Large deviations for the Navier–Stokes equations driven by a white-in-time noise

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Large deviations for the Navier–Stokes equations driven by a white-in-time noise

Vahagn Nersesyan

^∗

February 25, 2019

Abstract

In this paper, we consider the 2D Navier–Stokes system driven by a white-in-time noise. We show that the occupation measures of the trajectories satisfy a large deviations principle, provided that the noise acts directly on all Fourier modes. The proofs are obtained by developing an approach introduced previously for discrete-time random dynamical sys- tems, based on a Kifer-type criterion and a multiplicative ergodic theorem.

AMS subject classifications: 35Q30, 60B12, 60F10, 37H15

Keywords: Stochastic Navier–Stokes system, large deviations principle, occupation measures, multiplicative ergodicity

0 Introduction

We study the large deviations principle (LDP) for the 2D Navier–Stokes system for incompressible fluids:

∂tu+〈u,∇〉u−ν∆u+∇p=f(t, x), divu= 0, (0.1) whereν >0 is the viscosity of the fluid,u= (u1(t, x), u2(t, x)) andp=p(t, x) are unknown velocity field and pressure,f is an external (random) force, and

〈u,∇〉=u1∂1+u2∂2. Throughout this paper, we assume that the space vari- ablex= (x1, x2) belongs¹ to the standard torusT²=R²/2πZ². The problem is considered in the space of divergence-free vector fields with zero mean value

H =

󰀝

u∈L²(T²,R²) : divu= 0 inT²,

󰁝

T²u(x)dx= 0

󰀞

(0.2) endowed with theL²-norm󰀂·󰀂. We assume that the force is of the form

f(t, x) =h(x) +η(t, x),

where h ∈ H¹ := H¹(T²,R²)∩H is a given function and η is a white-in- time noise

η(t, x) =∂tW(t, x), W(t, x) =

󰁛∞ j=1

bjβj(t)ej(x). (0.3) Here{bj} is sequence of real numbers such that

B1=

󰁛∞ j=1

αjb²_j <∞, (0.4)

{βj} is a sequence of independent standard Brownian motions defined on a filtered probability space² (Ω,F,{F^t},P), and {ej} is an orthonormal basis in H consisting of the eigenfunctions of the Stokes operator L = −∆ with eigenvalues {αj}. As usual, projecting (0.1) to H, we eliminate the pressure

1The periodic boundary conditions are chosen to simplify the presentation. Similar results can be established in the case of a bounded domain with smooth boundary and Dirichlet boundary conditions.

2We assume that this space satisfies the usual conditions (see Definition 2.25 in [15]).

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and obtain an evolution equation for the velocity field³ (e.g., see Section 6 in Chapter 1 of [21]):

˙

u+B(u) +Lu=h(x) +η(t, x), (0.5) where B(u) = Π(〈u,∇〉u) and Π is the orthogonal projection onto H in L². This system is supplemented with the initial condition

u(0) =u0. (0.6)

Under these assumptions, problem (0.5), (0.6) admits a unique solution and defines a Markov family (ut,Pû) parametrised by the initial conditionu=u0∈H. The ergodic properties of this family are now well understood. In particular, it is known that (ut,Pû) admits a unique stationary measure, which is exponen- tially mixing, provided that sufficiently many coefficients bj are non-zero (see the papers [7, 18,6,19, 1, 12, 24] and the book [20]). A central limit theorem (CLT) for problem (0.5), (0.6) is established in [16, 25]. The LDP proved in the present paper is a natural extension of the CLT. Indeed, while the CLT describes the probability of small deviations of a time average of a functional from its mean value, the LDP quantifies the probability of large deviations.

Before formulating the main result of this paper, let us introduce some notation and definitions. We denote by P(H) the space of Borel probability measures onH endowed with the topology of weak convergence. Given a mea- sureν∈P(H), we setP^ν(Γ) =󰁕

HP^u(Γ)ν(du) for anyΓ∈F and consider the following family ofoccupation measures

ζt= 1 t

󰁝 t 0

δusds, t >0 (0.7)

defined on the probability space (Ω,F,P^ν). Hereδuis the Dirac measure concentrated atu∈H. We shall say that a mappingI:P(H)→[0,+∞] is agood rate functionif the level set{σ∈P(H) :I(σ)≤α}is compact for anyα≥0. A good rate functionIis nontrivial if its eﬀective domainDI ={σ∈P(H) :I(σ)<∞}

is not a singleton. For any numbersκ>0 andM >0, we denote Λ(κ, M) =

󰀝

ν ∈P(H) :

󰁝

H

e^κ󰀂^v^󰀂²ν(dv)≤M

󰀞 .

Main Theorem. Assume that (0.4) holds and bj > 0 for all j ≥ 1. Then for any numbers κ > 0 and M > 0, the family {ζt, t > 0} satisfies an LDP, uniformly with respect to ν ∈ Λ(κ, M), with a non-trivial good rate function I:P(H)→[0,+∞]not depending on κ and M. More precisely, the following two bounds hold.

Upper bound. For any closed subsetF ⊂P(H), we have lim sup

t→∞

1 t log sup

ν∈ΛP^ν{ζt∈F}≤ − inf

σ∈FI(σ).

3To simplify the notation, we shall assume thatν= 1.

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Lower bound. For any open subsetG⊂P(H), we have lim inf

t→∞

1 t log inf

ν∈ΛP^ν{ζt∈G}≥ − inf

σ∈GI(σ).

Furthermore,I is given by

I(σ) = sup

V∈Cb(H)

󰀕󰁝

H

V(u)σ(du)−Q(V)

󰀖

, σ∈P(H), (0.8) whereQ:Cb(H)→R is a1-Lipschitz convex function such thatQ(C) =C for anyC∈R.

