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

VA H A G N N E R S E S Y A N

L A R G E D E V I AT IO N S F O R T H E N AV IE R – S T O K E S E Q U AT IO N S

D R I V E N B Y A W H I T E - I N - T I M E N OIS E

G R A N D E S D É V I AT IO N S P O U R L E S É Q U AT IO N S D E N AV IE R – S T O K E S

P E R T U R B É E S PA R U N B R UI T B L A N C E N T E M P S

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 systems, based on a Kifer-type criterion and a multiplicative ergodic theorem.

Résumé. — Dans cet article, nous étudions le système de Navier–Stokes en dimension deux perturbé par un bruit blanc en temps. Nous montrons un principe de grandes déviations pour les mesures empiriques des trajectoires sous l’hypothèse que tous les modes de Fourier sont excités par le bruit. La preuve utilise une approche introduite précédemment pour des Keywords:Stochastic Navier–Stokes system, large deviations principle, occupation measures, mul- tiplicative ergodicity.

2010Mathematics Subject Classification:35Q30, 60B12, 60F10, 37H15.

DOI:https://doi.org/10.5802/ahl.23

(*) The author thanks Davit Martirosyan and Armen Shirikyan for many discussions. This research was supported by the ANR grant NONSTOPS ANR-17-CE40-0006-02 and CNRS PICS grant Fluctuation theorems in stochastic systems.

systèmes dynamiques aléatoires à temps discret, basée sur un critère de type Kifer et un théorème ergodique multiplicatif.

1. Introduction

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

(1.1) ∂_tu+hu,∇iu−ν∆u+∇p=f(t, x), divu= 0,

where ν >0 is the viscosity of the fluid, u= (u₁(t, x), u₂(t, x)) and p= p(t, x) are unknown velocity field and pressure, f is an external (random) force, and hu,∇i= u₁∂₁+u₂∂₂. Throughout this paper, we assume that the space variable x= (x₁, x₂) belongs⁽¹⁾ to the standard torus T² = R²/2πZ². The problem is considered in the space of divergence-free vector fields with zero mean value

(1.2) H =

u∈L²(T²,R²) : divu= 0 inT²,

Z

T²

u(x)dx= 0

endowed with theL²-norm k · k. 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 (1.3) η(t, x) = ∂_tW(t, x), W(t, x) =

∞

X

j=1

b_jβ_j(t)e_j(x).

Here{b_j} is sequence of real numbers such that

(1.4) B₁ =

∞

X

j=1

αjb²_j <∞,

{βj} is a sequence of independent standard Brownian motions defined on a filtered probability space⁽²⁾ (Ω,F,{F_t},P), and{e_j}is an orthonormal basis inH consisting of the eigenfunctions of the Stokes operatorL=−∆ with eigenvalues{α_j}. As usual, projecting (1.1) to H, we eliminate the pressure and obtain an evolution equation for the velocity field⁽³⁾ (e.g., see [Lio69, Chapter 1, Section 6]):

(1.5) u˙ +B(u) +Lu=h(x) +η(t, x),

where B(u) = Π(hu,∇iu) and Π is the orthogonal projection onto H in L². This system is supplemented with the initial condition

(1.6) u(0) =u₀.

Under these assumptions, problem (1.5), (1.6) admits a unique solution and defines a Markov family (u_t,P^u) parametrised by the initial condition u = u0 ∈ H. The

(1)The 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.

(2)We assume that this space satisfies the usual conditions (see [KS91, Definition 2.25]).

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

ergodic properties of this family are now well understood. In particular, it is known that (u_t,P^u) admits a unique stationary measure, which is exponentially mixing, provided that sufficiently many coefficients bj are non-zero (see the papers [FM95, KS00, EMS01, KS02, BKL02, HM06, Oda08] and the book [KS12]). A central limit theorem (CLT) for problem (1.5), (1.6) is established in [Kuk02, Shi06]. 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 byP(H) the space of Borel probability measures on H endowed with the topology of weak convergence. Given a measure ν ∈ P(H), we set Pν(Γ) = ^R_HPu(Γ)ν(du) for any Γ ∈ F and consider the following family of occupation measures

(1.7) ζ_t = 1

t

Z t 0

δ_usds, t >0

defined on the probability space (Ω,F,P^ν). Hereδ_u is the Dirac measure concentrated atu∈H. We shall say that a mapping I :P(H)→[0,+∞] is a good rate function if the level set {σ ∈ P(H) : I(σ) 6 α} is compact for any α > 0. A good rate functionI is nontrivial if its effective domainD_I ={σ∈ P(H) :I(σ)<∞} is not a singleton. For any numbersκ >0 and M > 0, we denote

Λ(κ, M) =

ν ∈ P(H) :

Z

H

e^κkvk2ν(dv)6M

.

