Sergei Kuksin

Huilin Zhang

EXPONENTIAL MIXING FOR DISSIPATIVE PDES WITH BOUNDED NON-DEGENERATE NOISE

Keywords: 2010 Mathematics Subject Classification. 35Q59, 35Q56, 35K59, 37A25, 37L55, 60F05, 60H15, 76D05 exponential mixing, 2d Navier-Stokes system, Ginzburg-Landau equation, dissipative PDEs, Markovian dynamic system, Haar colored noise

We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.

Introduction

In this paper we consider nonlinear PDEs, perturbed by random forces, which we write as [START_REF] Federer | Geometric Measure Theory[END_REF] ∂ t u t + νLu t + F (u t ) = η t , u t ∈ H.

Here H is a Hilbert space of functions of x, ν ∈ (0, 1], L is a self-adjoint operator with compact resolvent, F is a nonlinearity and η t = η ω t is a random process in H. We are concerned with the question when a solution u t of (1) is a mixing random process, i.e. when its distribution D(u t ) converges to a unique measure in H, independent from the initial data u 0 , while t → ∞. The problem of mixing in equations ( 1) is well motivated by modern physics, and it has received much attention during the last two decades, see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]. In the corresponding papers (discussed in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]) the authors prove the mixing for various classes of equations [START_REF] Federer | Geometric Measure Theory[END_REF], assuming that the random force η t has the structure which we will now discuss.

We suppose that

η t = ∞ i=1 b i η i t φ i , b 2 i < ∞,
where {φ i } i is an orthonormal basis of H and {η i t , i ≥ 1} are i.i.d. real processes, distributed as a certain etalon process η 0 t . Concerning the latter it was assumed that either this is a) a kick-process, η 0 t = ∞ k=1 δ(t -kT )ξ k , where T > 0 and {ξ k } are i.i.d. real random variables; or that b) η 0 t is a white noise; or c) η 0 t is a compound Poisson process (see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]). Concerning the coefficients b i it was usually assumed that either all of them are non-zero, or that b i = 0 for all i ≤ N ν , where N ν grows to infinity as ν → 0. In the paper [START_REF] Hairer | Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing[END_REF] the mixing was established for the case when [START_REF] Federer | Geometric Measure Theory[END_REF] is the 2D Navier-Stokes system on the two-dimensional torus, perturbed by a white in time random force (see below eq. ( 50)), and only a ν-independent finite system of these coefficients do not vanish. The proof of [START_REF] Hairer | Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing[END_REF] uses an infinite-dimensional version of the Malliavin calculus, and all attempts to generalize it to equations with kick-forces or compound Poisson processes have failed. Instead, in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF] equations of the form (1) were considered, where η 0 (t) is a random Haar series "of time-width one." 1 This means the following: The processes η 0 | [k-1,k) , k = 1, 2, . . . , are i.i.d., so it suffices to define the process η 0 | [0,1] . The latter is a random Haar series

η 0 t = ∞ j=0 c j 2 j -1 l=0 ξ jl h jl (t) ∀ 0 ≤ t ≤ 1, c j = 0 ∀ j,
where {h jl } is the Haar base in L 2 [0, 1] (see in [START_REF] Lamperti | Probability[END_REF] and see (47) below), and ξ jl are independent random variables, |ξ jl | ≤ 1, whose density functions ρ jl (x) are Lipschitz-continuous and do not vanish at x = 0. Let us denote E = L 2 ([0, 1], H) and consider the mapping

(2) S : H × E → H, S(u 0 , η) = u 1 ,
where u t is a solution of (1), equal u 0 at t = 0.

Assuming that (B0) j 2 j/2 |c j | < ∞, so the process η t is bounded uniformly in t and ω (see below in Section 2.1); (B1) the mapping S is well defined. Moreover, there exists a compactly embedded Banach subspace V ⊂ H such that S(H × E) ⊂ V , the mapping S : H × E → V is analytic and its derivatives are bounded on bounded sets; (B2) 0 ∈ H is an asymptotically stable equilibrium for eq. (1) with η = 0, and relation [START_REF] Ladyzhenskaja | Linear and quasilinear equations of parabolic type[END_REF] holds; (B3) b j = 0 for j ∈ J ⊂ N, where the set J, finite or infinite, is such that the linearised equation

∂ t v + νLv + dF (u t )v = j∈J b j ξ j t φ j , 0 ≤ t ≤ 1,
is approximately controllable by controls ξ j t , j ∈ J, provided that u t is a solution of (1) with arbitrary u 0 and with η = η ω , where ω does not belong to a certain null-set depending on u 0 , it was proved in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF] that eq. ( 1) is exponentially mixing. 2 The condition (B3) has been intensively studied, and for many important equations it is now established that (B3) holds if J is a finite set, satisfying certain explicit conditions, independent from ν. See in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF].

Our work continues the research in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF]. Namely, making the assumptions (B0 ) -(B3 ), where (B0 ) j c 2 j 2 j < ∞; (B1 ) the mapping S : H × E → V is C 2 -smooth and its derivatives up to second order are bounded on bounded sets; (B2 ) = (B2); (B3 ) for any u ∈ H and η ∈ E [START_REF] Kuksin | Diffeomorphisms of function spaces that correspond to quasilinear parabolic equations[END_REF] the operator d η S(u, η) : E → H has dense image, we prove in Theorem 1.5, Section 1, the following result:

there exists a unique Borel measure µ ν in H such that if u t is a solution of (1) with the initial data u ω 0 , satisfying u ω 0 ∈ {u ∈ H : u ≤ R} almost surely for some R > 0, then (4)

Du k -µ ν * L ≤ Cκ k , ∀ k ∈ N. Here • *
L is the Lipschitz-dual distance in the space of Borel measures in H (see [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]), and

C = C(R) > 0, κ = κ(R) ∈ (0, 1). Moreover, if u ω 0 is any random variable in H, then Du k weakly converges to µ ν as k → ∞.
A special case of systems (2) appears when E is a subspace of H and the mapping S has the form

S(u 0 , η) = S 1 (u 0 ) + η.
Such systems correspond to equations (1) with kick-forces η as in a) above (see Remark 2.7 in Sectuion 2.2). They are easier than general systems (2) and for them the mixing can be proved if the map S 1 is Lipschitz-continuous on bounded sets, see [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF].

In Section 2 we show that the result applies to the 2D Navier-Stokes system on torus as well as to equations (1) who are random perturbations of quasilinear parabolic systems which are -well posed for any bounded force η(t, x), sufficiently smooth is x; -satisfy the dissipativity assumption (B2). The proof uses some ideas from [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF][START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF] and is significantly shorter than that in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF] since the nondegeneracy assumption (B3 ) which we assume now is stronger than the assumption (B3); essentially it holds if the force η t is non-degenerate, i.e. all coefficients b i = 0, (unfortunately, we are not aware of a weaker sufficient condition which would guarantee (B3 )). As in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], the exponential convergence (4) follows from Doeblin's coupling, enhanced with the quadratic convergence in the form, close to that in the KAM-theory. 3 In particular, our work shows that the analyticity assumption in (B1) is not an intrinsic feature of the approach of [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], but is needed to work under the very weak nondegeneracy assumption (B3) and may be replaced by the C 2 -smoothness if the equation is non-degenerate in the sense (B3 ). Furthermore, our result is easier to apply then that in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF] since it is easier to check assumption (B3 ) than (B3) in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF].

Notation. As usual, by C, C 1 , . . . we denote various constants which change from line to line. By B F (R) we denote the closed R-ball in a Banach space F , centered at the origin; by D(ξ) -a law of a r.v. ξ; µ, f = f, µ stands for the integral of a measurable function f against a measure µ, and L(X, Y ) -for the space of bounded linear operators between Banach spaces X and Y . A complete separable metric space (i.e. a Polish space) X always is equipped with its Borel σ-algebra B X . By P(X) we denote the space of probability measures on (X, B X ). We provide it with the Lipschitz-dual distance [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF] µ

-ν * L = µ -ν * L(X) = sup | µ -ν, f |, where the supremum is taken over all Lipschitz functions f on X such that |f | L(X) := sup x∈X |f (x)| ∨ Lip(f ) ≤ 1, with the Kantorovich distance (6) µ -ν K = sup Lip(f )≤1 | µ -ν, f |,
and the distance of total variation • var . If X, Y are Polish spaces and f : X → Y is a Borelmeasurable mapping, then f * : P(X) → P(Y ) denotes the corresponding push-forward of measures.

A pair of random variables ξ, ζ defined on a probability space is a coupling for given measures

µ, ν ∈ P(X) if D(ξ) = µ, D(ζ) = ν.
Acknowledgements: S.K. thanks the Russian Science Foundation for support through the project 18-11-00032. H.Z. is supported by the National Postdoctoral Program for Innovative Talents, No: BX20180075 and China Postdoctoral Science Foundation.