This type of large deviations results have been first established by Donsker and Varadhan [4] and later generalised by many others (see the books [9, 3, 2]

and the references therein). There are only a few works studying the large deviations behaviour of solutions of randomly forced PDEs as time goes to infinity. The case of the stochastic Burgers and Navier–Stokes equations is first studied in [10,11]. In these papers, the random perturbation is of the form (0.3) with the following restriction on the coeﬃcients

cj⁻^α≤bj ≤Cj⁻¹²⁻^ε, 1

2 <α<1, ε∈

󰀕 0,α−1

2

󰀘

. (0.9)

Notice that the lower bound does not allow the sequence {bj} to converge to zero suﬃciently fast, so the external forcefisirregularwith respect to the space variable. This is not very natural from the physical point of view. The proof is based on a general suﬃcient condition established in [26], and essentially uses thestrong Feller property. The main novelty of our Main Theorem is that it proves an LDP without any lower bound on{bj} (so, in particular, we do not have a strong Feller property).

We use an approach introduced in the papers [13, 14], where an LDP is established for a family of dissipative PDEs perturbed by arandom kick force.

The proofs of these papers are based on a Kifer type criterion for LDP and a study of the large-time behaviour of generalised Markov semigroups. These results have been later extended in [22] to the case of the stochastic damped nonlinear wave equation driven by a spatially regular white noise. The main result of that paper is an LDP of local type. In the case of the Navier–Stokes system (0.5), although we follow a similar scheme, there are important diﬀer- ences in all the steps of the argument, coming from both the continuous-time nature of the system and the globalness of the LDP. Here we study the large- time asymptotics of the Feynman–Kac semigroup without any restriction on the smallness of the potential. One of the most important diﬃculties arises in the proof of the uniform Feller property. To establish this, we construct coupling processes usinga new two parameter auxiliary equation(see (3.1)) which allows to have an appropriate Foia¸s–Prodi estimate for the trajectories and a rapid exponential stabilisation for finite-dimensional projections.

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Let us also mention that the multiplicative ergodic theorem we obtain for system (0.5) is of more general form and works for a larger class of functionals and initial measures (see Theorem1.1).

It is a challengingopen problem whether an LDP still holds for (0.5), (0.6) when the driving noise ishighly degenerate(i.e., only a finite number ofbj are non-zero in (0.3)). For the Navier–Stokes system in this degenerate situation, exponential mixing is established in [12] for white-in-time noise and in [17] for a bounded noise satisfying some decomposability and observability hypothe- ses. Using these results and literally repeating the arguments of the proof of Theorem 5.4 in [23], one can prove a level-1 LDP of local type.

The paper is organised as follows. In Section1, we state a multiplicative ergodic theorem for the Navier–Stokes system and combine it with Kifer’s criterion for non-compact spaces to prove the Main Theorem. In Sections 2 and 3, we check the conditions of an abstract result on large-time behaviour of generalised Markov semigroups. Section 4 is devoted to the proof of the multiplicative ergodicity. In the Appendix, we prove various a priori estimates for the solutions and recall the statement of the above-mentioned result for generalised Markov semigroups.

Acknowledgments

The author thanks Davit Martirosyan and Armen Shirikyan for many discus- sions. This research was supported by the ANR grant NONSTOPS ANR-17- CE40-0006-02 and CNRS PICS grant Fluctuation theorems in stochastic sys- tems.

Notation

We shall use the following standard notation.

H is the space defined by (0.2), BH(a, R) is the closed ball inH of radius R centred ata. Whena= 0, we writeBH(R).

H¹ =H¹(T²,R²)∩H, where H¹(T²,R²) is the space of vector functions u= (u1, u2) with components in the usual Sobolev space of order 1 onT².

L^∞(H) is the space of bounded Borel-measurable functions f : H → R endowed with the norm󰀂f󰀂∞= sup_u∈H|f(u)|.

Cb(H) is the space of continuous functionsf ∈L^∞(H).

Lb(H) is the space of functionsf ∈Cb(H) for which the following norm is finite

󰀂ψ󰀂^L=󰀂ψ󰀂∞+ sup

u∕=v

|ψ(u)−ψ(v)|

󰀂u−v󰀂 .

V is the space of functions V :H →Rfor which there is an integer N ≥1 and a functionF∈Lb(HN) such that

V(u) =F(PNu), u∈H. (0.10)

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HerePN is the orthogonal projection inH onto the space

HN = span{e1, . . . , eN} (0.11) and{ej} is the orthonormal basis entering (0.3).

For a given Borel-measurable functionw:H →[1,+∞], we denote byCw(H) (respectively,L^∞_w(H)) the space of continuous (Borel-measurable) functionsf : H→Rsuch that

󰀂f󰀂^L^∞w = sup

u∈H

|f(u)| w(u) <∞.

M⁺(H) is the set of non-negative finite Borel measures on H endowed with the topology of the weak convergence. For µ ∈ M⁺(H) and f ∈ Cb(H), we denote〈f, µ〉=󰁕

Hf(u)µ(du).

P(H) is the set of Borel probability measures onH, and P^w(H) is the set of measuresµ∈P(H) such that〈w, µ〉<∞.

1 Proof of the Main Theorem

In this section, we state a multiplicative ergodic theorem for the Navier–Stokes system (0.5) and apply it to prove the Main Theorem. Let us start by intro- ducing the following twoweight functions

m_κ(u) = exp(κ󰀂u󰀂²), (1.1)

wm(u) = 1 +󰀂u󰀂^2m, u∈H (1.2) for positive numbersκandm. To avoid triple subscripts, we shall writeCm(H) and P^m(H) instead of Cm_κ(H) and P^mκ(H). Recall that the Feynman–Kac semigroupassociated with the family (ut,P^u) is defined by

P^Vt f(u) =E^u󰀋

Ξ^V_tf(ut)󰀌 , where

Ξ^V_t = exp

󰀕󰁝 t 0

V(us)ds

󰀖

. (1.3)

From estimate (5.21) it follows that, for suﬃciently smallκand anyV ∈Cb(H), the applicationP^V_t mapsCm(H) into itself. Let P^V_t^∗ :M⁺(H)→M⁺(H) be its dual. Then a measureµ∈P(H) and a functionh∈Cm(H) areeigenvectors corresponding to an eigenvalueλ>0 if

P^Vt^∗µ=λ^tµ, P^Vth=λ^th for any t >0.

We have the following result.

