|²_L2(0,T ,H₀¹):=

T

0

∇u(t )²

2dt= T

0

D_ξu(t )²

2dt,