Mixing for a class of Markov chains

1.1. Settings and assumptions. Let H, E be two separable Hilbert spaces and V be a Banach space, compactly and densely embedded in H. Let S : H × E → V be a continuous mapping. We consider the following random dynamical system (RDS) in the space H:

(7) u k = S(u k-1 , η k ), k ≥ 1,
where {η k } k is a sequence of i.i.d. random variables in E. We denote by the law of η k ,

= D(η k ) ∈ P(E) ∀ k,
and suppose that is supported by a compact set K ⊂ E,

K := supp E.
By (u k (v), k ≥ 0), we denote a trajectory of [START_REF] Lamperti | Probability[END_REF] such that u 0 (v) = v. The process {u k (v)} is a Markov chain in H, whose transition probability function after k steps is

P k (u, Γ) = P(u k (u) ∈ Γ).
It defines a semigroup of Markov operators in the space of functions

P k : C b (H) → C b (H), f (•) → E f (u k (•)) , k ≥ 0,
and a semigroup of operators in the space of measures

P * k : P(H) → P(H), µ → D(u k (v)), k ≥ 0,
where v is a r.v. in H, independent from the noise η, such that D(v) = µ (e.g. see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]).

We make the following assumptions concerning regularity of our system: (A1) (Regularity). The mapping S : H × E → V is twice continuously differentiable, and its derivatives up to second order are bounded on bounded sets.

Concerning the noise η we assume the following:

(H1) (Decomposability and non-degeneracy). There exists an orthonormal basis {e j , j ≥ 1} of E such that

η k = ∞ j=1 b j ξ jk e j , b j = 0 ∀ j.
Here ξ jk are independent random variables and b j are real numbers, satisfying

(8) |ξ jk | ≤ 1 a.s., D(ξ jk ) = ρ j (r)dr, R 2 η := ∞ j=1 b 2 j < ∞,
where ρ j : R → R are Lipschitz functions and ρ j (0) = 0 for all j. (H2) (Dissipativity). For any u ∈ H, η ∈ K we have

(9) S(u, η) H ≤ γ u H + β, S(u, 0) H ≤ γ u H with some 0 < γ < 1, β > 0.
(H3) (Non-degeneracy). For any u ∈ H and η ∈ K the image of the operator D η S(u, η) : E → V is dense in H. Note that by (8) supp = K belongs to the Hilbert brick [START_REF] Villani | Topics in Optimal Transportation[END_REF]

{η = η k e k : |η k | ≤ b k } =: K, so indeed K is a compact set, and K ⊂ B E (R η ).
It is easy to see that since ρ j (0) = 0 for all j, then according to relations [START_REF] Ladyzhenskaja | Linear and quasilinear equations of parabolic type[END_REF] and assumption (A1), for any a > γ and δ > 0, In what follows, by some confusion of notations, η stays for an element of K or for a random variable with the law , depending on the context. We will use a modified distance in the space H which depends on a parameter d 0 ∈ (0, 1]:

P ( S(u, η) H < a u H ) > p δ > 0, if u H ≥ δ. Remark 1. (11)
ξ 1 -ξ 2 d0 := ξ 1 -ξ 2 H ∧ d 0 . Clearly ξ 1 -ξ 2 d0 ≤ ξ 1 -ξ 2 H and (12) ξ 1 -ξ 2 H ≤ 2R * d 0 ξ 1 -ξ 2 d0 if ξ 1 , ξ 2 ∈ B H (R * ), with 2R * ≥ d 0 .
Theorem 1.3. Under the assumptions (A1), (H1), (H2) and (H3), for any R * > 0 and any u, u ∈ B H (R * ),

(13) P k (u, •) -P k (u , •) * L ≤ C u -u H κ k , where 0 < κ < 1 and C > 0 depend on R * . Corollary 1.4. If µ, ν ∈ P(B H (R * )), then (14) P * k µ -P * k ν * L ≤ C 1 µ -ν * L κ k ∀ k ≥ 1, where 0 < κ < 1 and C 1 > 0 depend on R * .
Proof. Let (u ω , u ω ) be a coupling for (µ, ν).

Then P * k (µ) = Du k (u), P * k (ν) = Du k (u ). Take any f ∈ C b (H) with |f | L(H) ≤ 1. Then | f, P * k (µ) -f, P * k (ν) | = E ω E ω (f (u k (u ω )) -f (u k (u ω ))) ≤ Cκ k E ω u ω -u ω H
, where the inequality follows from the theorem. This inequality remains true if in the r.h.s. we take infimum over all couplings (u, u ) for (µ, ν). By the Kantorovich-Rubinshtein theorem (see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF])

inf u,u E ω u ω -u ω H = µ -ν K , where • K is the Kantorovich distance (6). Since ν, µ supported on B H (R * ), then µ -ν K ≤ R * µ -ν * L
, and the result follows. Clearly without loss of generality we may assume that in Theorem 1.3

(15) R * ≥ β 1 -γ =: R 0 * ;
this relation is always assumed from now on. Then in view of (H2)

S(u, η) ∈ B H (R * ) if u ∈ B H (R * ), η ∈ K. By (A1) if u ∈ B H (R * ) and η ∈ K, then S(u, η) V ≤ C(K, R * )
, where C(K, R * ) is a constant depending only on K, R * . We see that the set

X = X K,R * = completion in H of B H (R * ) ∩ B V (C(K, R * )), is a compact subset of H such that S : B H (R * ) × K → X.
In particular, (16)

S : X × K → X.
That is, the RDS (7) defines a Markov chain in X and

S(u, η) ∈ X a.s. if u ∈ B H (R * ).
From here for u, u as in the theorem's assumption we have

u k (u), u k (u ) ∈ X for k ≥ 1, a.s.
The proof of Corollary 1.4 and this relation show that (14) holds if (13) is established only for u, u ∈ X. But (14) implies the validity of the assertion of Theorem 1.3 for all u, u ∈ B H (R * ). So proving the theorem we may assume that u, u ∈ X and regard (7) as a system in the compact set X. (17) Choosing in ( 14) k sufficiently big we find that the operator P * k defines a contraction of the complete metric space (P(X), • * L ). So it has a unique fixed point µ * ∈ P(X). Then P * k P * 1 µ * = P * 1 µ * , so P * 1 µ * = µ * by the uniqueness, i.e. µ * is a stationary measure for the Markov chain, defined by ( 7) on X. Due to (14) it is unique. Since X = X K,R * , then formally this measure depends on R * ,

µ * = µ * (R * ). But the measure µ * (R 0 * ) (see (15)) is stationary for the system on X K,R * for any R * ≥ R 0 * , so µ * (R 0 * ) = µ * (R *
) by the uniqueness. Denoting this measure by µ * we see that it is stationary for the system [START_REF] Lamperti | Probability[END_REF], considered on H, and derive from (14) (with a suitable R * ) that ( 18)

P k (u, •) -µ * * L → 0 as k → ∞, for every u ∈ H.
Assume that µ is another stationary measure for the system on H. Then for any bounded Lipschitz function f on H we have

f, µ = f, P * k µ = P k f, µ = H f, P k (u, •) µ (du) → f, µ *
by the Lebesgue theorem and (18). So µ = µ * . We finalize our analysis of eq. ( 7) in the following theorem:

Theorem 1.5. Under the assumptions (A1), (H1), (H2) and (H3) equation [START_REF] Lamperti | Probability[END_REF] defines in H a Markov chain which has a unique stationary measure µ * . This measure is supported by the ball

B H (R 0 * ), and if µ ∈ P(H) is supported by a ball B H (R * ), R * ≥ R 0 * , then (19) P * k µ -µ * * L ≤ Cκ k , ∀ k ≥ 1, where C > 0 and κ ∈ (0, 1) depend on R * . If ν is any measure in P(H), then (20) P * k ν µ * as k → ∞,
where signifies the weak convergence of mesures.

Proof. The first part of the theorem follows from what was said above if we note that the measure P * 1 µ is supported by the compact set X K,R * ∨R 0 * , so (19) is a consequence of (14) with µ := P * 1 µ and ν := µ * . Now consider any ν ∈ P(H). For R large, define a probability measure supported by

B H (R): ν R (•) := 1 ν(B H (R)) ν(• ∩ B H (R)).
In view of Ulam's theorem ν R converges to ν in the total variation distance. We have

P * k ν -µ * * L ≤ P * k ν -P * k ν R * L + P * k ν R -µ * * L = sup |f | L(H) ≤1 P k f, ν -ν R + C R κ k R ≤ 2 ν -ν R var + C R κ k R .
Choosing first R so big that the first term in the r.h.s. goes to zero and then k so large such that the second term goes to null we see that the l.h.s. converges to zero with k. This implies (20).

