Exponential mixing for the white - forced damped nonlinear wave equation

The paper is devoted to studying the stochastic nonlinear wave (NLW) equation in a bounded domain D $\subset$ R3. We show that the Markov process associated with the flow of solution has a unique stationary measure $\mu$, and the law of any solution converges to $\mu$ with exponential rate in the dual-Lipschitz norm


Introduction
We consider the stochastic NLW equation Here we only mention that they hold for functions f (u) = sin u and f (u) = |u| ρ u − λu, where λ and ρ ∈ (0, 2) are some constants. These functions correspond to the damped sine-Gordon and Klein-Gordon equations, respectively. The force η(t) is a white noise of the form Here {β j (t)} is a sequence of independent standard Brownian motions, {e j } is an orthonormal basis in L 2 (D) composed of the eigenfunctions of the Dirichlet Laplacian, and {b j } is a sequence of positive numbers that goes to zero sufficiently fast (see (2.4)). The initial point [u 0 , u 1 ] belongs to the phase space H = H 1 0 (D) × L 2 (D). Finally, h(x) is a function in H 1 0 (D). The following theorem is the main result of this paper. for any 1-Lipschitz function ψ : H → R, and any initial point y ∈ H.
Thus, the limit of the average of ψ(y(t)) is a quantity that does not depend on the initial point. Before outlining the main ideas of the proof of this result, let us discuss some of the earlier works concerning the ergodicity of the stochastic nonlinear PDE's and the main difficulties that occur in our case. In the context of stochastic PDE's, the initial value problem and existence of a stationary measure was studied by Vishik-Fursikov-Komech [28] for the stochastic Navier-Stokes system and later developed for many other problems (see the references in [7]). The uniqueness of stationary measure and its ergodicity are much more delicate questions. First results in this direction were obtained in the papers [15,21,13,4] devoted to the Navier-Stokes system and other PDE's arising in mathematical physics (see also [24,16] and Part III in [8] for some 1D parabolic equations). They were later extended to equations with multiplicative and very degenerate noises [25,17]. We refer the reader to the recent book [22] and the review paper [9] for a detailed account of the main results obtained so far.
We now discuss in more details the case of dispersive equations, for which fewer results are known. One of the first results on the ergodicity of dispersive PDE's was stablished in the paper of E, Khanin, Mazel and Sinai [14], where the authors prove the existence and uniqueness of stationary measure for the one dimensional inviscid Burgers equation perturbed by a space-periodic white noise. The qualitative study of stationary solutions is also carried out, and the analysis relies on the Lax-Oleinik variational principle. The ergodicity of a white-forced NLW equation was studied by Barbu and Da Prato [3], where the authors prove the existence of stationary distribution for a nonlinearity which is a non-decreasing function satisfying the growth restriction |f ′′ (u)| ≤ C(|u| + 1), and some standard dissipativity conditions. Uniqueness is established under the additional hypotheses, that f satisfies (2.1) with ρ < 2, and sup{|f ′ (u)| · |u| −ρ , u ∈ R} is sufficiently small. In the paper by Debussche and Odasso [10], the authors establish the convergence to the equilibrium with polynomial speed at any order (polynomial mixing) for weakly damped nonlinear Schrödinger equation. The proof of this result relies on the coupling argument. The main difficulty in establishing the exponential rate of convergence is due to the complicated Lyapunov structure and the fact that the Foaş-Prodi estimates hold in average and not path-wise. In [12], Dirr and Souganidis study the Hamilton-Jacobi equations perturbed by additive noise. They show, in particular, that under suitable assumptions on the Hamiltonian, the stochastic equation has a unique up to constants space-periodic global attracting solution, provided the unperturbed equation possesses such solution. In the recent paper by De-bussche and Vovelle [11] the existence and uniqueness of stationary measure is studied for scalar periodic first-order conservation laws with additive noise in any space dimension. It generalizes to higher dimensions the results established in [14] (see also [19]). In another recent paper [2] by Bakhtin, Cator and Khanin, the authors study the ergodicity of the Burgers equation perturbed by a space-time stationary random force. It is proved, in particular, that the equation possesses space-time stationary global solutions, and that they attract all other solutions. The proof uses the Aubry-Mather theory for action-minimizing trajectories, and weak KAM theory for the Hamilton-Jacobi equations.