Theorem 1.1. Under the conditions of the Main Theorem, for any V ∈ V, there are numbersm=m(V)≥1 and γ0=γ0(B0)>0, whereB0 =󰁓

j≥1b²_j, such that the following assertions hold for anyκ∈(0,γ0).

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Existence and uniqueness. There is a unique pair(µV, hV)∈P^m(H)×Cw(H) of eigenvectors corresponding to an eigenvalueλV >0 normalised by the condition〈hV, µV〉= 1.

Convergence. For any f ∈Cm(H),ν∈P(H), andR >0, we have

λ^−t_V P^V_tf → 〈f, µV〉hV in Cb(BH(R))∩L¹(H, µV)ast→ ∞, (1.4) λ⁻_V^tP^V_t^∗ν→ 〈hV,ν〉µV inM⁺(H)as t→ ∞. (1.5) Moreover, for any M >0 andκ^′ ∈(κ,γ0), the convergence

λ⁻_V^tEν

󰀝

f(ut) exp

󰀕 󰁝 t 0

V(us)ds

󰀖󰀞

→ 〈f, µV〉〈hV,ν〉 ast→ ∞ (1.6) holds uniformly in ν∈Λ(κ^′, M).

This theorem is established in Section4. Here we combine it with some arguments from [14,22], to prove the Main Theorem.

Proof of the Main Theorem. Step 1: Reduction. It suﬃces to prove the Main Theorem for smallκ, so we shall assume thatκ∈(0,γ0). Let us take anyM >0 and endow the set

Θ=R^∗+×Λ(κ, M)

with an order relation≺defined by (t1,ν1)≺(t2,ν2) if and only ift1≤t2. Then a family{xθ∈R,θ∈Θ}converges if and only if it converges uniformly with respect toν∈Λ(κ, M) ast→ ∞. Assume that the following three properties hold.

Property 1. For anyV ∈Cb(H) andν ∈Λ(κ, M), the following limit exists Q(V) = lim

t→∞

1

t logE^νexp

󰀕 󰁝 t 0

V(us)ds

󰀖 .

Moreover, it does not depend and is uniform inν∈Λ(κ, M).

Property 2. There is a vector space V ⊂ Cb(H) such that its restriction to any compact setK⊂H is dense inC(K), and for anyV ∈V, there is a uniqueσV ∈P(H) satisfying the relation

Q(V) =〈V,σV〉 −I(σV), (1.7) whereI(σ) is the Legendre transform ofQgiven by (0.8).

Property 3. There is a function Φ : H → [0,+∞] with compact level sets {u∈H :Φ(u)≤α} for anyα≥0 such that

E^νexp

󰀕 󰁝 t 0

Φ(us)ds

󰀖

≤Ce^ct, ν ∈Λ(κ, M), t >0 (1.8) for some positive constantsC andc.

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For anyθ= (t,ν)∈Θ, let us setrθ:=t andζθ :=ζt, whereζt is the random probability measure given by (0.7) defined on the probability space (Ωθ,F^θ,P^θ) :=

(Ω,F,P^ν). The definition of the relation ≺and Properties 1-3 imply that the family {ζθ} satisfies the conditions of the Kifer type criterion established in Theorem 3.3 in [14]. Hence (0.8) defines a good rate function I and for any closed setF ⊂P(H) and open setG⊂P(H), we have

lim sup

θ∈Θ

1

rθ logP^θ{ζθ∈F}≤ − inf

σ∈FI(σ), lim inf

θ∈Θ

1 rθ

logP^θ{ζθ∈G}≥ − inf

σ∈GI(σ).

These two inequalities imply the upper and lower bounds in the Main Theorem, since we have the following equalities

lim sup

θ∈Θ

1

rθlogP^θ{ζθ∈F}= lim sup

t→∞

1 t log sup

ν∈ΛP^ν{ζt∈F}, lim inf

θ∈Θ

1 rθ

logP^θ{ζθ∈G}= lim inf

t→∞

1 t log inf

ν∈ΛP^ν{ζt∈G}. Now we turn to the proofs of Properties 1-3.

Step 2: Proof of Properties 1-3. Property 3 is the easiest one. It is verified forΦ(u) = κ󰀂u󰀂²1 if we choose κ∈(0,γ0). Indeed, (1.8) follows from inequality⁴ (5.20), andΦ has compact level sets, since it is continuous fromH¹ toR and the embeddingH¹⊂H is compact.

Properties 1 and 2 are proved using the same methods as in the case of the discrete-time model considered in [14]. The restriction of V to any compact set K ⊂ H is dense in C(K). Taking f = 1 in (1.6), we get Property 1 for any V ∈V withQ(V) = logλV. In the case of an arbitrary V ∈Cb(H), this property is established by using a buc-approximating sequence Vn ∈ V of V (i.e., sup_n_≥₁󰀂Vn󰀂∞<∞and 󰀂Vn−V󰀂^L^∞^(K) →0 as n→ ∞for any compact K in H) and the exponential tightness of the family {ζθ} (which holds by Property 3). The reader is referred to Section 5.6 of [14] for the details.

To prove Property 2, for any V ∈ V and F ∈ Cb(H), we consider the following auxiliary Markov semigroup

St^V,Fg(u) =λ⁻_V^th⁻_V¹P^Vt^+F(hVg)(u), g∈Cb(H), t≥0.

By Property 1, the following limit exists Q^V(F) := lim

t→∞

1

t log(S_t^V,F1)(u).

LetI^V :M(H)→[0,+∞] be the Legendre transform ofQ^V. The arguments of Section 5.7 in [14] show thatσ∈P(H) satisfies (1.7) if and only ifI^V(σ) = 0.

4We shall see in the proof of Theorem1.1, thatγ0is the number in Lemma5.3.

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On the other hand, by Proposition 1.3 in [22] (whose proof is the same in our case), the measureσV =hVµV is the unique zero ofI^V.

It remains to show that the good rate function I is non-trivial. Assume, by contradiction, that DI is a singleton. Then I(µ) = 0 and I(σ) = +∞ for σ∈ P(H)\ {µ}, where µ is the stationary measure of (ut,P^u). On the other hand, as the Legendre transform is its own inverse, we derive from (0.8) that

Q(V) = sup

σ∈P(H)

󰀃〈V,σ〉 −I(σ)󰀄

forV ∈Cb(H).