Main Theorem. — Assume that (1.4) 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∈ΛPν{ζ_t∈F}6−inf

σ∈FI(σ).

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

(1.8) I(σ) = sup

V∈Cb(H)

Z

H

V(u)σ(du)−Q(V)

, σ ∈ P(H),

where Q : C_b(H) → R is a 1-Lipschitz convex function such that Q(C) = C for any C ∈R.

This type of large deviations results have been first established by Donsker and Varadhan [DV75] and later generalised by many others (see the books [FW84, DS89, DZ00] 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 [Gou07a, Gou07b]. In these papers, the random perturbation is of the form (1.3) with the following restriction on the coefficients

(1.9) cj^−α 6b_j 6Cj^−12−ε, 1

2 < α <1, ε∈

0, α− 1 2

.

Notice that the lower bound does not allow the sequence {b_j} to converge to zero sufficiently fast, so the external forcef isirregular with 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 [Wu01], and essentially uses the strong 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 [JNPS15, JNPS18], where an LDP is established for a family of dissipative PDEs perturbed by a random 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 [MN18a] to the case of the stochastic damped nonlinear wave equation driven by aspatially regular white noise. The main result of that paper is an LDP of local type. In the case of the Navier–Stokes system (1.5), although we follow a similar scheme, there are important differences 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 using a new two parameter auxiliary equation (see (4.1)) which allows to have an appropriate Foiaş–Prodi estimate for the trajectories and a rapid exponential stabilisation for finite-dimensional projections.

Let us also mention that the multiplicative ergodic theorem we obtain for sys- tem (1.5) is of more general form and works for a larger class of functionals and initial measures (see Theorem 2.1).

It is a challengingopen problem whether an LDP still holds for (1.5), (1.6) when the driving noise ishighly degenerate(i.e., only a finite number ofb_j are non-zero in (1.3)).

For the Navier–Stokes system in this degenerate situation, exponential mixing is established in [HM06] for white-in-time noise and in [KNS18] for a bounded noise satisfying some decomposability and observability hypotheses. Using these results and literally repeating the arguments of the proof of Theorem 5.4 in [MN18b], one can prove a level-1 LDP of local type.

The paper is organised as follows. In Section 2, 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 3 and 4, we check the conditions of an abstract result on large-time behaviour of generalised Markov semigroups. Section 5 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.

Notation We shall use the following standard notation.

• H is the space defined by (1.2), B_H(a, R) is the closed ball in H of radius R centred at a. When a= 0, we write B_H(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 on T².

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

• C_b(H) is the space of continuous functions f ∈L^∞(H).

• Lb(H) is the space of functions f ∈ Cb(H) for which the following norm is finite

kψk_L =kψk∞+ sup

u6=v

|ψ(u)−ψ(v)|

ku−vk .

• V is the space of functions V : H → R for which there is an integer N > 1 and a function F ∈Lb(HN) such that

(1.10) V(u) =F(P_Nu), u∈H.

Here PN is the orthogonal projection in H onto the space (1.11) H_N = span{e₁, . . . , e_N}

and {ej} is the orthonormal basis entering (1.3).

• For a given Borel-measurable function w : H → [1,+∞], we denote by C_w(H) (respectively, L^∞_w(H)) the space of continuous (Borel-measurable) functions f :H →R such that

kfk_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 hf, µi=^R_H f(u)µ(du).

• P(H) is the set of Borel probability measures on H, andP_w(H) is the set of measures µ∈ P(H) such that hw, µi<∞.