In the present paper we extend the results established in [3], proving that the hypotheses f ′ ≥ 0 and sup{|f ′ (u)| · |u| −ρ , u ∈ R} is small are not needed, and that the convergence to the equilibrium has exponential rate. We also show that the conclusion of the Main Theorem remains true for a force that is nondegenerate only in the low Fourier modes (see Theorem 5.3). The proof mainly relies on the coupling argument.
Of course, one of the main difficulties when dealing with dispersive PDE's comes from the lack of the regularizing property, and with it, of some wellknown compactness arguments. As a consequence, this changes the approach when showing the stability of solutions. In particular, this is the case, when establishing the Foiaş-Prodi estimate for NLW (Proposition 4.1). Moreover, this estimate (which shows that the large time behavior of solutions is determined by finitely many modes and enables one to use the Girsanov theorem) differs from the classical one, since the growth of the intermediate process should be controlled (see inequality (4.4)). Due to the last fact, the coupling constructed through the projections of solutions (cf. [27,25]) does not ensure exponential rate of convergence. We therefore introduce a new type of coupling constructed via the intermediate process (see (2.9)-(2.14)). The same difficulty occurs when showing the recurrence of solutions, i.e. that the trajectory of the solution enters arbitrarily small ball with positive probability in a finite time (Proposition 4.4). The standard argument to show this property is the use of the portmanteau theorem. However, due to the lack of the smoothing effect, the portmanteau technique is not applicable, and another approach is proposed.
Without going into details, we give an informal description of our approach. The proof of the existence of stationary measure is rather standard and relies on the Bogolyubov-Krylov argument, which ensures the existence, provided the process y(t) = [u(t),u(t)] has a uniformly bounded moment in some H-compact space. To obtain such a bound, we follow a well-known argument coming from the theory of attractors (e.g., see [1,18]). Namely, we split the function u to the sum u = v + z, where, roughly speaking, v takes the Brownian of equation, and z-nonlinearity. We then show that the corresponding flows have uniformly bounded moments in H s = H 1+s (D) × H s (D) for s > 0 sufficiently small (Proposition 3.4). The bound for |[v(t),v(t)]| H s follows from the Itô formula, while that of |[z(t),ż(t)]| H s is based on the argument similar to the one used in [29]. The proof of exponential mixing relies on Theorem 3.1.7 in [22], which gives a general criterion that ensures the convergence to the equilibrium with exponential rate. Construction of a coupling that satisfies the hypotheses of the mentioned theorem is based on four key ingredients: the Foiaş-Prodi estimate for NLW, the Girsanov theorem, the recurrence property of solutions, and the stopping time technique.
Finally, we make some comments on the hypotheses imposed on the nonlinear term f and the coefficients b j entering the definition of the force η. Inequalities (2.2)-(2.3) are standard in the study of NLW equation, they ensure that the Cauchy problem is well-posed (e.g., see [6] and [23] for deterministic cases). The hypothesis ρ < 2 is needed to prove the stability of solutions. The fact that the coefficients b j are not zero ensures that η is non-degenerate in all Fourier modes, which is used to establish the recurrence of solutions and exponential squeezing. As was mentioned above, we show that this condition could be relaxed.
The paper is organized as follows. In Section 2 we announce the main result and outline the scheme of its proof. Next, the large time behavior and stability of solutions are studied in Sections 3 and 4, respectively. Finally, the complete proof of the main result is presented in Section 5.
Acknowledgments. I am grateful to my supervisor Armen Shirikyan, for attracting my attention to this problem, and for many fruitful discussions. This research was carried out within the MME-DII Center of Excellence (ANR 11 LABX 0023 01) and partially supported by the ANR grant STOSYMAP (ANR 2011 BS01 015 01).