This implies that Q(V) = 〈V, µ〉 for any V ∈ Cb(H). Let us take any non- constantV ∈Vsuch that〈V, µ〉= 0. ThenQ(V) = 0, and from limit (1.4) with f =1andν =µwe getλV =e^Q(V⁾= 1 and

sup

t≥0E^µexp

󰀕󰁝 t 0

V(us)ds

󰀖

<∞. (1.9)

Combining the latter with the central limit theorem (see Proposition 4.1.4 in [20]), we getV =0. This contradicts the assumption thatV is non-constant and completes the proof of the Main Theorem.

2 Checking conditions of Theorem 5.6

Theorem1.1is proved by applying a convergence result for generalised Markov semigroups obtained in [14,22] and restated here as Theorem 5.6. In this and next sections, we show that the conditions of that theorem are satisfied for the generalised Markov family of transition kernels defined by

P_t^V(u,Γ) = (P^V_t^∗δu)(Γ), Γ∈B(H), u∈H,

if we takeX=H,XR=BH¹(R), andw=wmwith suﬃciently largem≥1.

2.1 Growth estimates

Estimate (5.24) implies that the measureP_t^V(u,·) is concentrated on the space H¹=∪^∞R=1XR=X_∞ for anyV ∈Cb(H),t >0, andu∈H. The boundedness of V implies that sup_t∈[0,1]󰀂P^V_t1󰀂∞ <∞. So the following proposition gives the growth condition in Theorem5.6.

Proposition 2.1. For any V ∈Cb(H), there are positive numbers m andR0

such that

sup

t≥0

󰀂P^V_tw󰀂^L^∞w

󰀂P^V_t1󰀂^R⁰ <∞, (2.1) wherew=wm and󰀂·󰀂^R⁰ is theL^∞ norm on XR0.

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Proof. Replacing V byV −infHV, we can assume thatV is non-negative.

Step 1. Let us show that there are integersm, R0≥1 such that sup

t≥0

󰀂P^V_t 1󰀂^L^∞w

󰀂P^V_t1󰀂^R⁰ <∞. (2.2) Indeed, letτ(R) be the first hitting time of the setXRdefined by (5.26), and let mandR0be the integers in Proposition5.4forγ=󰀂V󰀂∞. Then for anyu∈H, we have

P^V_t1(u) =EuΞ^V_t =Eu󰀋 IGtΞ^V_t󰀌

+Eu󰀋

IG^c_tΞ^V_t 󰀌

=:I1+I2, (2.3) where Ξ^V_t is given by (1.3) and Gt ={τ(R0) > t}. As V is non-negative, we haveP^V_t 1(u)≥1. This and (5.27) imply that

I1≤EuΞ^V_τ(R₀₎≤Euexp󰀃

γτ(R0)󰀄

≤Cw(u)≤Cw(u)󰀂P^V_t1󰀂^R⁰. (2.4) By the strong Markov property and (5.27),

I2≤E^u󰀋

I^G^tΞ^V_τ(R₀₎E^u(τ(R0))Ξ^V_t 󰀌

≤E^u{e^γτ(R⁰⁾}󰀂P^V_t1󰀂^R0≤Cw(u)󰀂P^V_t 1󰀂^R0. (2.5) Inequalities (2.3)-(2.5) imply (2.2).

Step 2. It suﬃces to prove (2.1) for integer timesk≥1:

sup

k≥0

󰀂P^V_kw󰀂^L^∞w

󰀂P^V_k1󰀂^R0

<∞. (2.6)

Indeed, the semigroup property and the fact thatV is non-negative and bounded imply that

󰀂P^V_t w󰀂^L^∞w =󰀂P^V_t₋_[t](P^V_[t]w)󰀂^L^∞w ≤C0󰀂P^V_[t]w󰀂^L^∞w,

󰀂P^V_t1󰀂^R⁰ ≥ 󰀂P^V_[t]1󰀂^R⁰,

where [t] is the integer part of t and C0 := sup_s_∈_[0,1]󰀂P^V_sw󰀂^L^∞w. By (5.23), we have

C0≤e^γ sup

s∈[0,1]󰀂Psw󰀂^L^∞w <∞, wherePt=P⁰_t is the Markov operator associated with (0.5).

Step 3. To prove (2.6), we use the Markov property and (5.23):

P^V_kw(u)≤e^γE^u󰀋

Ξ^V_k−1w(uk)󰀌

=e^γE^u󰀋

Ξ^V_k₋₁E^uk−1w(u1)󰀌

≤e^γE^u󰀋

Ξ^V_k₋₁[e⁻^mα¹w(uk−1) +C1]󰀌

≤qP^V_k−1w(u) +e^γC1P^V_k−11(u),

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where we choose m >γ/α1, so thatq :=e^γ⁻^mα¹ <1. Iterating this inequality and using the fact that the sequence{󰀂P^V_k1󰀂^R0} is a non-decreasing in k, we obtain

P^V_kw(u)≤q^kw(u) + (1−q)⁻¹e^γC1P^V_k1(u).

This and (2.2) imply (2.6).

We shall also need the following growth estimates with two other weights.

Proposition 2.2. LetV ∈Cb(H)and letR0 andγ0 be the numbers in Propo- sition2.1and Lemma5.3, respectively. Then for any κ∈(0,γ0), we have

sup

t≥0

󰀂P^V_tm󰀂^L^∞m

󰀂P^V_t 1󰀂^R0

<∞, (2.7)

sup

t≥1

󰀂P^V_tF󰀂^L^∞m

󰀂P^V_t1󰀂^R0

<∞, (2.8)

wherem=m_κ andF(u) =󰀂u󰀂²1.