2. Proof of the Main Theorem

In this section, we state a multiplicative ergodic theorem for the Navier–Stokes system (1.5) and apply it to prove the Main Theorem. Let us start by introducing the following twoweight functions

m_κ(u) = exp(κkuk²), (2.1)

w_m(u) = 1 +kuk^2m, u∈H (2.2)

for positive numbersκ and m. To avoid triple subscripts, we shall writeC_m(H) and Pm(H) instead of Cm_κ(H) and Pm_κ(H). Recall that the Feynman–Kac semigroup associated with the family (u_t,P^u) is defined by

P^V_t f(u) = Eu

nΞ^V_t f(u_t)^o, where

(2.3) Ξ^V_t = exp

Z _t

0

V(u_s)ds

.

From estimate (6.21) it follows that, for sufficiently smallκ and anyV ∈C_b(H), the application P^V_t maps C_m(H) into itself. Let P^V_t^∗ : M₊(H)→ M₊(H) be its dual.

Then a measureµ∈ P(H) and a functionh∈C_m(H) areeigenvectors corresponding to an eigenvalueλ >0 if

P^V_t ^∗µ=λ^tµ, P^V_t h=λ^th for any t >0.

We have the following result.

Theorem 2.1. — Under the conditions of the Main Theorem, for any V ∈ V, there are numbers m=m(V)>1 andγ0 =γ0(B0)>0, where B0 =^P_j>1b²_j, such that the following assertions hold for any κ∈(0, γ₀).

Existence and uniqueness. — There is a unique pair (µ_V, h_V)∈ P_m(H)×C_w(H) of eigenvectors corresponding to an eigenvalue λ_V >0 normalised by the condition

hh_V, µ_Vi= 1.

Convergence. — For anyf ∈C_m(H), ν ∈ P(H), and R >0, we have λ^−t_V P^V_t f → hf, µ_Vih_V in C_b(B_H(R))∩L¹(H, µ_V) ast → ∞, (2.4)

λ^−t_V P^V_t^∗ν → hh_V, νiµ_V inM₊(H) ast → ∞.

(2.5)

Moreover, for any M > 0 and κ⁰ ∈(κ, γ₀), the convergence (2.6) λ^−t_V Eν

(

f(u_t) exp

Z t 0

V(u_s)ds

!)

→ hf, µ_Vi hh_V, νi ast→ ∞ holds uniformly inν ∈Λ(κ⁰, M).

This theorem is established in Section 5. Here we combine it with some arguments from [JNPS18, MN18a], to prove the Main Theorem.

Proof of the Main Theorem.

Step 1: Reduction. — It suffices to prove the Main Theorem for small κ, 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 if t₁ 6t₂. Then a family {xθ ∈R, θ ∈Θ}converges if and only if it converges uniformly with respect toν ∈Λ(κ, M) as t → ∞. Assume that the following three properties hold.

(1) For any V ∈Cb(H) and ν ∈Λ(κ, M), the following limit exists Q(V) = lim

t→∞

1

t logEνexp

Z t 0

V(u_s)ds

!

.

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

(2) There is a vector space V ⊂C_b(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

(2.7) Q(V) = hV, σ_Vi −I(σ_V),

whereI(σ) is the Legendre transform ofQ given by (1.8).

(3) There is a functionΦ:H →[0,+∞] with compact level sets{u∈H :Φ(u)6 α} for any α >0 such that

(2.8) E^νexp

Z t 0

Φ(u_s)ds

!

6Ce^ct, ν ∈Λ(κ, M), t >0 for some positive constants C and c.

For anyθ = (t, ν)∈Θ, let us setrθ :=tandζθ := ζt, whereζtis the random probabil- ity measure given by (1.7) defined on the probability space (Ω_θ,Fθ,P^θ) := (Ω,F,P^ν).

The definition of the relation≺and Properties (1)–(3) imply that the family{ζ_θ}sat- isfies the conditions of the Kifer type criterion established in Theorem 3.3 in [JNPS18].

Hence (1.8) defines a good rate function I and for any closed set F ⊂ P(H) and open setG⊂ P(H), we have

lim sup

θ∈Θ

1

r_θ logP^θ{ζ_θ ∈F}6− inf

σ∈FI(σ), lim inf

θ∈Θ

1

r_θ logPθ{ζ_θ ∈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

r_θ logPθ{ζ_θ ∈F}= lim sup

t→∞

1

t log sup

ν∈ΛPν{ζ_t ∈F}, lim inf

θ∈Θ

1

r_θ logPθ{ζ_θ ∈G}= lim inf

t→∞

1

t log inf

ν∈ΛPν{ζ_t∈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 for Φ(u) = κkuk²₁ if we choose κ ∈ (0, γ₀). Indeed, (2.8) follows from inequality⁽⁴⁾ (6.20), and Φhas compact level sets, since it is continuous from H¹ to Rand the embedding H¹ ⊂H is compact.