Note that since the theorem deals with initial data, supported by a ball B H (R * ), then Remark 1.2 applies and it suffices to check Assumption (H2) in the weaker form, specified in that remark.

1.3. Main lemma and proof of Theorem 1.3. In this section we state the main technical lemma and derive from it Theorem 1.3. Based on (17) we regard (7) as a system on the compact set X = X K,R * .

Lemma 1.6. Under the assumptions (A1), (H1), (H2), (H3), there exist constants C > 0, δ ∈ (0, 1] and a continuous mapping Φ :

X × H × K → E such that Φ(u, u ; η) = 0 if u = u
, and the mapping

Ψ(u, u ; η) := η + Φ(u, u ; η) satisfies -Ψ * (u, u ; •) var ≤ C u -u H , (21)
S(u, η) -S(u , Ψ(u, u ; η)) H ≤ 1 2 u -u H , ( 22
)
for all u, u ∈ D δ := {(u, u ) ∈ X × H : u -u H ≤ δ} and all η ∈ K.
Lemma 1.6 will be proved in the next section. Now, supposing that we have this result, we will first establish a coupling lemma and then derive from it Theorem 1.3.

Lemma 1.7. For δ as in Lemma 1.6 there exists C 1 = C 1 (δ), a probability space (Ω 0 , F 0 , P 0 ) and a measurable mapping

D δ × Ω 0 → K × K, (u, u , ω 0 ) → (η ω0 1 (u, u ), η 1 ω0 (u, u )) such that D(η 1 (u, u )) = D(η 1 (u, u )) =
, and u 1 := S(u, η 1 ), u 1 := S(u , η 1 ) satisfy

(23) P u 1 -u 1 H ≤ 1 2 d > 1 -C 1 d, d = u -u H ≤ δ. Proof.
Recall that η has the law . Denote η = Ψ(u, u , η) and = Dη . Then according to Lemma 1.6,

S(u, η) -S(u , η ) H ≤ 1 2 d, a.s. (24) -var ≤ C 1 d. ( 25
)
The law of η need not to be . To improve this, note that in view of (22) and the Dobrushin lemma (see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]), there exists a coupling (η, η ) for ( , ), defined on a probability space (Ω 1 , F 1 , P 1 ), satisfying (26)

P 1 (η = η ) = -var ≤ C 1 d.
Moreover, the mapping (ω 1 , u, u ) → (η, η ) may be chosen to be measurable, see [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF], Section 1.2.4.

For the pair of measures ( , ) we have obtained two couplings -(η, η ) and (η, η ).

According to the Gluing lemma (see [START_REF] Villani | Topics in Optimal Transportation[END_REF]), there exist random variables (ζ 1 , ζ 2 , ζ 3 ) defined on a probability space (Ω 0 , F 0 , P 0 ) such that

D(ζ 1 , ζ 2 )=D(η, η ), D(ζ 2 , ζ 3 )=D(η , η). The triplet (ζ 1 , ζ 2 , ζ 3
) may be chosen to be a measurable function of (ω 0 , u, u ), see [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], Appendix 3.

In particular, D(ζ 1 ) = D(ζ 3 ) = and by inequality ( 24)

S(u, ζ 1 ) -S(u , ζ 2 ) H ≤ 1 2 d, P 0 -a.s. Furthermore, in view of inequality (26), P 0 (ζ 2 = ζ 3 ) = P 1 (η = η) ≤ C 1 d, which implies that P 0 S(u, ζ 1 ) -S(u , ζ 3 ) H ≤ 1 2 d ≥ 1 -C 1 d. To complete the proof it remains to choose η 1 = ζ 1 and η 1 = ζ 3 .
To obtain the exponential mixing, claimed by Theorem 1.3, we will apply the method of Kantorovich functional (see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]). A Kantorovich functional for measures on X is a symmetric function K : P(X) × P(X) → R + , defined via its density f K . The latter is a symmetric measurable function f 12)), and K is defined in terms of f K by the relation

K on X × X, satisfying f K (ξ 1 , ξ 2 ) ≥ ξ 1 -ξ 2 d0 ∀ ξ 1 , ξ 2 ∈ X, (see (
K(µ 1 , µ 2 ) = inf{Ef K (ξ 1 , ξ 2 )},
where the infimum is taken over all couplings (ξ 1 , ξ 2 ) for (µ 1 , µ 2 ).

For R * as above and u, u

∈ B H (R * ) we denote d = u -u H and set (27) (u 1 , u 1 )(ω 0 ) = (S(u, η), S(u , η)), d > d 0 , (S(u, η 1 ), S(u , η 1 )), d ≤ d 0 ,
where η 1 (ω 0 ), η 1 (ω 0 ) are as in Lemma 1.7, and η is a r.v. on Ω 0 , distributed as , and independent from η 1 , η 1 . Clearly (u 1 , u 1 ) is a measurable function of (u, u , ω 0 ) and D(u 1 ) = P 1 (u, •) and D(u 1 ) = P 1 (u , •). Lemma 1.8. There exist d 0 > 0 and a function

(28) f : ( 1 2 d 0 , R * ] → (2d 0 , 3d 0 ] non-decreasing, such that for any ξ 1 , ξ 2 ∈ X, (29) f K (ξ 1 , ξ 2 ) = ξ 1 -ξ 2 H , if ξ 1 -ξ 2 H ≤ d 0 , f ( ξ 1 H ∨ ξ 2 H ), if ξ 1 -ξ 2 H > d 0 , is a Kantorovich density. It satisfies (30) E [f K (u 1 , u 1 )] ≤ κf K (u, u ) for any u, u ∈ B H (R * )
, where κ ∈ (0, 1) is a constant, depending on R * .

Everywhere below in the proof f K always stays for the Kantorovich density, defined in (29).

Proof. We may assume that d 0 ≤ δ, where δ is introduced in Lemma 1.6. The fact that f K is a Kantorovich density is obvious, as well as that it satisfies

(31) f K (ξ 1 , ξ 2 ) ≤ 3 ξ 1 -ξ 2 H .

It remains to verify (30). Let us denote

R 1 (ω 0 ) := S(u, η) H ∨ S(u , η ) H , R := u H ∨ u H . Recall that d = u -u H . Case 1: d > d 0 . Now 1 2 d 0 ≤ R ≤ R * .
Let as take a = 1+γ 2 . According to (11) and ( 9),

P [R 1 ≤ aR] > p d0 =: p > 0, R 1 ≤ γR + β, a.s. Therefore E[f K (u 1 , u 1 )] ≤ pf (aR) + (1 -p)f (γR + β).
To obtain inequality (30), we need to find f of the form (28), satisfying

(32) pf (aR) + (1 -p)f (γR + β) ≤ κ 1 f (R) for R ∈ ( 1 2 d 0 , R * ], with some κ 1 ∈ (0, 1). Let us first consider f (R) when R ∈ J := ( 1 2 d 0 , R 0 * ].
We define the segments

I 1 = (aR 0 * , R 0 * ], I 2 = (a 2 R 0 * , aR 0 * ], . . . , I n = (a n R 0 * , a n-1 R 0 * ], . . . Then ∪ N0 j=1 I j ⊃ J, where N 0 = ln 2R0+ln d -1 0 ln a -1 + 1.
We define f on J by the relation

f (x) = a j , x ∈ I j , 1 ≤ j ≤ N 0 , where (33) 2.5d 0 = a 1 > a 2 > • • • > a N0 > 0 is a sequence to be defined. If R ∈ I 1 , then (32) is valid if a 1 > pa 2 + (1 -p)a 1 , which holds true since a 1 > a 2 . If R ∈ I n , n ≥ 2, then (32) holds if (34) a n-1 > pa n + (1 -p)a 1 .
Let us take some p 1 ∈ (0, p) and define the sequence {a j } as follows:

a n := a 1 - a 1 -a 2 p n 1 .
If a 2 < a 1 is chosen sufficiently close to a 1 , then (33) holds. Relation (34) holds as well and (30) is proved for d > d 0 with some κ = κ 1 < 1. Choosing a 1 -a 2 > 0 sufficiently small we achieve that also

2d 0 ≤ f (x) ≤ 2.5d 0 on J. If R * = R 0 * , we are done. If R * > R 0 * , we extend f (x) to the interval (R 0 * , R * ] as an affine function l(x) such that l(R * 0 ) = 2.6d 0 and l(R * ) = 3d 0 . Then for R > R 0 * we have f (R) ≥ 2.6d 0 . From the other hand, f (γR + β) ≤ f (R) = l(R), while f (aR) is ≤ 2.5d 0 if aR ≤ R 0
* and is l(aR) if aR > R 0 * . In both cases (32) holds with a suitable κ 1 < 1.