Proof. Step 1: Proof of (2.7). As in the previous proof, we can assume thatV is non-negative andt=kis integer. We take anyA >0 and write

P^V_km(u) =Eu

󰀋I{󰀂uk󰀂²≤A}Ξ^V_km(uk)󰀌 +Eu

󰀋I{󰀂uk󰀂²>A}Ξ^V_km(uk)󰀌

=:Ik+Jk. (2.9)

By (2.2), we have

󰀂P^V_k1󰀂^L^∞m ≤C2󰀂P^V_k1󰀂^R⁰, hence

󰀂Ik󰀂^L^∞m ≤e^κ^A󰀂P^V_k1󰀂^L^∞m ≤C2e^κ^A󰀂P^V_k1󰀂^R⁰. (2.10) To estimateJk, we use the Markov property and (5.22)

Jk(u)≤A⁻¹E^u󰀋

󰀂uk󰀂²Ξ^V_km(uk)󰀌

≤A⁻¹e^γE^u󰀋

󰀂uk󰀂²Ξ^V_k−1m(uk)󰀌

=A⁻¹e^γE^u󰁱

Ξ^V_k₋₁E^uk−1

󰁱󰀂u1󰀂²m(u1)󰁲󰁲

≤A⁻¹C3P^V_k₋₁m(u).

Combining this with (2.9) and (2.10), and choosing A >0 so large thatq :=

A⁻¹C3<1, we get

󰀂P^V_km󰀂^L^∞m ≤C2e^κ^A󰀂P^V_k1󰀂^R0+q󰀂P^V_k−1m󰀂^L^∞m. Iterating, we obtain

󰀂P^V_km󰀂^L^∞m ≤C2e^κ^A(1−q)⁻¹󰀂P^V_k1󰀂^R0+q^k. AsP^V_k1(u)≥1, we arrive at the required inequality (2.7).

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Step 2: Proof of (2.8). For anyt≥1, we have

P^V_tF =P^V_t₋₁(P^V₁F)≤e^γP^V_t₋₁(P1F).

So (5.24) and (2.7) imply that

P^V_tF(u)≤C4P^V_t−1w8(u)≤C5P^V_t−1m(u)≤C6󰀂P^V_t1󰀂^R0m(u).

This proves (2.8).

2.2 Time-continuity

The following lemma proves the time-continuity property.

Lemma 2.3. The function t 󰀁→ P^V_t g(u) is continuous from R⁺ toR for any V ∈Cb(H),g∈Cw(H),u∈H, andw=wm with anym≥1.

Proof. Let us show the continuity at the point T ≥0. For anyt≥0, we write P^V_Tg(u)−P^V_tg(u) =E^u󰀋󰀅

Ξ^V_T −Ξ^V_t󰀆 g(ut)󰀌

+E^u󰀋

[g(uT)−g(ut)]Ξ^V_T󰀌

=:S1+S2.

AsV is bounded andg∈Cw(H), we have

|S1|≤E^u󰀫󰀏󰀏

󰀏󰀏

󰀏exp

󰀣󰁝 T t

V(us)ds

󰀤

−1

󰀏󰀏

󰀏Ξ^V_t|g(ut)|

󰀬

≤ 󰀂g󰀂^L^∞w

󰀓e^|^T⁻^t^|󰀂^V^󰀂^∞−1󰀔

e^T^󰀂^V^󰀂^∞Euw(ut).

Combining this with (5.23), we get S1→0 ast→T. To estimateS2, we take anyR >0 and write

e⁻^T^󰀂^V^󰀂^∞|S2|≤E^u|g(uT)−g(ut)|

=E^u󰀋

I^G^cR|g(uT)−g(ut)|󰀌

+E^u{I^G^R|g(uT)−g(ut)|}

=:S3+S4,

whereGR:={ut, uT ∈BH(R)}. Fromg∈Cw(H) and (5.23) we derive S3≤C1E^u󰀋

I^G^cR(w(uT) +w(ut))󰀌

≤C1R⁻¹E^u󰀋

w²(uT) +w²(ut)󰀌

≤C2R⁻¹w²(u).

On the other hand, by the Lebesgue theorem on dominated convergence, for anyR >0, we haveS4→0 ast→T. ChoosingR >0 suﬃciently large and t suﬃciently close toT, we see thatS3+S4 can be made arbitrarily small. This shows thatS2→0 ast→T and completes the proof of the lemma.

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2.3 Uniform irreducibility

AsV is a bounded function, we have

P_t^V(u,dv)≥e^−t󰀂V^󰀂^∞Pt(u,dv), u∈H,

wherePt(u,·) is the transition function of the Markov family (ut,P^u). Thus to show the uniform irreducibility of{P_t^V}, it suﬃces to prove the following result.

Proposition 2.4. The family{Pt} is uniformly irreducible with respect to the sequence{XR}, i.e., for any integers ρ, R≥1 and anyr >0, there are positive numbersl=l(ρ, r, R)andp=p(ρ, r)such that

Pl(u, BH(ˆu, r))≥p, u∈BH(R),uˆ∈Xρ. (2.11) Proof. Step 1. There is a numberd >0 such that for anyR≥1, we have

Pt(u, Xd)≥1

2, u∈BH(R) (2.12)

for suﬃciently larget=t(R). Indeed, combining (5.23), (5.24), and the Markov property, we get

E^u󰀂ut󰀂²1≤C(e^−8α¹^tR⁸+ 1), u∈BH(R),t≥1.

Taking t so large that e⁻^8α¹^tR⁸ <1 and d > 2√

C and using the Chebyshev inequality, we arrive at

Pt(u, Xd)≥1−d⁻²C(e^−8α¹^tR⁸+ 1)≥ 1 2.

Step 2. By Lemma 3.3.11 in [20], for any non-degenerate ballB⊂H, there isp1=p1(d, B)>0 such that

P1(u, B)≥p1, u∈Xd.

Combining this with a simple compactness and continuity argument, we get P1(u, BH(ˆu, r))≥p2, u∈Xd,uˆ∈Xρ

for some p2 = p2(d,ρ, r) > 0. This estimate, (2.12), and the Kolmogorov–

Chapman relation imply (2.11) withl=t+ 1 and p=p2/2.

2.4 Existence of an eigenvector

Here we show that the dual operator P^V_t^∗ has an eigenvector and give some decay estimates for it.