Notation
For an open set D of a Euclidean space and separable Banach spaces X and Y , we introduce the following function spaces: L p = L p (D) is the Lebesgue space of measurable functions whose p th power is integrable. In the case p = 2 the corresponding norm is denoted by · . H s = H s (D) is the Sobolev space of order s with the usual norm · s . H s 0 = H s 0 (D) is the closure in H s of infinitely smooth functions with compact support. H 1,p = H 1,p (D) is the Sobolev space of order 1 with exponent p, that is, the space of L p functions whose first order derivatives remain in L p . L(X, Y ) stands for the space of linear continuous operators from X to Y endowed with the natural norm. C b (X) is the space of continuous bounded functions ψ : X → R endowed with the norm of uniform convergence: B X (R) stands for the ball in X of radius R and centered at the origin.
B(X) is the Borel σ-algebra of subsets of X. P(X) denotes the space of probability Borel measures on X. Two metrics are defined on the space P(X): the metric of total variation and the dual Lipschitz metric where (ψ, µ) denotes the integral of ψ over X with respect to µ.

Exponential mixing
We start this section by a short discussion of the well-posedness of the Cauchy problem for equation (1.1). We then state the main result and outline the scheme of its proof.

Existence and uniqueness of solutions
Before giving the definition of a solution of equation (1.1), let us make the precise hypotheses on the nonlinearity and the coefficients entering the definition of η(t).
We suppose that the function f satisfies the growth restriction where C and ρ < 2 are positive constants, and the dissipativity conditions where F is the primitive of f , ν ≤ (λ 1 ∧ γ)/8 is a positive constant, and λ j stands for the eigenvalue corresponding to e j . The coefficients b j are supposed to be positive numbers satisfying Let us introduce the functions Definition 2.1. Let y 0 = [u 0 , u 1 ] be a H-valued random variable defined on a complete probability space (Ω, F , P) that is independent ofζ(t). A random process y(t) = [u(t),u(t)] defined on (Ω, F , P) is called a solution (or a flow ) of equation (1.1) if the following two conditions hold: • Almost every trajectory of y(t) belongs to the space C(R + ; H), and the process y(t) is adapted to the filtration F t generated by y 0 andζ(t).
Let us endow the space H with the norm where α > 0 is a small parameter. Introduce the energy functional and let E u (t) = E(y(t)). We have the following theorem.
Theorem 2.2. Under the above hypotheses, let y 0 be an H−valued random variable that is independent ofζ and satisfies EE(y 0 ) < ∞. Then equation (1.1) possesses a solution in the sense of Definition 2.1. Moreover, it is unique, in the sense that ifỹ(t) is another solution, then with P-probability 1 we have y(t) =ỹ(t) for all t ≥ 0. In addition, we have the a priori estimate We refer the reader to the book [7] for proofs of similar results. We confine ourselves to the formal derivation of inequality (2.7) in the next section.

Main result and scheme of its proof
Let us denote by S t (y, ·) the flow of equation (1.1) issued from the initial point y ∈ H. A standard argument shows that S t (y, ·) defines a Markov process in H (e.g., see [7,22]). We shall denote by (y(t), P y ) the corresponding Markov family. In this case, the Markov operators have the form where P t (y, Γ) = P y (S t (y, ·) ∈ Γ) is the transition function. The following theorem on exponential mixing is the main result of this paper. Moreover, there exist positive constants C and κ such that for any λ ∈ P(H) we have Scheme of the proof. We shall construct an extension for the family (y(t), P y ) that satisfies the hypotheses of Theorem 3.1.7 in [22], providing a general criterion for exponential mixing. To this end, let us fix an initial point y = (y, y ′ ) in H = H × H, and let ξ u = [u, ∂ t u] and ξ u ′ = [u ′ , ∂ t u ′ ] be the flows of equation (1.1) that are issued from y and y ′ , respectively. Consider an intermediate process v, which is the solution of Let us denote by λ(y, y ′ ) and λ ′ (y, y ′ ) the laws of the processes Thus, λ and λ ′ are probability measures on C(0, T ; H). Let (V(y, y ′ ), V ′ (y, y ′ )) be a maximal coupling for (λ(y, y ′ ), λ ′ (y, y ′ )). By Proposition 1.2.28 in [22], such a pair exists and can be chosen to be a measurable function of its arguments. For any s ∈ [0, T ], we shall denote by V s and V ′ s the restrictions of V and V ′ to the time s. Denote by [ṽ, ∂ tṽ ] and [ũ ′ , ∂ tũ ′ ] the corresponding flows. Then we have where ψ satisfies Introduce an auxiliary processũ, which is the solution of Let us note that u satisfies the same equation, where ψ should be replaced by In view of (2.11), we have (see the appendix for the proof) on the half-line t ≥ 0 (we do not recall here the procedure of construction, see the paper [27] for the details). With a slight abuse of notation, we shall keep writing [ũ, ∂ tũ ] and [ũ ′ , ∂ tũ ′ ] for the extensions of these two processes, and write ξṽ(t) = V s (S kT (y)) for t = s + kT, 0 ≤ s < T . This will not lead to a confusion. For any continuous process y(t) with range in H, we introduce the functional 15) and the stopping time where L, M and r are some positive constants to be chosen later. In the case when y is a process of the form y = [z,ż], we shall write, F z and τ z instead of F [z,ż] and τ [z,ż] , respectively. Introduce the stopping times: Suppose that we are able to prove the following.
where τ d stands for the first hitting time of the set B H (d).
(Exponential squeezing): For any y ∈ B H (d), we have In view of Theorem 3.1.7 in [22], this will imply Theorem 2.3. We establish Theorem 2.4 in Section 5. The proof of recurrence relies on the Lyapunov function technique, while the proof of exponential squeezing is based on the Foiaş-Prodi type estimate for equation (1.1), the Girsanov theorem and the stopping time argument.   (1.1) issued from a non-random point y 0 ∈ H, the following statements hold. Strong law of large numbers. For any ε > 0 there is an almost surely finite random constant l ≥ 1 such that