Properties (1) and (2) are proved using the same methods as in the case of the discrete-time model considered in [JNPS18]. The restriction of V to any compact setK ⊂H is dense inC(K). Takingf =1in (2.6), we get Property 1 for anyV ∈ V with Q(V) = logλ_V. In the case of an arbitrary V ∈ C_b(H), this property is established by using abuc-approximatingsequenceV_n ∈ V ofV (i.e., sup_n>1kV_nk∞<

∞andkV_n−Vk_L^∞_(K) →0 asn → ∞for any compactK in H) and the exponential tightness of the family{ζ_θ} (which holds by Property 3). The reader is referred to Section 5.6 of [JNPS18] for the details.

(4)We shall see in the proof of Theorem 2.1, thatγ0is the number in Lemma 6.3.

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

St^V,Fg(u) = λ^−t_V h⁻¹_V P^V_t^+F(h_Vg)(u), g ∈C_b(H), t>0.

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

t→∞

1

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

Let I^V : M(H) → [0,+∞] be the Legendre transform of Q^V. The arguments of Section 5.7 in [JNPS18] show thatσ ∈ P(H) satisfies (2.7) if and only if I^V(σ) = 0.

On the other hand, by Proposition 1.3 in [MN18a] (whose proof is the same in our case), the measureσV =hVµV is the unique zero of I^V.

It remains to show that the good rate functionI is non-trivial. Assume, by contra- diction, thatD_I is a singleton. ThenI(µ) = 0 and I(σ) = +∞ forσ ∈ P(H)\ {µ}, where µ is the stationary measure of (u_t,Pu). On the other hand, as the Legendre transform is its own inverse, we derive from (1.8) that

Q(V) = sup

σ∈P(H)

hV, σi −I(σ) for V ∈C_b(H).

This implies that Q(V) =hV, µi for any V ∈C_b(H). Let us take any non-constant V ∈ V such that hV, µi = 0. Then Q(V) = 0, and from limit (2.4) with f = 1 and ν=µ we get λ_V =e^Q(V)= 1 and

(2.9) sup

t>0 Eµexp

Z _t

0

V(u_s)ds

<∞.

Combining the latter with the central limit theorem (see [KS12, Proposition 4.1.4]), we getV =0. This contradicts the assumption thatV is non-constant and completes

the proof of the Main Theorem.

3. Checking conditions of Theorem 6.6

Theorem 2.1 is proved by applying a convergence result for generalised Markov semigroups obtained in [JNPS18, MN18a] and restated here as Theorem 6.6. 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^∗δ_u)(Γ), Γ∈ B(H), u∈H,

if we take X =H, X_R =B_H¹(R), and w=w_m with sufficiently large m>1.

3.1. Growth estimates

Estimate (6.24) implies that the measure P_t^V(u,·) is concentrated on the space H¹ =^S∞_R=1X_R=X∞ for any V ∈C_b(H),t >0, and u∈H. The boundedness of V implies that sup_t∈[0,1]kP^V_t 1k∞ <∞. So the following proposition gives the growth condition in Theorem 6.6.

Proposition 3.1. — For any V ∈C_b(H), there are positive numbers m and R₀ such that

sup

t>0

kP^V_t wk_L^∞_w kP^V_t 1k_R0 <∞, (3.1)

where w=wm and k · kR0 is the L^∞ norm on XR0.

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

Step 1. — Let us show that there are integers m, R₀ >1 such that

(3.2) sup

t>0

kP^V_t 1k_L^∞_w kP^V_t 1k_R0 <∞.