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Proposition 2.5. For anyV ∈Cb(H)andt >0, the operatorP^V_t^∗has at least one eigenvectorµt,V ∈P(H) with a positive eigenvalueλt,V:

P^V_t^∗µt,V =λt,Vµt,V. (2.13) Moreover, any such eigenvector satisfies

󰁝

H

(󰀂u󰀂ⁿ1+m_κ(u))µt,V(du)<∞, (2.14)

󰀂P^V_t wm󰀂^XR

󰁝

X_R^c

wm(u)µt,V(du)→0 asR→ ∞ (2.15) for anyκ∈(0,γ0) andn, m≥1.

Proof. Step 1: Estimate (2.14). Let us fix t > 0, and let µ ∈ P(H) be an eigenvector of the operator P^V_t^∗ corresponding to an eigenvalue λ > 0. Let us show that µ ∈ P^m(H) with m = m_κ for any κ ∈ (0,γ0). Indeed, for any measurable functionf :H →R⁺∪{+∞}, we have

〈f, µ〉=λ⁻¹〈P^V_t f, µ〉 ≤λ⁻¹e^t󰀂V^󰀂^∞〈Ptf, µ〉. (2.16) Takingf =m_κ, any numberA >0, and setting⁵ C1=λ⁻¹e^t^󰀂^V^󰀂^∞, we obtain

󰁝

H

m_κ(u)µ(du)≤C1

󰁝

HE^u{m_κ(ut)}µ(du)

=C1

󰁝

H

󰀓E^u󰀋

I{󰀂ut󰀂²≤A}m_κ(ut)󰀌 +E^u󰀋

I{󰀂ut󰀂²>A}m_κ(ut)󰀌󰀔

µ(du)

≤C1

󰁝

H

󰀃exp(κA) +A⁻¹Eu󰀋

󰀂ut󰀂²m_κ(ut)󰀌 󰀄 µ(du)

≤C1

󰁝

H

󰀃exp(κA) +C2A⁻¹m_κ(u)󰀄 µ(du), where we used inequality (5.22). ChoosingA > C1C2, we get

󰁝

H

m_κ(u)µ(du)≤C1(1−C1C2A⁻¹)⁻¹exp(κA)<∞, (2.17) so⁶ µ ∈ P^m(H). Taking f(u) = 󰀂u󰀂ⁿ1 in (2.16) and using (5.24) and (2.17), we obtain

󰁝

H󰀂u󰀂ⁿ1µ(du)≤C1

󰁝

HE^u{󰀂ut󰀂ⁿ1}µ(du)≤C3

󰁝

H

(1 +󰀂u󰀂⁸ⁿ)µ(du)<∞ for anyn≥1. This proves (2.14).

5We do not indicate the dependence of diﬀerent constants onV, t, m, n,andκ.

6Note that this proof is formal. A rigorous proof can be obtained by applying the above arguments to bounded approximations ofm.

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Step 2: Limit (2.15). From (5.23) it follows that

󰀂P^V_twm󰀂^XR ≤e^t^󰀂^V^󰀂^∞ sup

u∈XR

E^uwm(ut)

≤C4 sup

u∈BH(R)

wm(u) =C4(1 +R^2m). (2.18) Using the Cauchy–Schwarz inequality, (2.14), and the Chebyshev inequality, we see that 󰁝

X_R^c

wm(u)µ(du)≤ 〈w²_m, µ〉^1/2µ󰀃 X_R^c󰀄1/2

≤C5R⁻ⁿ. Combining this with (2.18) and choosingn >2m, we obtain (2.15).

Step 3: Construction of an eigenvector. Let us take anyA >0 andm≥1 and define the convex set

DA,m:=󰀋

ν ∈P(H) :〈wm,ν〉 ≤A󰀌 .

By the Fatou lemma,DA,mis closed inP(H). Consider the continuous mapping G:=G(t, V) :DA,m→P(H), ν󰀁→ P^Vt^∗ν

P^V_t^∗ν(H).

Let us show thatG(DA,m)⊂DA,m for an appropriate choice ofA andm, and that G(DA,m) is compact in P(H). In view of the Leray–Schauder theorem, this will imply the existence of an eigenvectorµ∈DA,m satisfying (2.13) with eigenvalueλ=P^V_t^∗µ(H)>0. From (5.23) we derive that

󰀍wm, G(ν)󰀎

≤exp{tOsc(V)}〈wm,P^∗_tν〉

≤exp{t(Osc(V)−mα1)}〈wm,ν〉+C6,

where Osc(V) := sup_u∈HV(u)−infu∈HV(u) is the oscillation of V. Choos- ingAand mso large that exp{t(Osc(V)−mα1)}≤1/2 andA≥2C6, we get that G(DA,m) ⊂ DA,m. In view of the Prokhorov compactness criterion (see Theorem 11.5.4 in [5]), to prove thatG(DA,m) is relatively compact, it suﬃces to check that

󰁝

H󰀂u󰀂²1P^V_t^∗ν(du)≤C7 for anyν∈DA,m. Using (5.24) and the fact that V is bounded, we get

󰁝

H󰀂u󰀂²1P^V_t^∗ν(du)≤exp(t󰀂V󰀂∞)

󰁝

H󰀂u󰀂²1(P^∗_tν)(du)

≤C8

󰁝

H󰀂u󰀂⁸ν(du)

≤C9

󰁝

H

wm(u)ν(du)≤C9A=:C7. Thus there is an eigenvectorµ∈DA,m.

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3 Uniform Feller property

In this section, we establish the following result.

Theorem 3.1. For any V ∈ V, the family {P_t^V} satisfies the uniform Feller property with respect to the sequence{XR}, i.e., there is an integerR0≥1 such that the family{󰀂P^V_t 1󰀂⁻¹R P^V_t ψ, t≥0} is uniformly equicontinuous on XR for anyψ∈V andR≥R0.

See the papers [13, 14, 23] for similar results in the case of a discrete-time random dynamical system and [22] for the case of the stochastic damped nonlinear wave equation. The main diﬃculty in the proof of Theorem 3.1 comes from the fact that the oscillation of the potential V can be arbitrarily large.

To overcome this, we introduce a new auxiliary equation in the construction of the coupling processes and choose carefully the parameters in order to have a stabilisation property with an appropriate rate.