Law of large numbers and central limit theorem
(2.23) Central limit theorem. If (ψ, µ) = 0, there is a constant a ≥ 0 depending only on ψ, such that for any ε > 0, we have where we set

Large time estimates of solutions
The goal of this section is to analyze the dynamics of solutions and to obtain some a priori estimates for them.

Proof of inequality (2.7)
Let us apply the Itô formula to the function G(y) = |y| 2 H . Recall that for the process of the form (2.5), the Itô formula gives where we set Here (∂ y G)(y; v) and (∂ 2 y G)(y; v, v) stand for the values of the first-and secondorder derivatives of G on the vector v. Since for G(y) = |y| 2 H we have ∂ y G(y;ȳ) = 2(y,ȳ) H , ∂ 2 y G(y;ȳ,ȳ) = 2|ȳ| 2 H , relation (3.1) takes the form Let us note that By the Young and Poincaré inequalities, we have Note also that, thanks to inequality (2.3), we have Now, by substituting (3.3) into (3.2), using inequalities (3.4)-(3.8), and noting that we obtain that for α > 0 sufficiently small where K > 0 depends only on γ, B and h , and M (t) is the stochastic integral Taking the mean value in inequality (3.9) and using the Gronwall comparison principle, we arrive at (2.7).

Exponential moment of the flow
In the following proposition we establish the uniform boundedness of exponential moment of |ξ u (t)| H .
It remains to use the inequality and the Gronwall lemma, to conclude.

Exponential supermartingale-type inequality
The following result provides an estimate for the rate of growth of solutions.
+r ≤ e −βr for any r > 0, (3.12) where K is the constant from inequality (3.9), and β = α/8 · (sup b 2 j ) −1 . Proof. Let us first note that It follows that the stochastic integral M (t) defined in (3.10) is a martingale, and its quadratic variation M (t) equals Combining this with inequality inequality (3.9), we obtain We conclude that where we used the exponential supermartingale inequality.
We recall that for a process of the form y(t) = [u(t),u(t)], F u ≡ F y stands for the functional defined by (2.15), and τ u ≡ τ y stands for the stopping time defined by (2.16).  This result follows from Proposition 3.2 and the fact that, due to inequality

Existence of stationary measure
In this subsection we show that the process y(t) = [u(t),u(t)] has a bounded second moment in the more regular space H s = H s+1 (D) × H s (D), with s = s(ρ) > 0 sufficiently small. By the Bogolyubov-Krylov argument, this immediately implies the existence of stationary distribution for the corresponding Markov process.
The standard argument shows that for any s ∈ [0, 1], we have so that it remains to bound the average of |ξ z (t)| 2 H s . In view of (1.1) and (3.13), z(t) is the solution of We now follow the argument used in [29]. Let us differentiate (3.15) in time, and set θ = ∂ t z. Then θ solves Let us fix s ∈ (0, 1 − ρ/2), multiply this equation by (−∆) s−1 (θ + αθ) and integrate over D. We obtain where we setẼ

By the Hölder and Sobolev inequalities
where we used the embedding H 1−s ֒→ L 6/(3−ρ) . Substituting this estimate in (3.17) and taking the mean value we obtain Applying the Gronwall lemma and using Proposition 3.1, we see that where the constant C 5 depends only on α and |y(0)| H . Moreover, by (3.16) we haveẼ In view of (3.15) . Taking the mean value in this inequality and using (3.18), we obtain This completes the proof of Proposition 3.4.