Indeed, letτ(R) be the first hitting time of the set XR defined by (6.26), and let m andR₀ be the integers in Proposition 6.4 for γ = kVk_∞. Then for any u ∈ H, we have

(3.3) P^V_t 1(u) =EuΞ^V_t =Eu

n

IGtΞ^V_t ^o+Eu

n

IG^c_tΞ^V_t ^o=:I₁+I₂,

where Ξ^V_t is given by (2.3) and G_t = {τ(R₀) > t}. As V is non-negative, we have P^V_t 1(u)>1. This and (6.27) imply that

(3.4) I₁ 6E^uΞ^V_τ(R

0)6E^uexpγτ(R₀)6Cw(u)6Cw(u)kP^V_t 1k_R0. By the strong Markov property and (6.27),

I₂ 6Eu

n

IGtΞ^V_τ(R0)Eu(τ(R0))Ξ^V_t ^o

6Eu{e^γτ(R0)} kP^V_t 1k_R0 6Cw(u)kP^V_t 1k_R0. (3.5)

Inequalities (3.3)–(3.5) imply (3.2).

Step 2. — It suffices to prove (3.1) for integer times k >1:

(3.6) sup

k>0

kP^V_kwk_L^∞_w kP^V_k1kR0

<∞.

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

kP^V_t wk_L^∞_w =kP^V_t−[t](P^V_[t]w)k_L^∞_w 6C₀kP^V_[t]wk_L^∞_w, kP^V_t 1k_R0 >kP^V_[t]1k_R0,

where [t] is the integer part of t and C₀ := sup_s∈[0,1]kP^V_swk_L^∞_w.By (6.23), we have C₀ 6e^γ sup

s∈[0,1]

kP_swk_L^∞_w <∞,

where P_t =P⁰_t is the Markov operator associated with (1.5).

Step 3. — To prove (3.6), we use the Markov property and (6.23):

P^V_kw(u)6e^γEu

nΞ^V_k−1w(u_k)^o

=e^γE^u

nΞ^V_k−1E^uk−1w(u₁)^o 6e^γEu

nΞ^V_k−1[e^−mα1w(uk−1) +C₁]^o 6qP^V_k−1w(u) +e^γC1P^V_k−11(u),

where we choose m > γ/α₁, so that q :=e^γ−mα1 < 1. Iterating this inequality and using the fact that the sequence{kP^V_k1k_R0} is a non-decreasing ink, we obtain

P^V_kw(u)6q^kw(u) + (1−q)⁻¹e^γC₁P^V_k1(u).

This and (3.2) imply (3.6).

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

Proposition 3.2. — Let V ∈ C_b(H) and let R₀ and γ₀ be the numbers in Proposition 3.1 and Lemma 6.3, respectively. Then for anyκ ∈(0, γ₀), we have

sup

t>0

kP^V_t mk_L^∞_m kP^V_t 1k_R0 <∞, (3.7)

sup

t>1

kP^V_t Fk_L^∞_m kP^V_t 1k_R0 <∞, (3.8)

where m=m_κ and F(u) =kuk²₁. Proof.

Step 1: Proof of (3.7). — As in the previous proof, we can assume that V is non-negative andt=k is integer. We take any A >0 and write

P^V_km(u) = E^u

n

I^{kukk²6A}Ξ^V_km(u_k)^o+E^u

n

I^{kukk²>A}Ξ^V_km(u_k)^o

=:Ik+Jk. (3.9)

By (3.2), we have

kP^V_k1k_L^∞_m 6C₂kP^V_k1k_R0, hence

(3.10) kI_kk_L^∞_m 6e^κAkP^V_k1k_L^∞_m 6C₂e^κAkP^V_k1k_R0. To estimate J_k, we use the Markov property and (6.22)

J_k(u)6A⁻¹Eu

nku_kk²Ξ^V_km(u_k)^o6A⁻¹e^γEu

nku_kk²Ξ^V_k−1m(u_k)^o

=A⁻¹e^γE^u

Ξ^V_k−1E^uk−1

ku₁k²m(u₁)

6A⁻¹C₃P^V_k−1m(u).

Combining this with (3.9) and (3.10), and choosingA so large that q :=A⁻¹C₃ <1, we get

kP^V_kmk_L^∞_m 6C₂e^κAkP^V_k1k_R0 +qkP^V_k−1mk_L^∞_m. Iterating, we obtain

kP^V_kmk_L^∞_m 6C₂e^κA(1−q)⁻¹kP^V_k1k_R0 +q^k.