3.1 Construction of coupling processes

The coupling processes are constructed following the arguments of [22]. Let us take anyz, z^′∈H and denote byutandu^′_tthe solutions of (0.5) issued fromz andz^′. For any integer N ≥1 and numberλ>0, letv be the solution of the following problem

˙

v+B(v) +Lv+PN[λ(v−u) +B(u)−B(v)] =h+η(t), v(0) =z^′, (3.1) whereηis defined by (0.3). We denote byν(z, z^′) andν^′(z^′) the laws of processes {v(t), t∈J} and{u^′(t), t∈J}, respectively, whereJ = [0,1]. We shall use the following result.

Proposition 3.2. There exists an integer N1 ≥ 1 such that if N ≥ N1 and λ≥N²/2, then for anyε>0 andz, z^′∈H, we have

󰀂ν(z, z^′)−ν^′(z^′)󰀂^var ≤ε^a+ 2󰁫 exp󰀓

Cλ,Nε^a⁻²󰀂z−z^′󰀂²e^C(^󰀂^z^󰀂²⁺^󰀂^z^′^󰀂²⁾󰀔

−1󰁬1/2

, (3.2) where󰀂·󰀂^var denotes the total variation distance onP(C(J;H))anda <2, C, andCλ,N are positive constants not depending onε, z, z^′.

See Section5.2for the proof. By Proposition 1.2.28 in [20], there is a probability space ( ˆΩ,Fˆ,Pˆ) and measurable functionsZ,Z^′:H×H×Ωˆ →C(J;H) such that (Z(z, z^′),Z^′(z, z^′)) is a maximal coupling for (ν(z, z^′),ν(z^′)) for any z, z^′ ∈H. We denote by ˜v and ˜u^′_t the restrictions of Z andZ^′ to time t∈J. Then ˜vtis a solution of

˙˜

v+B(˜v) +L˜v+PN[λ˜v−B(˜v)] =h+ψ(t), ˜v(0) =z^′, where the process{󰁕t

0ψ(s)ds, t∈J}has the same law as

󰀝 W(t)−

󰁝 t 0

PN[B(us)−λus]ds, t∈J

󰀞 .

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Let ˜utbe a solution of

u˙˜+B(˜u) +L˜u+PN[λ˜u−B(˜u)] =h+ψ(t), u(0) =˜ z.

Then{u˜t, t∈J}has the same law as{ut, t∈J}. Now the coupling operatorsR andR^′ are defined by

R^t(z, z^′,ω) = ˜ut, R^′t(z, z^′,ω) = ˜u^′_t, z, z^′∈H,ω∈Ω, tˆ ∈J.

By Proposition3.2, for anyε>0,N ≥N1, andλ≥N²/2, we have P{∃ˆ t∈J s.t. ˜vt∕= ˜u^′_t}

≤ε^a+ 2󰁫 exp󰀓

Cλ,Nε^a⁻²󰀂z−z^′󰀂²e^C(^󰀂^z^󰀂²⁺^󰀂^z^′^󰀂²⁾󰀔

−1󰁬1/2

. (3.3)

Let (Ω^k,F^k,P^k), k ≥ 0 be a sequence of independent copies of ( ˆΩ,Fˆ,Pˆ) and (Ω,F,P) the direct product of (Ω^k,F^k,P^k). For anyω = (ω¹,ω², . . .)∈Ωand z, z^′∈H, we set ˜u0=z, ˜u^′₀=z^′, and

˜

ut(ω) =R^s(˜uk(ω),u˜^′_k(ω),ω^k), u˜^′_t(ω) =R^′s(˜uk(ω),u˜^′_k(ω),ω^k),

˜

vt(ω) =Z^s(˜uk(ω),u˜^′_k(ω),ω^k),

wheret =s+k, s∈[0,1). We shall say that (˜ut,u˜^′_t) is acoupled trajectory at level(N,λ) issued from (z, z^′).

3.2 Proof of Theorem 3.1

Step 1: Stratification. Let us take any V,ψ ∈ V and z, z^′ ∈ XR such that d:=󰀂z−z^′󰀂 ≤1. We need to prove the uniform equicontinuity of the family {gt, t≥0}onXR, where

gt=󰀂P^Vt1󰀂⁻R¹P^Vtψ.

Without loss of generality, we can assume thatψandV are non-negative,ψ≤ 1, and the integerN in representation (0.10) is the same for ψandV (we denote it byN0). Let (ut, u^′_t) := (˜ut,u˜^′_t) be a coupled trajectory at level (N,λ) issued from (z, z^′) and letvt:= ˜vtbe the associated process. The parametersN ≥N0

andλ≥N²/2 will be chosen later.

Following [22,14], for any integersr≥0 andρ≥1, we introduce the events⁷ G¯r=

󰁟r j=0

Gj, Gj={vt=u^′_t,∀t∈(j, j+ 1]}, Fr,0=∅,

Fr,ρ=

󰀝 sup

t∈[0,r]

󰀕󰁝 t 0

󰀃󰀂us󰀂²1+󰀂u^′_s󰀂²1

󰀄ds−Kt

󰀖

≤ 󰀂z󰀂²+󰀂z^′󰀂²+ρ;

󰀂ur󰀂²+󰀂u^′_r󰀂²≤ρ

󰀞 ,

7The event ¯Gr is well defined also forr= +∞.