Stability of solutions
In this section we establish the stability and the recurrence property of solutions of equation (1.1).

The Foiaş-Prodi estimate
Here we establish an estimate which will allow us to use the Girsanov theorem. Let us consider the following two equations: where g(t) is a function in C(R + ; H 1 0 (D)), and P N stands for the orthogonal projection from L 2 (D) to its N -dimensional subspace spanned by the functions e 1 , e 2 , . . . , e N . Proposition 4.1. Suppose that for some non-negative constants K, l, s and T the inequality holds for z = u and z = v, where u and v are solutions of (4.1) and (4.2), respectively. Then, for any ε > 0 there is an integer N * ≥ 1 depending only on ε and K such that for all N ≥ N * we have Proof. Let us set w = v − u. Then w(t) solves and we need to show that the flow y(t) = ξ w (t) satisfies The function y(t) satisfies We first note that and that where p ∈ (6/5, 2) will be chosen later. Further, For J 1 we have where we used the Hölder and Sobolev inequalities and chose p = 6(3 + ρ) −1 . And finally, for J 2 we have where we once again used the Hölder inequality. Combining inequalities (4.8)-(4.12) together, we obtain Substituting this inequality in (4.7), we see that (4.14) By the Sobolev embedding theorem, the space H 1,p (D) is compactly embedded in L 2 (D) for p > 6/5. This implies that the sequence |I − P N | L(H 1,p →L 2 ) goes to zero as N goes to infinity. Combining this fact with the Gronwall lemma applied to (4.14) and using (4.3), we arrive at (4.6).

Controlling the growth of intermediate process
The goal of this subsection is to show that inequality (4.3) (and therefore (4.4)) holds with high probability, for g(t) = ζ(t). Proposition 4.2. Let u and v be solutions of (4.1) and (4.2) where g(t) = ζ(t), that are issued from initial points y, y ′ ∈ B 1 , respectively. Then

15)
where β is the constant from Proposition 3.2.
Proof. To prove this result, we follow the arguments presented in Section 3.3 of [22] and Section 4 of [20]. First, note that since inequality (4.15) concerns only the law of v and not the solution itself, we are free to choose the underlying probability space (Ω, F , P). We assume that it coincides with the canonical space of the Wiener process {ζ(t)} t≥0 . More precisely, Ω is the space of continuous functions ω : R + → H endowed with the metric of uniform convergence on bounded intervals, P is the law ofζ and F is the completion of the Borel σalgebra with respect to P. Let us define vectorsê j = [0, e j ] and their vector span which is an N -dimensional subspace of H. The space Ω = C(R + , H) can be represented in the form H N ) and Ω ⊥ N = C(R + , H ⊥ N ). We shall write ω = (ω (1) , ω (2) ) for ω = ω (1)+ ω (2) .
Let u ′ be a solution of equation (4.1) that has the same initial data as v. Introduce the stopping timeτ and a transformation Φ : Ω → Ω given by (4.17) where P N is the orthogonal projection from H to H N .

Lemma 4.3.
For any initial points y and y ′ in B 1 , we have where Φ * P stands for the image of P under Φ.
Proof of lemma 4.3.
Step 1. Let us note that by the definition ofτ we have for all t ≤τ . We claim that there is an integer N = N (α, L, M ) such that for all t ≤τ we have 20) where θ = |E u (0)| ∨ |E u ′ (0)| + r. Indeed, in view of inequality (2.2), for any y = [y 1 , y 2 ] in H, we have Combining this inequality with (4.19), we see that for all t ≤τ for z = u and z = v. Using this inequality and applying Proposition 4.1 with ε = α/2 we arrive at (4.20).
Step 2. Let us note that the transformation Φ can be represented in the form where Ψ : Ω → Ω N is given by It is straightforward to see that (2) ), (4.22) where P N and P ⊥ N are the images of P under the natural projections P N : Ω → Ω N and Q N : Ω → Ω ⊥ N , respectively. Define the processes It follows that P N = Dz and Ψ * (P, ω (2) ) = Dz. By Theorem A.10.1 in [22], we have provided the Novikov condition holds. In view of inequalities (4.19) and (4.20) we have where So not only the Novikov condition holds, but also there is a positive constant C M,r = C(α, L, M, r) such that the term on the right-hand side of inequality (4.23) does not exceed C M,r |y − y ′ | H . Combining this with inequality (4.22), we arrive at (4.18). Now we are ready to establish (4.15). Introduce auxiliary H-continuous processes y u , y u ′ and y v defined as follows: for t ≤τ they coincide with processes ξ u , ξ u ′ and ξ v , respectively, while for t ≥τ they solve where λ > 0 is a large parameter. By construction, with probability 1, we have Let us note that Moreover, in view of (4.24) where we used the fact that for t ≥τ the norms of auxiliary processes decay exponentially. Combining these two inequalities we obtain