As P^V_k1(u)>1, we arrive at the required inequality (3.7).

Step 2: Proof of (3.8). — For any t>1, we have

P^V_t F =P^V_t−1(P^V₁F)6e^γP^V_t−1(P₁F).

So (6.24) and (3.7) imply that

P^V_t F(u)6C4P^V_t−1w8(u)6C5P^V_t−1m(u)6C6kP^V_t 1kR0m(u).

This proves (3.8).

3.2. Time-continuity

The following lemma proves the time-continuity property.

Lemma 3.3. — The function t 7→ P^V_t g(u) is continuous from R+ to R for any V ∈C_b(H), g ∈C_w(H), u∈H, and w=w_m with any m>1.

Proof. — Let us show the continuity at the point T >0. For any t >0, we write P^V_Tg(u)−P^V_t g(u) = E^u

nhΞ^V_T −Ξ^V_t ⁱg(u_t)^o+E^u

n[g(u_T)−g(u_t)] Ξ^V_T^o

=:S₁+S₂. As V is bounded andg ∈C_w(H), we have

|S₁|6Eu

(

exp

Z T t

V(u_s)ds

!

−1

Ξ^V_t |g(u_t)|

)

6kgkL^∞_w

e^{|T−t|kVk∞} −1e^TkVk∞E^uw(ut).

Combining this with (6.23), we get S₁ → 0 as t → T. To estimate S₂, we take any R >0 and write

e^−TkVk∞|S₂|6Eu|g(u_T)−g(u_t)|

=Eu

n

IG^c_R|g(u_T)−g(u_t)|^o+Eu{IGR|g(u_T)−g(u_t)|}

=:S₃+S₄,

where G_R:={u_t, u_T ∈B_H(R)}. From g ∈C_w(H) and (6.23) we derive S3 6C1E^u

n

I^Gc_R(w(u_T) +w(u_t))^o 6C₁R⁻¹Eu

nw²(u_T) +w²(u_t)^o6C₂R⁻¹w²(u).

On the other hand, by the Lebesgue theorem on dominated convergence, for any R >0, we haveS₄ →0 ast →T. Choosing R >0 sufficiently large andt 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.

3.3. Uniform irreducibility AsV is a bounded function, we have

P_t^V(u,dv)>e^−tkVk∞P_t(u,dv), u∈H,

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

Proposition 3.4. — The family {P_t} is uniformly irreducible with respect to the sequence {X_R}, i.e., for any integers ρ, R>1 and anyr >0, there are positive numbers l =l(ρ, r, R)and p=p(ρ, r) such that

(3.11) P_l(u, B_H(ˆu, r))>p, u∈B_H(R), uˆ∈X_ρ. Proof.

Step 1. — There is a number d >0 such that for any R>1, we have (3.12) P_t(u, X_d)> 1

2, u∈B_H(R)

for sufficiently large t = t(R). Indeed, combining (6.23), (6.24), and the Markov property, we get

Euku_tk²₁ 6C(e^−8α1tR⁸+ 1), u∈B_H(R),t >1.

Takingtso large thate^−8α1tR⁸ <1 andd >2√

Cand using the Chebyshev inequality, we arrive at

P_t(u, X_d)>1−d⁻²C(e^−8α1tR⁸+ 1) > 1 2.

Step 2. — By Lemma 3.3.11 in [KS12], for any non-degenerate ball B ⊂H, there isp₁ =p₁(d, B)>0 such that

P₁(u, B)>p₁, u∈X_d.

Combining this with a simple compactness and continuity argument, we get P₁(u, B_H(ˆu, r))>p₂, u∈X_d, uˆ∈X_ρ

for some p₂ =p₂(d, ρ, r)>0. This estimate, (3.12), and the Kolmogorov–Chapman relation imply (3.11) withl =t+ 1 andp=p₂/2.

3.4. Existence of an eigenvector

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

Proposition 3.5. — For any V ∈ C_b(H) and t > 0, the operator P^V_t^∗ has at least one eigenvector µ_t,V ∈ P(H) with a positive eigenvalueλ_t,V:

(3.13) P^V_t^∗µ_t,V =λ_t,Vµ_t,V.