We have constructed a function f K , satisfying (28), (31) and (32), if d > d 0 .

Case 2. d ≤ d 0 . Denote D = u 1 -u 1 H . Since d 0 ≤ δ, then according to Lemma 1.7,

P D ≤ 1 2 d ≥ 1 -C 1 d if d ≤ d 0 .
In view of (28),

E[f K (u 1 , u 1 )] ≤ C 1 d(3d 0 ) + (1 -C 1 d)( 1 2 d) ≤ 1 2 d + ( 5 2 C 1 d)d 0 . Choosing d 0 ≤ 1/(10C 1 )
we achieve that (30) holds with κ = 3/4. Now the lemma is proved by choosing [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]) and any measures µ, ν ∈ P(H) we have

d 0 ≤ 1/(10C 1 ) ∧ 2β/(1 -γ) with κ = κ 1 ∨ (3/4). Proof of Theorem (1.3). Note that for any f ∈ C b (H) with |f | L(H) ≤ 1 (see
H f (x)(µ -ν)(dx) = |E [f (ξ) -f (ζ)]| ≤ E ξ -ζ H , where (ξ, ζ) is a coupling for µ, ν. Therefore µ -ν * L ≤ inf (ξ,ζ) E ξ -ζ H ,
where the infimum is taken over all couplings (ξ, ζ) for (µ, ν). If µ, ν are supported by X, then the couplings are valued in B H (R * ) a.s., and due to (29) and (28) we have that

ξ ω -ζ ω H ≤ R * d 0 f K (ξ ω , η ω ) a.s.
For u, u ∈ X and k ≥ 1 let (u k , u k ) be any fixed coupling for (P k (u, •), P k (u , •). Then (35)

P k (u, •) -P k (u , •) * L ≤ inf ξ,ζ E ξ -ζ H ≤ R * d 0 inf ξ,ζ Ef K (ξ, ζ) ≤ R * d 0 Ef K (u k , u k ),
where, as above, (ξ, ζ) is a coupling for the two measures. Denote µ k = P k (u, •) and µ k = P k (u , •). We claim that for any k ≥ 1 there exists a coupling (u k , u k ) for the measures (µ k , µ k ), defined on a probability space (Ω k , F k , P k ), such that

(36) E k [f K (u k , u k )] ≤ κ k f K (u, u ),
where κ ∈ (0, 1).

For k = 1 this is the assertion of Lemma 1.8. Now let k ≥ 2. Consider a coupling

(u k-1 (ω k-1 ), u k-1 (ω k-1 )) for (µ k-1 , µ k-1
), defined on a probability space (Ω k-1 , F k-1 , P k-1 ) and satisfying (36) with k := k -1, existing by the base of induction. Let (Ω 0 , F 0 , P 0 ) be the probability space from Lemma 1.7.

We take for Ω k the direct product of these two spaces:

Ω k = {ω k = (ω k-1 , ω 0 )}, and set (u k , u k ) = (u 1 , u 1 )(ω k-1 , ω 0 )
, where the pair (u 1 , u 1 ) is defined as in (27

) with u = u k-1 (ω k-1 ), u = u k-1 (ω k-1 )
. By (30) and the base of induction we have:

Ef K (u k , u k ) = E k-1 E 0 f K u 1 , u 1 (u k-1 (ω k-1 ), u k-1 (ω k-1 )) ≤ κE k-1 f K (u k-1 , u k-1 ) ≤ κ k f K (u, u ), so (36)
is proved. By (35), ( 36) and (31)

P k (u, •) -P k (u , •) * L ≤ R * d 0 f K (u, u )κ k ≤ 3R * d 0 u -u H κ k ,
which proves Theorem 1.3.

1.4. Proof of the main lemma. We start with a result on almost-inverse linear operators, depending on a parameter, following closely Section 2.2 of [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF]. Let H, E and V be the spaces as above and X, K be the compact sets as above. For the Hilbert base {e j } of E as in (H1) and any M ≥ 1 we denote by E M the subspace of E, generated by the first M vectors of the base, and denote by P M the orthogonal projection E → E M .

Proposition 1.9. Let A : X ×K → L(E, H) be a continuous mapping. Assume that for any u ∈ X and η ∈ K the operator A(u, η) has dense image in H. Then for any ε > 0 there exists M ε ∈ N, C(ε) > 0 and a continuous mapping R ε : X × K → L(H, E) such that for any u ∈ X, η ∈ K,

Im (R ε (u, η)) ⊆ E Mε , R ε (u, η) L(H,E) ≤ C(ε); (37) A(u, η)R ε (u, η)f -f H ≤ ε f V ∀f ∈ V. ( 38
)
Proof. For any u ∈ X and η ∈ K define G(u, η) := A(u, η)A * (u, η), where A * is the adjoint operator. This is a non-negative self-adjoint operator. Its kernel equals to that of A * and is trivial since A has dense image. So G = G * > 0, and the operator (G + rI) -1 is well-defined and smooth in {r > 0}.

For M ∈ N denote G r (u, η) := A * (u, η)(G(u, η) + rI) -1 , G r,M (u, η) := P M • G r (u, η).
For any r and M the operator G r,M is continuous in u and η. Below we will choose M = M ε and r = r ε in such a way that R ε := G r,M satisfies (37),(38). Since sup X×K A(u, η) < ∞ and (G + rI) -1 ≤ r -1 , then sup X×K G r,M (u, η) ≤ Cr -1 , which implies (37). Now we will prove (38). Doing that we may assume that f ∈ B V (1). Since AG r -AG r,M = A(I -P M )G r , then

AG r f -AG r,M f E ≤ C (I -P M )G r f H =: CF M (u, η, f ), for all (u, η, f ) ∈ X × K × B V (1) =: A.
The functions F M are continuous on A, and F M 0 as M → ∞, pointwise. Since the set A is compact, this convergence is uniform by Dini's theorem. That is, sup

A AG r f -AG r,M f → 0 as M → ∞,
for any r > 0. So to obtain inequality (38) it remains to show that (39) sup

A AG r f -f → 0 as r → 0.
To prove this note that since

AG r = AA * (G + rI) -1 = G(G + rI) -1 , then AG r f -f E = r (G + rI) -1 f E =: Φ r (u, η, f ).
Consider the family of continuous functions

Φ r : A → R + , r > 0.
Writing the operator r(G + rI) -1 and the vector f in terms of the spectral decomposition for the positive self-adjoint operator G(u, η), we get that the functions Φ r pointwise monotonically converge to zero as r → 0 (see [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], Lemma 2.4, for details). Since A is a compact set, then evoking again Dini's theorem 4 we see that the convergence is uniform. So (39) follows and the proposition is proved. Now we are ready to prove the main lemma.

Proof of Lemma 1.6.

Step 1: construction of the mapping Φ. By the regularity assumption (A1), for any

ζ ∈ E (40) S(u , η + ζ) = S(u, η) + D u S(u, η)(u -u) + D η S(u, η)ζ + O( u -u 2 H + ζ 2 E ).
To obtain (22), consider the relation above with ζ = Φ(η) = Φ(u, u ; η). If we can construct a measurable mapping Φ :

D δ × K → E Mε such that D η S(u, η)Φ(η) + (D u S(u, η))(u -u) H ≤ 1 4 u -u H , (41)
Φ(η) E ≤ C u -u H , (42)
for all (u, u , η) with a uniform constant C, then (22) should follow if δ 1. For a fixed (u, u , η) ∈ D δ × K consider the following equation on ζ : By assumption (H3) and Proposition 1.9, for any ε > 0 there exists an M = M ε ∈ N and a linear operator R ε (u, η) : H → E Mε whose norm is bounded uniformly in u, η ∈ X × K, such that for any f ∈ V,

D η S(u, η)ζ = -(D u S)(u, η)(u -u).
(D η S)R ε f -f H ≤ ε f V . Take f = -(D u S)(u -u). By (A1) f belongs to the space V and f V ≤ C u -u H , uniformly in (u, η) ∈ X × K. Define Φ : D δ × K → E Mε , Φ(u, u ; η) = R ε (u, η)f = -R ε (u, η)(D u S(u, η))(u -u).
For any ε > 0 this is a continuous mapping, and due to assumption (A1)

Φ(u, u ; η) E ≤ C ε u -u H , (43)
Φ(u, u ; η) -Φ(u, u ; η ) E ≤ C ε u -u H η -η E , (44) for any (η, η ) ∈ B E (R η ) × B E (R η ) and (u, u ) ∈ D δ .
Step 2: estimate (22). According to identity (40) and estimate (43), for any η ∈ K and u, u ∈ D δ we have

S(u , η + Φ(u, u ; η)) -S(u, η) H ≤ C u -u 2 H + Φ(u, u ; η) 2 E + ε (D u S)(u -u) V ≤ u -u H Cε + (1 + C 2 ε )C u -u H ≤ 1
2 u -u H , where to get the last inequality we choose ε so small that Cε < 1 4 next choose δ ≤ δ(ε) so small that (1 + C 2 ε )Cδ < 1 4 . This proves (22).