Hitting a non-degenerate ball
Here we show that the trajectory of the process y(t) = [u(t),u(t)] issued from arbitrarily large ball hits any non-degenerate ball centered at the origin, with positive probability, at a finite non-random time. We denote by B d the ball of radius d in H, centered at the origin. where P t (y, Γ) = P y (S t (y, ·) ∈ Γ) is the transition function of the Markov process corresponding to (1.1).
Proof. Let us first split u to the sumũ +ū, whereū is the solution of In view of (4.29),ũ solves Now note that this equation is equivalent tõ and thereforeỹ(T ) continuously depends onζ (in the sense that the small perturbation ofζ in C(0, T ; H) will result in a small perturbation ofỹ(T ) in H). Let us consider equation (4.33) with the right-hand sidẽ which is a non-random force (the notationζ y is justified by the fact that it is uniquely determined by the initial point y). Then the functionũ ≡ 0 solves that equation. It follows that there exists ε = ε(d) > 0 such that Combining this with inequality (4.31) we obtain We need the following lemma. It is established in the appendix.
Let us suppose that we have (4.32), and let y j (0) = [u j (0),u j (0)] be a minimizing sequence. This sequence is bounded in H 1 × L 2 , so it has a converging subsequence in H 1−s × H −s (s is the constant from the previous lemma). Moreover, a standard argument coming from theory of m-dissipative operators shows that the resolving operator of (4.29) generates a continuous semigroup in H 1−s × H −s (e.g., see [5]). It follows that for all t ≥ 0 the corresponding sequence of solutionsȳ j (t) issued from y j (0) converges in that space. In particularū j (t) converges in H 1−s . Denoting byû(t) its limit and using Lemma 4.5 together with inequality Inequality (4.38) implies that (4.41) Let us fix j 0 ≥ 1 so large that for all j ≥ j 0 |ζ j −ζ| C(0,T ;L 2 ) ≤ ε/2.

Proof of Theorem 2.4
In this section we establish Theorem 2.4. As it was already mentioned, this will imply Theorem 2.3. We then show that the non-degeneracy condition imposed on the force can be relaxed to allow forces that are non-degenerate only in the low Fourier modes (see Theorem 5.3).