Moreover, any such eigenvector satisfies

Z

H

(kukⁿ₁ +m_κ(u))µ_t,V(du)<∞, (3.14)

kP^V_t w_mk_XR

Z

X_R^c

w_m(u)µ_t,V(du)→0 asR → ∞ (3.15)

for any κ∈(0, γ₀) and n, m>1.

Proof.

Step 1: Estimate(3.14). — Let us fixt >0, and letµ∈ P(H) be an eigenvector of the operatorP^V_t^∗ corresponding to an eigenvalueλ >0. Let us show thatµ∈ Pm(H) with m = m_κ for any κ ∈ (0, γ₀). Indeed, for any measurable function f : H → R⁺∪ {+∞}, we have

(3.16) hf, µi=λ⁻¹hP^V_t f, µi6λ⁻¹e^tkVk∞hP_tf, µi.

Takingf =m_κ, any numberA >0, and setting⁽⁵⁾ C₁ =λ⁻¹e^tkVk∞, we obtain

Z

H

m_κ(u)µ(du)6C₁

Z

HEu{m_κ(u_t)}µ(du)

=C₁

Z

H

Eu

n

I^{kutk²6A}m_κ(u_t)^o+Eu

n

I^{kutk²>A}m_κ(u_t)^o

µ(du)

6C₁

Z

H

exp(κA) +A⁻¹Eu

nku_tk²m_κ(u_t)^o µ(du) 6C₁

Z

H

exp(κA) +C₂A⁻¹m_κ(u)µ(du), where we used inequality (6.22). ChoosingA > C₁C₂, we get (3.17)

Z

H

m_κ(u)µ(du)6C₁(1−C₁C₂A⁻¹)⁻¹exp(κA)<∞,

so⁽⁶⁾ µ∈ P_m(H). Takingf(u) =kukⁿ₁ in (3.16) and using (6.24) and (3.17), we obtain

Z

H

kukⁿ₁µ(du)6C₁

Z

HEu{ku_tkⁿ₁}µ(du)6C₃

Z

H

(1 +kuk⁸ⁿ)µ(du)<∞ for any n>1. This proves (3.14).

Step 2: Limit (3.15). — From (6.23) it follows that kP^V_t w_mk_XR 6e^tkVk∞ sup

u∈XR

Euw_m(u_t) 6C₄ sup

u∈B_H(R)

wm(u) = C₄(1 +R^2m).

(3.18)

Using the Cauchy–Schwarz inequality, (3.14), and the Chebyshev inequality, we see

that Z

X_R^c

w_m(u)µ(du)6hw²_m, µi^1/2µX_R^c1/2 6C₅R⁻ⁿ. Combining this with (3.18) and choosingn >2m, we obtain (3.15).

(5)We do not indicate the dependence of different constants on V, t, m, n,andκ.

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

Step 3: Construction of an eigenvector. — Let us take any A >0 and m>1 and define the convex set

D_A,m :=ⁿν ∈ P(H) :hw_m, νi6A^o.

By the Fatou lemma, D_A,m is closed in P(H). Consider the continuous mapping G:=G(t, V) :D_A,m → P(H), ν 7→ P^V_t^∗ν

P^V_t ^∗ν(H).

Let us show that G(D_A,m) ⊂ D_A,m for an appropriate choice of A and m, and thatG(D_A,m) is compact inP(H). In view of the Leray–Schauder theorem, this will imply the existence of an eigenvector µ ∈ DA,m satisfying (3.13) with eigenvalue λ=P^V_t^∗µ(H)>0. From (6.23) we derive that

Dw_m, G(ν)^E6exp{tOsc(V)}hw_m,P^∗_tνi

6exp{t(Osc(V)−mα₁)}hw_m, νi+C₆,

where Osc(V) := sup_u∈HV(u)−inf_u∈HV(u) is the oscillation of V. Choosing A and m so large that exp{t(Osc(V) −mα₁)} 6 1/2 and A > 2C₆, we get that G(D_A,m) ⊂ D_A,m. In view of the Prokhorov compactness criterion (see [Dud02, Theorem 11.5.4]), to prove that G(D_A,m) is relatively compact, it suffices to check

that _Z

H

kuk²₁P^V_t^∗ν(du)6C₇ for any ν ∈D_A,m. Using (6.24) and the fact thatV is bounded, we get

Z

H

kuk²₁P^V_t^∗ν(du)6exp(tkVk∞)

Z

H

kuk²₁(P^∗_tν)(du) 6C₈

Z

H

kuk⁸ν(du)

6C₉

Z

H

w_m(u)ν(du)6C₉A =:C₇.