(A) = h A (e) (de) = K⊥ M ⊥ M (dw) KM h A (w + v) M (dv) = K⊥ M ⊥ M (dw) KM h A (w + v)ρ M (v)dv. Similar, Ψ * ( )(A) = h A (e + Φ(u, u ; e)) (de) = K⊥ M ⊥ M (dw) KM h A (w + v + Φ(u, u ; w + v))ρ M (v)dv. Since -Ψ * var = sup A∈B E | (A) -Ψ * ( )A|, then (21) would follow if we prove that (45) | KM h A (w + v)ρ M (v)dv - KM h A (w + v + Φ(u, u ; w + v))ρ M (v)dv| ≤ C u -u ∀w, u, u ,
where C does not depend on A as well as on w, u, u . To do this let us consider the mapping

Φ w : KM v → Φ(u, u ; v + w) ∈ E M .
By ( 43) and ( 44) the map's norm and the Lipschitz constant are bounded by

C ε u -u H ≤ C ε δ.
We will assume that δ is so small that (46)

C ε δ ≤ 1/2
(where ε was fixed at Step 2 of the proof). By the Kirszbraun theorem (see [START_REF] Federer | Geometric Measure Theory[END_REF]), for each fixed u, u and w, Φ w extends to a mapping Φw : E M → E M with the same Lipschitz constant. Now consider the mapping

Ξ : E M → E M , v → ξ = v + Φw (v).
Due to (46) its Lipschitz constant is

≤ 1 + C ε δ ≤ 3/2. The inverse mapping Ξ -1 : E M → E M exists and Lip (Ξ -1 -id) ≤ C ε u -u .
Let us consider the second integral in (45) and write it as an integral over E M with an integrand which vanishes outside KM :

KM h A (w + v + Φ(u, u ; w + v))ρ M (v)dv = E M h A (w + v + Φ(u, u ; w + v))ρ M (v)dv.
Passing there from the variable v to ξ we write the integral as

E M h A (w + ξ)ρ M (v(ξ)) det ∂ξ ∂v -1 dξ = Ξ( KM ) h A (w + ξ)ρ M (v(ξ)) det ∂ξ ∂v -1
dξ (concerning Lipschitz changes of variables in integrals over R N see e.g. Theorem 3.2.5 in [START_REF] Federer | Geometric Measure Theory[END_REF]).

Writing the first integral in (45) as K M h A (w + ξ)ρ M (ξ) dξ and using that the mapping v → ξ and its inverse both are C ε u -u -close to the identity in the Lipschitz norm, we get (45). The main lemma is proved.

Remark 1.10. If, as in Remark 1.1, the mapping S is defined only on a subdomain Q of H × E, containing H × K, and the assumption of Remark 1.2 holds, then assumption (46) should be strengthened to C ε δ ≤ min(1/2, r(R + )), where r is the positive functions from Remark 1.1. Then the extension Φw can be chosen to be bounded in norm by r(R + ), so the points (u , Ψ) stay in the domain Q, where S is well defined.

Applications

In this section we apply the abstract theorems above to the two-dimensional Navier-Stokes system on the torus T 2 = R 2 /(2π)Z ⊕ (2π)Z and to well posed quasilinear parabolic systems on T d , perturbed by random forces, and prove that these systems are exponentially mixing. To do that we will pass from the corresponding PDE (1) to a discrete time system (2), will show that the latter satisfies the assumptions (A1), (H1), (H2), (H3) and then will apply Theorem 1.5. We will do that in the situation when the space H is a Hilbert space of functions of x and E is the space L 2 ([0, 1]; H). Accordingly, the Hilbert base {e j } of E will be of the form {h r ⊗ φ i }, where {φ i } is a base of H and {h r } -a base of L 2 (0, 1). To apply the abstract theorems to the 2d Navier-Stokes system the base {φ i } may be arbitrary, while to apply them to the quasilinear parabolic systems its elements should be bounded functions. Below we restrict ourselves to the case when {φ i } is the Haar base.

2.1. Random Haar series ("colored noises"). We will apply Theorem 1.5 to equations (1), perturbed by random forces η t of the form

η t = ∞ k=1 1 [k-1,k) (t)η k (t -k + 1),
where {η k (τ ), 0 ≤ τ < 1} k is a sequence of i.i.d. random variables in L 2 ([0, 1], H). It suffices to define the process η 1 = η | [0,1) . To do this, let us denote by {h jl : j ∈ N, 0 ≤ l ≤ 2 j -1} the orthonormal Haar basis on [0, 1] :

(47) h 00 (t) = 1 0 ≤ t < 1 0 t < 0 or t ≥ 1, h jl (t) =    0 t < l2 -j or t ≥ (l + 1)2 -j , 2 j 2 l2 -j ≤ t < (l + 1 2 )2 -j , -2 j 2 (l + 1 2 )2 -j ≤ t < (l + 1)2 -j . Then {h jl φ i } ijl is an orthonormal basis of E = L 2 ([0, 1], H) (we recall that {φ i } i is a Hilbert basis of H). Now we define the process η 1 = η | [0,1) in H as (48) η 1 (t) = ∞ i=1 b i η i t φ i , b i = 0 ∀ i.
Here i b 2 i < ∞ and for i ≥ 1, η i t is the random process (49)

η i t = j=0 c i j 2 j -1 l=0 ξ i jl h jl , c i j = 0 ∀ i, j, where ijl (c i j b i ) 2 = i b 2 i j 2 j (c i j ) 2 =: R 2 η < ∞,
and {ξ i jl } i,j,l are independent real random variables with |ξ i jl | ≤ 1, whose density functions ρ ijl are Lipschitz-continuous and ρ ijl (0) = 0. Denote the basis {h jl φ i } i,j,l as {e k , k ≥ 1}, where

j(k) → ∞, i(k) → ∞ as k → ∞.
Then assumption (H1) obviously holds.

2.2. The 2D Navier-Stokes system. Consider the 2D Navier-Stokes system (50)

∂ t u -ν u + u, ∇ u + ∇p = η(t, x), div u = 0, u dx = 0, where x ∈ T 2 = R 2 /2πZ 2 , u = (u 1 , u 2 )
, p is an unknown scalar function (the pressure), ν > 0 is the viscosity and η is a random force as in the previous section. Denote by H the L 2 -space

H = u ∈ L 2 (T 2 , R 2 ) : div u = 0, T 2 u(x)dx = 0 with the norm • . For m ∈ N we set H m := H ∩ H m (T 2 , R 2 )
, where H m is the Sobolev space on the torus, equipped with the Sobolev norm • m , and for m ∈ -N denote by H m the closure of H in the Sobolev norm • m . Let {φ j , j ≥ 1} be the usual trigonometric Hilbert basis of H, (e.g. see Chapter 2 in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]). This also is an orthogonal basis of every space H m .

The map D η S: By Lemma 2.1 with j = 1, we have v . (0, ξ) ∈ U 3 . It remains to show v . (0, ξ) analytically depends on u, which would imply that D η S(u 0 , η)ξ = v 1 is an analytic mapping. To this end consider the mapping

F : U 2 × U 3 -→ E 1 × H 2 , (u, v) -→ A 2 (u)v = (∂ t v -ν∆v + Π ( u, ∇ v + v, ∇ u) , v(0)) .
To prove the required analyticity it suffices to apply Lemma 2.1 and the implicit function theorem to the equation F (u, v) = (ξ, 0). The map D u0 S: Similarly for any h ∈ H 1 , D u0 S(u 0 , η)h = v1 , where vt solves (54) with ξ = 0, v0 = h. By Lemma 2.1 with j = 0, v ∈ U 2 ; so v(1) ∈ H 1 . To improve the smoothness and show that v(1) ∈ H 2 we consider the function w = tv. Calculating ∂ t w and using that v satisfies (54) with ξ = 0, we find that in its turn, w satisfies (54) with v 0 = 0, ξ = v ∈ U 2 . Then by Lemma 2.1, w ∈ U 3 . It follows that v1 = w 1 ∈ H 2 analytically depends on (u 0 , η) for the same reason as above. This proves the required analyticity of the mapping S.