Recurrence: verification of (2.17)-(2.18)
In view of Proposition 3.1, it is sufficient to establish inequality (2.18). To this end, we shall use the existence of a Lyapunov function, combined with an auxiliary result established in [27]. Let S t (y, ω) be a Markov process in a separable Banach space H and let R t (y, ω) be its extension on an interval [0, T ]. Consider a continuous functional G(y) ≥ 1 on H such that lim |y|H→∞ G(y) = ∞.
Suppose that there are positive constants d, R, t * , C * and a < 1, such that We now show that for any d > 0 we can find an integer k ≥ 1 and T * ≥ 1 sufficiently large, such that we have (5.3) for any T ∈ {kT * , (k + 1)T * , . . .}. In what follows, we shall drop the subscript and write |y| instead of |y| H . So let us fix any d > 0, and consider the events where F y (t) is defined in (2.15), and L is the constant from Corollary 3.3.
Let us note that ifτ is finite, then we have |Sτ −T * (y, ·)| ≤ R and |Sτ (y, ·)| > d/2, where inequalities hold for any y in B R . Moreover where we used that forτ > kT * , we have |S kT * (y, ·)| H ≤ d/2. In view of (5.7) Sinceσ is a.s. finite, we can use the strong Markov property, and obtain where v = Sσ −T * (y, ·), and F t is the filtration generated by S t . In view of (5.5), the last term in this inequality does not exceed 1 − c d . Combining this with inequalities (5.8) and (5.9), we arrive at (5.6).
Step 2. It follows from the previous step that for any T ∈ {T * , 2T * , . . .} where we used that R t is an extension of S t . Further, by Corollary 3. By the symmetry, we can assume that where we set N = {V(y, y ′ ) = V ′ (y, y ′ )}. We claim that for any T ∈ {kT * , (k + 1)T * , . . .} with k ≥ 1 sufficiently large. To prove this, let us fix any ω in G d/2 E r N c , and note that it is sufficient to establish |R T (y, y ′ , ω) − R ′ T (y, y ′ , ω)| ≤ d/2, for any y, y ′ in B R . (5.15) Since ω ∈ N c , we have that V = V ′ , and therefore, in view of (2.10)-(2.14), R t (y, y ′ ) and R ′ t (y, y ′ ) are, respectively, the flows of equations and It follows that their difference w =ṽ −ũ solves Using the Foiaş-Prodi estimate established in Proposition 4.1 (see (4.5)-(4.6)) together with the fact that ω ∈ E r , we can find an integer N ≥ 1 depending only on L such that Since r is fixed, we can find k ≥ 1 sufficiently large, such that the right-hand side of this inequality is less than d 2 /4 for any T ∈ {kT * , (k + 1)T * , . . .}, so that we have (5.14).
Step 3. We now follow the argument presented in [27]. In view of (5.14) where we used the independence of V and V ′ conditioned on the event N . Combining this inequality with (5.11), we obtain We claim that the right-hand side of this inequality is no less than c 2 d /8. Indeed, if P y (G d/2 N c ) ≥ c 2 d /4, then the required result follows from inequality (5.12). If not, then by inequalities (5.10) and (5.13), we have . We have thus shown that for any y, y ′ in B R

Exponential squeezing: verification of (2.19)-(2.22)
Let u, u ′ , v,ũ,ũ ′ ,ṽ and ̺, τ, σ be the processes and stopping times constructed in Subsection 2.2. Consider the following events: Lemma 5.2. There exist positive constants d, r, L and M such that for any initial point y ∈ B H (d) and any T ≥ 1 sufficiently large Proof.
Step1. (Probability of Q ′ k ). Let L be the constant from Corollary 3.3. Then using second inequality of this corollary, we obtain for M ≥ 2β −1 , r ≥ 5β −1 + 4C. From now on, the constants L, M and r will be fixed.
Step2. (Probability of Q ′′ k ). Let us first note that by the Markov property we have (5.19) whereȳ(·) = y(kT, ·), and F t stands for the filtration corresponding to the process S t . Moreover, it follows from the definition of maximal coupling, that for any y in H, we have Combining this with inequality (5.19), we obtain Further, let us note that whereτ = τ u ∧ τ u ′ ∧ τ v , and the supremum is taken over all Γ ∈ B(C(0, T ; H)).
In view of (4.24) we have where Φ is the transformation constructed in Subsection 4.2, and we used the fact that forτ = ∞ we have y v ≡ ξ v and y u ′ ≡ ξ u ′ Further, in view (4.26) we have Combining inequalities (5.20)-(5.23), we get Let us note that for any ω ∈ {σ ≥ kT } we have Moreover, it follows from Proposition 4.1 (see the derivation of (4.20)) that for any ε > 0 there is N depending only on ε, α, L and M , such that for all kT ≤ t ≤ τ ∧ τṽ, on the set σ ≥ kT , we have where we set θ = ε · (|Eũ(kT )| ∨ |Eũ′ (kT )| + r).

Relaxed non-degeneracy condition
We finish this section with the following result that allows to relax the nondegeneracy condition imposed on the force.
Theorem 5.3. There exists N depending only on γ, f, h and B such that the conclusion of Theorem 2.3 remains true for any random force of the form (1.2), whose first N coefficients b j are not zero.
6 Appendix 6.1 Proof of (2.13) Let us consider the continuous map G from C(0, T ; H 1 0 (D)) to C(0, T ; H) defined by G(ϕ) =ỹ, whereỹ is the flow of equation