Thus there is an eigenvector µ∈D_A,m.

4. Uniform Feller property

In this section, we establish the following result.

Theorem 4.1. — For any V ∈ V, the family {P_t^V} satisfies the uniform Feller property with respect to the sequence{X_R}, i.e., there is an integerR₀ >1such that the family {kP^V_t 1k⁻¹_R P^V_t ψ, t>0} is uniformly equicontinuous onX_R for any ψ ∈ V and R >R₀.

See the papers [JNPS15, JNPS18, MN18b] for similar results in the case of a discrete-time random dynamical system and [MN18a] for the case of the stochastic damped nonlinear wave equation. The main difficulty in the proof of Theorem 4.1 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.

4.1. Construction of coupling processes

The coupling processes are constructed following the arguments of [MN18a]. Let us take any z, z⁰ ∈ H and denote by u_t and u⁰_t the solutions of (1.5) issued from z and z⁰. For any integer N > 1 and number λ > 0, let v be the solution of the following problem

(4.1) v˙ +B(v) +Lv+P_N[λ(v−u) +B(u)−B(v)] = h+η(t), v(0) =z⁰, where η is defined by (1.3). We denote by ν(z, z⁰) and ν⁰(z⁰) the laws of pro- cesses {v(t), t ∈ J} and {u⁰(t), t ∈ J}, respectively, where J = [0,1]. We shall use the following result.

Proposition 4.2. — There exists an integer N₁ > 1 such that if N > N₁ and λ >N²/2, then for any ε >0and z, z⁰ ∈H, we have

(4.2) kν(z, z⁰)−ν⁰(z⁰)k_var 6ε^a+ 2^hexpC_λ,Nε^a−2kz−z⁰k²e^{C(kzk2+kz0k2)}−1^i1/2, where k · k_var denotes the total variation distance on P(C(J;H)) and a < 2, C, and C_λ,N are positive constants not depending on ε, z, z⁰.

See Section 6.2 for the proof. By Proposition 1.2.28 in [KS12], there is a probability space ( ˆΩ,F,ˆ Pˆ) and measurable functions Z,Z⁰ : H × H × Ωˆ → C(J;H) such that (Z(z, z⁰),Z⁰(z, z⁰)) is a maximal coupling for (ν(z, z⁰), ν(z⁰)) for any z, z⁰ ∈H.

We denote by ˜v and ˜u⁰_t the restrictions of Z and Z⁰ to time t ∈ J. Then ˜v_t is a solution of

˙˜

v+B(˜v) +L˜v+P_N[λ˜v−B(˜v)] =h+ψ(t), v(0) =˜ z⁰, where the process {^R₀^tψ(s)ds, t∈J} has the same law as

W(t)−

Z t 0

P_N[B(u_s)−λu_s]ds, t∈J

. Let ˜u_t be a solution of

˙˜

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

Then {˜u_t, t ∈ J} has the same law as {u_t, t ∈ J}. Now the coupling operators R and R⁰ are defined by

R_t(z, z⁰, ω) = ˜u_t, R⁰_t(z, z⁰, ω) = ˜u⁰_t, z, z⁰ ∈H, ω∈Ω, tˆ ∈J.

By Proposition 4.2, for any ε >0,N >N₁, and λ>N²/2, we have (4.3) Pˆ{∃t ∈J s.t. ˜vt6= ˜u⁰_t}

6ε^a+ 2^hexpCλ,Nε^a−2kz−z⁰k²e^{C(kzk2+kz0k2)}−1^i1/2. 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 any ω = (ω¹, ω², . . .) ∈Ω and z, z⁰ ∈H, we set ˜u0 =z, ˜u⁰₀ =z⁰, and

˜

u_t(ω) =R_s(˜u_k(ω),u˜⁰_k(ω), ω^k), u˜⁰_t(ω) = R⁰_s(˜u_k(ω),u˜⁰_k(ω), ω^k),

˜

v_t(ω) =Z_s(˜u_k(ω),u˜⁰_k(ω), ω^k),