The last proposition implies (A1). To verify (H2) we note that since

1 2 d dt u 2 1 = Lu, ∂ t u = -ν u 2 2 + Lu, η ≤ - ν 2 u 2 2 + 1 2ν η 2 , then u 1 2
1 ≤ e -ν u 0 2 1 + β for η ∈ K, with some constant β. So (H2) follows. It remains to check assumption (H3):

Lemma 2.3. For any (u 0 , η) ∈ H 1 × E 1 the mapping D η S(u 0 , η) : E 1 → H 1 has dense image.
Proof. Denote by u = u(u 0 , η) ∈ U 2 the solution of equation ( 51) and by v t (v 0 , ξ) the solution of equation ( 54). Define L :

E 1 → H 1 by L(ξ) = v 1 (0, ξ). Then D η S(u, η)ξ = L(ξ)
, so we should show that the mapping L : E 1 → H 1 has dense image. According to Fredholm's alternative, we only need to verify that the adjoint operator L * : H 1 → E 1 has trivial kernel. In order to do that let us consider the dual problem (55)

-∂ t w + νLw -Π( u, ∇ w) + Π(du) * w = 0, w 1 = w,
where ((du) * w) j = 2 l=1 ∂u l ∂x j w l . It is dual to the problem (54) with ξ = 0 in the sense that if v solves (54) with ξ = 0 and w solves (55), then v t , w t = const.

For a fixed τ ∈ [0, 1], let S t τ : v → v t , τ ≤ t ≤ 1, be the resolving operator for equation (54) with ξ = 0 and the initial value v τ = v. Similar, let St 1 : w → w t be the resolving operator for eq. ( 55) with terminal value w 1 = w. Then St 1 is the dual operator for S 1 t with respect to the L 2 -scalar product. Accordingly, for any η ∈ E 1 and w ∈ H 2 , by Duhamel's principle,

1 0 η t , (L * w) t 1 dt = η, L * w E1 = L(η), w 1 = 1 0 S 1 t η t dt, w 1 = 1 0 η, St 1 Lw dt. (56)
If w ∈ H 1 , then we approximate w in H 1 by smooth functions w n , n ≥ 1, and substitute (57)

St 1 Lw n = L 1/2 ξ n t . Then ξ n 1 = L -1/2 Lw n = L 1/2 w n and -∂ t ξ n + νLξ n + L -1/2 Π -u, ∇ L 1/2 ξ n + (du) * L 1/2 ξ n = 0, ξ n 1 = L 1/2 w n .
Taking a scalar product of this equation with ξ n in H and using that the vectors ξ n 1 = L 1/2 w n are bounded uniformly in H we find that |ξ n | U1 ≤ C(|u| U2 , w 1 ), uniformly in n. So ξ nj ξ ∈ U 1 weakly in U 1 , for a suitable sequence n j → ∞. From this convergence, (57) and (56) we get that

(LL * w)(t) = L 1/2 ξ t ∈ C([0, 1], H -1 ), (LL * w)(1) = L 1/2 ξ 1 = Lw.
Now let w1 ∈ H 1 be such that L * w1 = 0. Then L w1 = (LL * w1 )(1) = 0, which implies w1 = 0 and completes the proof.

It follows that Theorem 1.5 applies and implies that Theorem 2.4. Suppose that the noise η has the form (48), (49), where {φ i } is the trigonometric Hilbert basis of H 1 . Then for any ν > 0, the Markov chain (u k , P k ), defined by the 2D Navier-Stokes equation (50) in H 1 , has a unique stationary measure µ ν ∈ P(H 1 ). Furthermore, for any R > 0 and any measure µ supported by the ball B H 1 (R), there exist C = C(R) > 0 and κ = κ(R) ∈ (0, 1) such that

P * k µ -µ ν * L(H 1 ) ≤ Cκ k , ∀ k ≥ 1.
Let u ν t be a solution of (50) such that D(

u ν 0 ) = µ ν . Then D(u ν k ) = µ ν for k ∈ N, while for t = k + τ , 0 ≤ τ < 1, we have D(u ν t ) = Σ τ D(u ν k ) = Σ τ µ
, where Σ τ is a Lipschitz operator in the space (P(X), • * L ). So we get Corollary 2.5. Under the assumptions of Theorem 2.4 let u t (x) be a solutions of (50) such that D(u 0 ) = µ. Then D(u t ) -D(u ν t ) * L(H 1 ) ≤ Cκ t , ∀ t ≥ 0. Note that the mapping S : H 0 × E 1 → H 1 is also well defined (see e.g. [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]), and for any initial distribution µ ∈ P(H 0 ), D(u 1 ) ∈ P(H 1 ). By applying Theorem 2.4 and Theorem 1.5, we have the following result.

Corollary 2.6. For any µ ∈ P(H 0 ), supported by the ball B H 0 (R), there exist C = C(R) > 0 and κ = κ(R) ∈ (0, 1) such that

P * k µ -µ ν * L(H 1 ) ≤ Cκ k , ∀ k ≥ 1. Furthermore, if ν ∈ P(H 0 ) is any measure, then P * k ν µ ν as k → ∞.
Remark 2.7. Now consider the 2D NSE (50), where η(t, x) is a random kick force (see [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF], Sec-

tion 2.3): η = ∞ k=1 η k δ(t -k).
Here {η k } is a sequence of i.i.d. random variables in H 2 . Solutions u t , t ≥ 0, of the equation define the map S : H 1 × H 2 → H 1 by the relation

S(u 0 , η 1 ) = u 1 = S 1 (u 0 ) + η 1 ,
where S 1 is the time-one flow-map for the free equation (50) η=0 . According to Proposition 2.2, the map S satisfies the regularity assumption (A1) with E := H 2 . Now as in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF], Section 3.2.4, set η k = ∞ j=1 b j ξ jk φ j , where {b j , j ≥ 1}, is an l 2 -sequence of nonzero real numbers, {φ j , j ≥ 1}, is the orthonormal trigonometric basis of the space H 2 , and {ξ jk } are i.i.d. real variables whose density function is Lipschitz-continuous and does not vanish at the origin. Then (H1), (H2) and (H3) hold trivially. Applying Theorem 1.5 we recover the well known result that the 2D NSE with a non-degenerate random kick force is exponentially mixing, see in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF].

Similarly Theorem 1.5 applies to the CGL equation (62) as below in Example 2.10, where η is a nondegenerate kick-force.

2.3.

Quasilinear parabolic systems on T d . In this part we consider quasilinear parabolic systems (58)

∂ t u = ∆u + f (x, u, ∇u) + η, u(0) = u 0 ,
where f (x, 0, 0) = 0, u = u t (x) ∈ R n , x ∈ T d and f :

T d × R n × R n×d → R n is a C ∞ -smooth function.
We restrict ourselves to the case when the solution u is sought in the space

H m = H m (T d ; R n ), m > d 2 + 1,
and η t is a process as in Section 2.1 with H = H m . We are going to apply Theorem 1.5 with H = H m . To do that we will make two assumptions concerning the well-posedness and regularity of eq. ( 58). To formulate them we define the following spaces, where k ∈ Z and T > 0:

E T k = L 2 ([0, T ]; H k ), U T k = {u ∈ E T k : ∂ t u ∈ E T k-2 } ⊂ C([0, T ]), H k-1
), and abbreviate

E 1 k = E k , U 1 k = U k . ( A0
) (Well-posedness). For any M, T > 0, u 0 ∈ H m and any η ∈ E T m-1 such that η t m-1 ≤ M for all t, the problem (58) has a unique solution u ∈ U T m+1 . It satisfies (59)

u t m ≤ C(M, u 0 m ) ∀ t.
(H1 ) (Random force). The force η has the form (48), (49), where {φ i } is a Hilbert base of H m , {h ij } is the Haar base and

M 2 = (sup i j 2 j 2 |c i j |) 2 ( i b 2 i ) < ∞.
(H2 ) (Dissipativity). There exists 0 ≤ r ≤ m and κ > 0, such that if u solves (58) with η = 0 and u 0 ∈ H m , then (60) u t r ≤ e -κt u 0 r .

Note that due to (H1 ) (61) η t m ≤ M ∀ t, ∀ ω.