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whereK is the constant in (5.19), and the pairwise disjoint events A0=G^c₀, Ar,ρ=󰀃G¯r−1∩G^c_r∩Fr,ρ󰀄

\Fr,ρ−1, r≥1,ρ≥1, A˜= ¯G+∞

We decompose as follows

P^V_t ψ(z)−P^V_tψ(z^′) =E󰀋 I^A0

󰀅Ξ^V_tψ(ut)−Ξ^V_t ψ(u^′_t)󰀆󰀌

+

󰁛∞ r,ρ=1

E󰀋 I^Ar,ρ

󰀅Ξ^V_tψ(ut)−Ξ^V_tψ(u^′_t)󰀆󰀌

+E󰀋 IA^˜

󰀅Ξ^V_tψ(ut)−Ξ^V_tψ(u^′_t)󰀆󰀌

=I₀^t+

󰁛∞ r,ρ=1

I_r,ρ^t + ˜I^t, (3.4)

where

I₀^t=E󰀋

I^A0[Ξ^V_tψ(ut)−Ξ^V_tψ(u^′_t)󰀆󰀌

, I_r,ρ^t =E󰀋

I^Ar,ρ[Ξ^V_t ψ(ut)−Ξ^V_t ψ(u^′_t)󰀆󰀌

, I˜^t=E󰀋

IA^˜

󰀅Ξ^V_tψ(ut)−Ξ^V_t ψ(u^′_t)󰀆󰀌

. In Steps 2 and 3, we estimateI₀^t, I_r,ρ^t , and ˜I^t.

Step 2: Estimates for I₀^t andI_r,ρ^t . We have following inequalities

|I₀^t|≤C1(R, V)󰀂P^Vt 1󰀂^RP(A0)^1/2, (3.5)

|I_r,ρ^t |≤C2(R, V)e^r^󰀂^V^󰀂^∞󰀂P^V_t1󰀂^RP(Ar,ρ)^1/2 (3.6) for any integersr,ρ≥1 andR≥R0, whereR0is the number in Proposition2.1.

Let us prove (3.6), the other inequality is proved in a similar way. First assume thatr+ 1≤t. Using ψ≤1, the positivity of Ξ^V_tψ, and the Markov property, we derive

I_r,ρ^t ≤E󰀋

I^Ar,ρΞ^V_tψ(ut)󰀌

≤E󰀋

I^Ar,ρΞ^V_t󰀌

=E󰀋 IAr,ρE󰀅

Ξ^V_t 󰀏󰀏F^r+1󰀆󰀌

≤e^r^󰀂^V^󰀂^∞E󰀋

IAr,ρ(P^V_t₋_r₋₁1)(ur+1)󰀌 , where{F^t}is the filtration generated by (ut, u^′_t). Then from the positivity ofV and (2.1) it follows that

P^V_t−r−11(y)≤P^V_t1(y)≤M󰀂P^V_t1󰀂^R0w(y), y∈H, so that

I_r,ρ^t ≤C3e^r󰀂V^󰀂^∞󰀂P^V_t 1󰀂^R0E󰀋

I^Ar,ρw(ur)󰀌

≤C3e^r^󰀂^V^󰀂^∞󰀂P^V_t 1󰀂^R⁰󰀋

P(Ar,ρ)Ew²(ur)󰀌^1/2 .

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Using this, (5.23), and the symmetry, we obtain (3.6). Ifr > t, then I_r,ρ^t ≤e^r^󰀂^V^󰀂^∞P(Ar,ρ)≤e^r^󰀂^V^󰀂^∞󰀂P^V_t1󰀂^RP(Ar,ρ)^1/2, which implies (3.6) by symmetry.

Step 3: Estimate for I˜^t. Let us show that

|I˜^t|≤C4(R, V,λ, N,ψ)󰀂P^V_t1󰀂^Rd. (3.7) Indeed, we write

I˜^t=E󰀋

IA^˜Ξ^V_t[ψ(ut)−ψ(u^′_t)]󰀌 +E󰀋

IA^˜[Ξ^V_t −Ξ^V_t^′]ψ(u^′_t)󰀌

=:J₁^t+J₂^t, whereΞ^V_t^′ := exp󰀓󰁕t

0V(u^′_s)ds󰀔

. Then by (5.3), on the event ˜Awe have

󰀂PN(us−u^′_s)󰀂 ≤e⁻^λsd, s∈[0, t].

Sinceψ∈Lb(H), we derive from this

|J₁^t|≤E󰀋

IA^˜Ξ^V_t |ψ(ut)−ψ(u^′_t)|󰀌

≤ 󰀂ψ󰀂^Le⁻^λt󰀂P^V_t1󰀂^Rd≤ 󰀂ψ󰀂^L󰀂P^V_t1󰀂^Rd.

Similarly, asV ∈Lb(H),

|J₂^t|≤E󰀋

IA^˜|Ξ^V_t −Ξ^V_t^′|󰀌

≤E

󰀝 IA^˜Ξ^V_t

󰀗 exp

󰀕󰁝 t

0 |V(us)−V(u^′_s)|ds

󰀖

−1

󰀘󰀞

≤󰀅 exp󰀃

Cλ,Nλ⁻¹󰀂V󰀂^Ld(1−e⁻^λt)󰀄

−1󰀆

󰀂P^V_t1󰀂^R

≤[exp (C5(R, V,λ, N)d)−1]󰀂P^V_t1󰀂^R.

Recalling thatd≤1 and combining the estimates forJ₁^tandJ₂^t, we get (3.7).

Step 4: Uniform equicontinuity ofgt. We use the following lemma, which is proved at the end of this subsection.

Lemma 3.3. For anyα>0, there is an integerN2(α)≥1and positive numbers aandβ such that

P{A0}≤C6(R,λ, N)d^a/2, (3.8)

P{Ar,ρ}≤C7(R)

󰀫󰀣

d^ae^−aαr+󰁫 exp󰀓

C8(R,λ, N)d^ae^C^′^ρ−aαr󰀔

−1󰁬^1/2󰀤

∧e^−βρ

󰀬

(3.9) for anyN ≥N2(α),λ≥N²/2,R≥1, and a universal constant C^′>0.

From (3.4)-(3.9) it follows that, for anyz, z^′∈XR,t≥0,R≥R0, andα>0, we have

󰀏󰀏g_t(z)−gt(z^′)󰀏󰀏≤C9(R, V,λ, N,ψ)

󰀣

d^a/4+d

+

󰁛∞ r,ρ=1

e^r^󰀂^V^󰀂^∞

󰀝󰀕

d^a/2e⁻^aαr/2+󰁫 exp󰀓

C8d^ae^C^′^ρ⁻^aαr󰀔

−1󰁬1/4󰀖

∧e⁻^βρ/2

󰀞󰀤

,

Large deviations for the Navier–Stokes equations driven by a white-in-time noise