domain: math nlin

text:
Guan Huang
email: huangguan@tsinghua.edu.cn

Sergei Kuksin
email: sergei.kuksin@imj-prg.fr

AVERAGING AND MIXING FOR STOCHASTIC PERTURBATIONS OF LINEAR CONSERVATIVE SYSTEMS

We study stochastic perturbations of linear systems of the form

where A is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field P (v) and the matrix-function B(v) are locally Lipschitz with at most a polynomial growth at infinity, that the equation is well posed and first few moments of norms of solutions v(t) are bounded uniformly in ε. We use the Khasminski approach to stochastic averaging to show that as ε → 0, a solution v(t), written in the interaction representation in terms of operator A, for 0 ≤ t ≤ Const ε -1 converges in distribution to a solution of an effective equation. The latter is obtained from ( * ) by means of certain averaging. Assuming that eq. ( * ) and/or the effective equation are mixing, we examine this convergence further.

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1. Introduction 1.1. The setting and problems. The goal of this paper is to present an averaging theory for perturbations of conservative linear differential equations by locally Lipschitz nonlinearities and stochastic terms. Namely, we will examine stochastic equations (1.1) dv(t) + Av(t) dt = εP (v(t))dt + √ ε B(v(t))dW (t), v ∈ R D ,

1
where 0 < ε ≤ 1, A is a linear operator with non-zero pure imaginary eigenvalues {iλ j } (so the dimension D is even), P is a locally Lipschitz vector field on R D , W (t) is the standard Wiener process in R N and B(v) is an D × N -matrix. We wish to study for small ε the behaviour of solutions for eq. (1.1) on time-intervals of order ε -1 , and under some additional restriction on the equation examine the limiting as ε → 0 behaviour of solutions uniformly in time.

1.2. Our results and their deterministic analogies. We tried to make our work "readerfriendly" and accessible to people with just a limited knowledge of stochastic calculus. To achieve that in the main part of the paper we restrict ourselves to the case of equations with additive noise √ ε B dW (t) and exploit there a technical convenience: we introduce in R D a complex structure, re-writing the phase-space R D as C D 2 (recall that D is even), in such a way that in the corresponding complex coordinates the operator A is diagonal, A =diag{iλ j }. General equations (1.1) are discussed in Section 8, where they are treated in parallel with earlier considered equations with additive noise.

As it is custom in the classical deterministic Krylov-Bogolyubov (K-B) averaging (e.g. see [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF], [START_REF] Arnold | Mathematical Aspects of Classical and Celestial Mechanics[END_REF] and [START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF]), to study solutions v(t) ∈ C D 2 we write them in the interaction representation which preserves the norms of the complex components v j (τ ), but amends their angles. See below substitution (2.8). The first principal result of the work is given by Theorem 4.7, where we assume uniform in ε and in t ≤ Cε -1 bounds on a first few moments of norms of solutions. The theorem states that as ε → 0, for t ≤ Cε -1 solutions v(t), written in terms of the interaction representation, weakly converge in distribution to solutions of an additional effective equation. The latter is obtained from eq. (1.1) by means of certain averaging of vector field P in terms of the spectrum {iλ j } and in many cases may be written down explicitly. Proof of Theorem 4.7, given in Section 4, is obtained by a synthesis of the K-B method (as it is presented e.g. in [START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF]) and the Khasminski approach to the stochastic averaging [START_REF] Khasminski | On the averaging principle for Ito stochastic differential equations[END_REF]; it may serve as an introduction to the latter. The number of works on the stochastic averaging is immense, see Section 1.3 for some references. We were not able to find there the result of Theorem 4.7, but do not insist on its novelty (and certainly related statements may be found in the literature).

In Section 5 we suppose that the bounds on the moments of norms of solutions, mentioned above, are uniform in time, and that eq. (1.1) is mixing. So its solutions, as time goes to infinity, converge in distribution to a unique stationary measure (which is a Borel measure in R D = C D 2 ). In Theorem 5.5, postulating that the effective equation as well is mixing, we prove that when ε → 0 the stationary measure for eq. (1.1) converges to that for the effective equation. Note that this convergence holds without passing to the interaction representation.

In short Section 6 we discuss non-resonant systems (1.1) (where the frequencies {λ j } are rationally independent). In particular, we show that then the actions I j (v(t)) of solutions v(t) (see below (1.2)), as ε → 0, converge in distribution to solutions of a system of stochastic equations, depending only on actions. The convergence hold on time-intervals 0 ≤ t ≤ Cε -1 .

In Section 7 we keep the assumption on the norms of solutions from Section 5. Supposing that the effective equation is mixing (and without assuming that for the original equation (1.1)) we prove there Theorem 7.4. It states that the convergence as in the principal Theorem 4.7 is uniform for t ≥ 0 (and not only for t ≤ Cε -1 ).