Remark 2.8. Here we assume that the random force η is bounded uniformly in ω and t, while in Section 2.2 we assumed that it is bounded in t in the L 2 -sense (i.e. in the norm of the space E). This is needed since the class of equations (58) for which it is possible to prove well-posedness of the initial value problem for u 0 ∈ H and η ∈ E T m is smaller than the class of equations for which we can prove the well-posedness for the problem with bounded in time η (e.g. we cannot prove that assumption (A0) holds for the CGL equations as in Example 2.10 without the additional restriction η t m-1 ≤ M ). So working with bounded in time random forces as in (61) we can apply our main theorem to a larger class of quasilinear equation (58). Lemma 2.9. Under the assumption (A0) there exists an open neighbourhood

Q of H m × K in H m × E m-1 which contains each point (u 0 , η) ∈ H m × K with its vicinity in H m × E m-1 of
radius which depends only on u m , such that for every (u 0 , η) ∈ Q the problem (58) has a unique solution u. Moreover, the mapping

Q (u 0 , η) → u ∈ U m+1
is C ∞ -smooth, and for any k ≥ 1 its C k -norm is bounded on bounded sets.

Proof. Consider the mapping

Φ : U m+1 → H m × E m-1 ; u → (u 0 , ∂ t u -∆u -f (x, u, ∇u)) .
It is C ∞ smooth (the smoothness of the nonlinear component follows from a much more general result in [START_REF] Runst | Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations[END_REF], pp. 14, 381), and by (A0), its image contains H m × K. For any (u 0 , η) ∈ H m × K and for u = Φ -1 (u 0 , η), the linear mapping

dΦ(u) : U m+1 → H m × E m-1 ; dΦ(u)(v) = (v 0 , ∆v + D u f (x, u, ∇u)v + D ∇u f (x, u, ∇u)∇v).
is an isomorpism of U m+1 and H m × E m-1 (see e.g. in [START_REF] Ladyzhenskaja | Linear and quasilinear equations of parabolic type[END_REF]), so by the inverse map theorem the point

(u 0 , η) has a neighbourhood Q (u0,η) ⊂ H m × E m-1
, where the inverse mapping Φ -1 exists and is C ∞ . By the constructive nature of the inverse map theorem, the size of the neighbourhood and the norms of derivatives of the inverse mapping are bounded in terms of the norms u 0 m and |η| Em-1 .

Since η belongs to the compact set K, these quantities may be chosen η-independent. Taking for Q the open set ∪ (u0,η) Q (u0,η) we arrive at the conclusion.

Due to the lemma and (61), Remarks 1.1, 1.10 apply to the equations (58) if the assumptions (A0) -(H2 ) hold.

Example 2.10. Consider the complex Ginzburg-Landau (CGL) equation:

(62) ∂ t u -(ν 1 + iν 2 )∆u + γu + i|u| 2r u = η, x ∈ T d , u(0) = u 0 ,
where γ > 0, ν 1 > 0 and ν 2 ≥ 0. This is an example of system (58) with n = 2. Assumption (H2 ) with r = 0 obviously holds for all equations (62). Assumption (A0) also is fulfilled for a large class of the equations. In particular, it holds if ν 2 > 0 and d ≤ 2, r ∈ N or d = 3 and r = 1; or if ν 2 = 0 and r ∈ N. See Appendix.

We may assume that u 0 ∈ B H m (R) a.s., for some R > 0. According to assumption (A0), the mapping S :

H m × E m-1 ⊃ Q R → H m is C 2 - smooth and its C 2 -norm is bounded. Here Q R is a neighbourhood of B H m (R) × K in H m × E m-1
which contains each point of B H m (R) × K with its neighbourhood of radius c R > 0. As above, S(u 0 , η) = u 1 where u t , 0 ≤ t ≤ 1, is a solution of (58). So (58) defines in H m the random dynamical system (52). By (61) and (A0) the trajectories of (52) with u 0 ∈ B H m (R) satisfy u k m ≤ R + , R + = R + (R, M ), for all k, a.s. In the following three steps we will check that the assumptions (A1), (H2) and (H3) hold with H = H m , E = E m and V = H m+1 . Verifying assumption (H2), we only need to take into account Remark 1.2.

Step 1. (Improvement of regularity) In view of (A0) to prove (A1) it remains to show that S restricted to Q R ⊂ H m × E m takes values in H m+1 and is smooth. Evoking equality (53) (where now S is the operator in (52)) we see that it suffices to examine the regularity of the maps D u0 S(u 0 , η)h and D η S(u 0 , η)ξ. Let u = u(u 0 , η) ∈ U m+1 solves (58) with u 0 ∈ H m and η ∈ E m . Since D η S(u 0 , η)ξ = v 1 (0, ξ; u) and D u0 S(u 0 , η)h = v 1 (h, 0; u), where v t (v 0 , ξ; u) solves the linear system (63)

∂ t v = ∆v + D u f (x, u, ∇u)v + D ∇u f (x, u, ∇u)∇v + ξ, v(0) = v 0 ,
we only need to examine the regularity of the mapping (v 0 , ξ, u) → v 1 (v 0 , ξ; u). Due to the classical results from the linear parabolic theory (see [START_REF] Ladyzhenskaja | Linear and quasilinear equations of parabolic type[END_REF], Part 7), if u ∈ U m+1 , then for j = m and j = m + 1, for v 0 ∈ H j , ξ ∈ E j-1 eq. ( 63) has a unique solution v ∈ U j+1 . By an argument similar to that in Proposition 2.2 (but using a smooth version of the implicit function theorem), we show that v smoothly depends on u ∈ U m+1 . So v 1 (0, ξ; u) ∈ H m+1 smoothly depends on u ∈ U m+1 , ξ ∈ E m .

To analyse v t = v t (h, 0; u), as in Section 2.2 we consider w = tv. This vector-function satisfies (63) with v 0 = 0, ξ = v(h, 0; u). So v 1 = w 1 ∈ H m+1 smoothly depends on ξ ∈ U m+1 , i.e. on h ∈ H m and u ∈ U m+1 . This implies the smoothness, required in (A1).

Remark 2.11. By a similar argument we can deduce that the mapping S is analytic, if f is.

Step 2. Below we follow [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], Section 4.2. Firstly we claim that for any R > 0 and u 0 ∈ H m with u 0 m < R,

(64) S(u 0 , 0) m ≤ C R u 0 r ,
where C R > 0 depends only on R. Indeed, since f (x, 0, 0) = 0, then S(u 0 , 0) = then D u S(tu 0 , 0)u 0 = v 1 . Choosing any points

0 < t r +1 < t r +2 < • • • < t m = 1
and arguing as at Step 1 we find that v t r +1 r +1 ≤ C R u 0 r , . . . , v tm m ≤ v tm-1 m-1 . So v 1 m ≤ C R u 0 r and (64) follows. Due to (64) and Assumption (H2 ), in the space H m there exists an equivalent norm • m such that for any u 0 ∈ H m with u 0 m ≤ R, (66) S(u 0 , 0) m ≤ q u 0 m , where q ∈ (0, 1) depends on R. Indeed, let • m := • r + ε • m , where ε > 0 is a parameter to be defined. We have S(u 0 , 0) m = S(u 0 , 0) r + ε S(u 0 , 0) m ≤ e -κ u 0 r + εC R u 0 r ≤ (e -κ + εC R ) u 0 m .

It remains to choose ε so small that q := e -κ + εC R < 1. Since the C 2 -norm of the map S : Q R → H m is bounded, then (66) implies [START_REF] Ladyzhenskaja | Linear and quasilinear equations of parabolic type[END_REF] for u m ≤ R + with the norm • m replaced by • m . This proves assumption (H2) in the weaker form, suggested in Remark 1.2.

Step 3. Now we verify (H3), i.e. check that for any u 0 ∈ H m , η ∈ E m , the linear operator So L * w = 0 implies that ξ 1 = 0 and w = 0, i.e. L * has trivial kernel. Now an application of Theorem 1.5 implies the validity for eq. ( 58) of obvious reformulations of Theorem 2.4 and Corollary 2.5.

2.4. Appendix. Case ν 2 > 0. We will only discuss equations with d ≥ 2 and to simplify notation take ν 1 = ν 2 = γ = 1:

(67) ∂ t u -i∆u + (u -∆u) + i|u| 2r u = η t , u 0 m =: R, where η t m-1 ≤ M for all η, with a fixed m > d/2. We start with apriori estimates, assuming that u is a smooth solution of (67).

Step 1. Noting that eq. ( 67) with η = 0 and with removed term u -∆u is hamiltonian with the Hamiltonian H(u) = By (68) and the Sobolev embedding,

|u t | Lq ≤ C 1 (R, M, ε) ∀ t ≥ 0, 1 q = d -2 + ε 2d . S.

4

  A version of the theorem for families of functions easily follow from the usual Dini's theorem for sequences of functions.

Step 3 :x

 3 estimate (21). Recall that E M =span {e 1 , . . . , e M }. Denote by E ⊥ M the complementary space of E M in E. For any A ∈ B E we will denote by A M and A ⊥ M its projections to E M and E ⊥ M , respectively. Consider the Hilbert brick K, defined in[START_REF] Villani | Topics in Optimal Transportation[END_REF]. Then KM is a parallelepiped, K⊥ M is a Hilbert brick in E ⊥ M , i e i |x i | ≤ b i , i = 1, ..., M }, and K = KM × K⊥ M . By assumption (H1) the measure is supported by K and = M ⊗ ⊥ M where M is supported by KM and ⊥ M -by K⊥ M . Besides, ρ M = ρ M (dv), where E M = {v} and ρ M is a Lipschitz function, supported by KM . For any fixed A ∈ B E , denote by h A the indicator function of A. According to Fubini's theorem,

1 0( 1 0D

 11 ∂/∂t)S(tu 0 , 0)u 0 dt = u S(tu 0 , 0)u 0 dt, and we only need to show that D u S(tu 0 , 0)u 0 m ≤ C R u 0 r if u 0 m ≤ R. Denote by ũt (tu 0 , 0) the solution of equation (58) with u 0 replaced by tu 0 and η = 0. By (61) and (59), ũt m ≤ C(R, r) =: R + for all t. Consider the following linear system (65)∂ t v = ∆v + D u f (x, ũ, ∇ũ)v + D ∇u f (x, ũ, ∇ũ)∇v, v(0) = u 0 ;

1 0 1 0 S 1 t η t dt, w m = 1 0L m w, S 1 t η t dt = 1 0 1 L

 11111 D η S(u 0 , η) : E m → H m has dense image. Define the operator L : E m → H m by L(ξ) = v 1 (0, ξ), where v t (v 0 , ξ) solves equation (63). By Fredholm's alternative it suffices to show that L * : H m → E m has trivial kernel. Denote S 1 t : v t → v 1 the resolving operator for equation (63) with ξ = 0, v(t) = v t . Consider the adjoint system-∂ t w = ∆w + D u f (x, u, ∇u) * w -div D ∇u f (x, u, ∇u) * w , w(1) = w 1 ,and denote by St 1 : w 1 → w t , 0 ≤ t ≤ 1, its resolving operator with initial condition at t = 1. Then v t , w t ≡ constant, so the operator St1 is the L 2 -dual of S 1 t . It follows that, for η ∈ E m and w ∈ H 2m , η t , (L * w) t m dt = η, L * w E 1 m = L(η), w m = St m w, η t dt, L = 1 -∆.If w ∈ H m , then arguing as in Section 2.2 we get that L m L * w = L m/2 ξ, ξ ∈ U m ; ξ 1 = L m/2 w.

1 2 ∂

 2 t u 2 m + u 2 m+1 ≤ | ∇ m |u| 2m u, ∇ m u | + | η t , ((-∆) m + 1)u |.The first term in the r.h.s. is bounded by a finite sum of the terms of the formU = C |v 1 . . . v 2r+2 | dx, v j = ∂ aj u or v j = ∂ aj ū, where (70) |a 1 | + • • • + |a 2r+2 | = 2m, |a j | ≤ m ∀ j.

  1. It is not necessary to define the map S on the whole space H × E: it suffices that it is defined on a neighbourhood Q of H × K which contains each point (u 0 × η) ∈ B H (R) × K with its vicinity in H × E of a positive radius r(R). The map S should be C 2 -smooth and its C 2 -norm should depend only on u H . Main results. Here we formulate the main results of this paper.

	Remark 1.2. Assume that the system possesses the following additional property: there exists a
	function R + (R) such that if u 0 H ≤ R and η 1 , η 2 , . . . are arbitrary points in K, then u k H ≤
	R + (R) for all k ≥ 0. Then in order to study solutions of (7) with u 0 H ≤ R we can do the
	following:
	-define O ⊂ H as a union of all trajectories of (7) with u 0 H ≤ R and η k ∈ K for all k; -consider the closure Ō. This is a closed subset of B H (R + ), invariant for (7) a.s. Then to study solutions with u 0 H ≤ R we can work with system's restriction to Ō. In particular
	it suffices to verify assumptions (A1) and (H3) for u ∈ B H (R + ). Even more, it suffices to check
	(A1) and (H3) for u ∈ B H (R + ) with the norm • H replaced by any equivalent Hilbert norm • H
	in the space H, depending on R + (i.e., depending on R). Applying Remark 1.1 we observe that it
	suffices to verify (A1) and (H3) on the set Q ∩ (B H (R + ) × E).
	1.2.

1 2

 1 |∇u| 2 + 1 2r+2 |u| 2r+2 dx and since ∇H = -∆u + |u| 2r u, we find that ∂ t H(u t ) + 1 2 u

	So
	(68)	H(u t ) + 1 2 u t	2 0 ≤ C(R, M ) ∀ t ≥ 0.
	Step 2. Now let us consider 1 2 ∂ t u 2 m , where u 2 m = u 2 0 + ∇ m u 2 0 . We have:
	(69)

t 2 0 = ∇H + u, ∆u -u + ∇H + u, η ≤ -C(H(u t ) + 1 2 u t 2 0 ) + u t 1 η t 1 + |u| 2r+1 |η| dx ≤ -C 2 (H(u t ) + 1 2 u t 2 0 ) + C 1 ( η t 1 ).

  Kuksin, Institut de Mathémathiques de Jussieu-Paris Rive Gauche, CNRS, Université Paris Diderot, UMR 7586, Sorbonne Paris Cité, F-75013, Paris, France; and School of Mathematics, Shandong University, Shanda Nanlu 27, 250100, PRC; and Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia, E-mail address: sergei.kuksin@imj-prg.fr H. Zhang, Institute of Mathematics, Fudan University, Handan Road 220, 200433, PRC, E-mail address: huilinzhang2014@gmail.com

The peculiarity of the used KAM-scheme is that now, in difference with the "traditional" KAM, the rate of convergence is not super-exponential, but only exponential. Cf.[START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF], Section 1.3.

Suppose that η(t, •) ∈ H for all t, a.s. Applying the Leray projection Π : L 2 (T 2 , R 2 ) → H to equation (50) we obtain the following nonlocal PDE:

(51) ∂ t u + νLu + B(u) = η(t, x), u(0) = u 0 , where L = -Π∆ and B(u) = Π( u, ∇ u).

For j ∈ Z denote E j = L 2 ([0, 1], H j ).

By the classical results (see e.g. Chapter 2 in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]), if m ≥ 1, then for any u 0 ∈ H m-1 and η ∈ E m-2 eq. ( 50) with 0 ≤ t ≤ 1 has a unique solution

We equip the space U m with the Hilbert norm

where u(t) is a solution of equation (51) with u(0) = u k-1 and η = η k . We are going to apply to system (52) Theorem 1.5 with H = H 1 and E = E 1 . Accordingly we rescale the basis {φ i } to be an orthonormal basis of the space H 1 . The validity of assumption (H1) was already checked in Section 2.1. In order to verify the the remaining assumptions we need two auxiliary results:

Lemma 2.1. For any u ∈ U 2 and j = 0 or 1, consider the mapping

Then A 2 (u) is a linear isomorphism. Furthermore, the norm of its inverse depends only on u U2 .

The proof easily follows by Galerkin' method.

Proposition 2.2. For u 0 ∈ H 1 and η ∈ E 1 , we have S(u 0 , η) ∈ H 2 . Furthermore, the mapping S :

Proof. Consider equation (51) with u 0 ∈ H 1 and η ∈ E 0 . It has a unique solution u ∈ U 2 . By Lemma 2.1 with j = 0 and the implicit function theorem, u analytically depends on (u 0 , η) (see [START_REF] Kuksin | Diffeomorphisms of function spaces that correspond to quasilinear parabolic equations[END_REF] for details), so the mapping

is analytic, as well as the mapping

It remains to improve the regularity and show that the map

so it suffices to show that D u0 S(u 0 , η)h and D η S(u 0 , η)ξ as mappings

stands for a solution of the following linear equation (54)

Here and below ε = 0 if d ≥ 3 and ε is any positive number if d = 2. So by the Gagliardo-Nirenberg inequality (72)

where

Denoting α = α j we get from the last relation and ( 70) that

From here

Step 3. The obtained estimates imply (A0) via Galerkin's method.

Case ν 2 = 0, d is any. In this case the function r(t, x) = |u(t, x)| satisfies a differential inequality with the maximum principle, e.g. see in [START_REF] Kuksin | Stochastic nonlinear Schrödinger equation 1. A priori estimates[END_REF], where the white in time stochastic force ζ t has to be replaced by the easier for analysis force η t . This implies that |u t | L∞ ≤ C(R, M ) for all t and any dimension d. Then (A0) follows by the same argument as at the Steps 2-3 above.