In Proposition 9.4 we present a simple sufficient condition on eq. (1.1), based on results from [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF], which insures that Theorems 4.7, 5.5 and 7.4 apply to it.

In Section 8 we pass to general equations (1.1), where the dispersion matrix B depends on v. Assuming the same estimates on solutions as in Section 4 we show that Theorem 4.7 remains valid if either the matrix B(v) is non-degenerate, or it is a C 2 -smooth function of v. Theorems 5.5 and 7.4 also remains true for general systems (1.1), but we do not discuss this, hoping that the corresponding modifications of the proofs should be clear after reading Section 8.

A deterministic analogy of our results, which deals with eq. (1.1) with W = 0 and describes the behaviour of its solutions on time-intervals of order ε -1 in the interaction representation in comparison with solutions for a corresponding effective equation, is given by the K-B averaging (see [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF][START_REF] Arnold | Mathematical Aspects of Classical and Celestial Mechanics[END_REF][START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF]). 1 Theorem 5.5 has no analogy for deterministic systems, but Theorem 7.4 has it. Namely, for the K-B averaging it is known that if the effective equation has a globally asymptotically stable equilibrium, then the convergence of solutions for eq. (1.1) W =0 , written in the interaction representation, to solutions of the effective equation, is uniform in time. This result is known in folklore as the second K-B theorem and may be found in [START_REF] Burd | The Averaging Method on Infinite Time-Interval and Some Problems of the Theory of Oscillations[END_REF].

The K-B method and Khasminski approach to averaging which we exploit, are flexible tools. They are applicable to various stochastic systems in finite and infinite dimensions, including stochastic PDEs, and the specific realization of the two methods which we use now is inspired by our previous work on averaging for stochastic PDEs. See [START_REF] Huang | Time-averaging for weakly nonlinear CGL equations with arbitrary potential[END_REF][START_REF] Kuksin | Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations[END_REF] for an analogy for SPDEs of Theorem 4.7, [START_REF] Huang | Time-averaging for weakly nonlinear CGL equations with arbitrary potential[END_REF] -for an analogy of Theorem 5.5 and [START_REF] Huang | On averaging and mixing for stochastic PDEs[END_REF] -for an analogy of Theorem 7.4 (also see [START_REF] Duan | Effctive Dynamics of Stochastic Partial Differential Equations[END_REF] for more results and references on averaging for SPDEs).

1.3. Relation with classical stochastic averaging. The averaging in stochastic systems is a well developed topic, usually dealing with fast-slow stochastic systems, e.g. see publications [START_REF] Khasminski | On the averaging principle for Ito stochastic differential equations[END_REF], [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF]Section 7], [START_REF] Skorokhod | Asymptotic Methods in the Theory of Stochastic Differential Equations[END_REF]Section II.3], [START_REF] Liu | Stochastic Averaging and Stochastic Extremum Seeking[END_REF], [START_REF] Yu | On the averaging principle for systems of stochastic differential equations[END_REF], [START_REF] Yu | Large Deviations and Adiabatic Transitions for Dynamical Systems and Markov Processes in Fully Coupled Averaging[END_REF] and references there. To explain relation of that theory with our work let us write eq. (1.1) in the complex form v(t) ∈ C n , n = D 2 (when the operator A is diagonal) and then pass to the slow time τ = εt and action-angle coordinates (I, ϕ) = (I 1 , . . . , I n ; ϕ 1 , . . . , ϕ n ) ∈ R n + × T n , where R + = {x ∈ R ∶ x ≥ 0}, T n = R n (2πZ n ) and

I k (v) = 1 2 v k 2 = 1 2 v k vk , ϕ k (v) = Arg v k ∈ S 1 = R 2πZ, k = 1, . . . , n (1.2)
(if v k = 0, we set ϕ k (v) = 0 ∈ S 1 ). In these coordinates equation (1.1) takes the form (1.3) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ dI(τ ) = P I (I, ϕ)dτ + Ψ I (I, ϕ)dβ(τ ), dϕ(τ ) + ε -1 Λdτ = P ϕ (I, ϕ)dτ + Ψ ϕ (I, ϕ)dβ(τ ).

Here β = (β 1 , . . . , β N ), where {β l } are independent standard real Wiener processes, and the coefficients of the system are given by Itô's formula. This is a fast-slow system with the slow variable I and fast variable ϕ. The stochastic averaging treats systems like (1.3), usually adding to the fast part of the ϕ-equation a non-degenerate stochastic term of order ε -1 2 . The (first) goal of a system's analysis usually is to prove that on time-intervals 0 ≤ τ ≤ T distributions of the I-components of solutions converge as ε → 0 to distributions of solutions for a suitably averaged I-equation. After that other goals may be pursued. 2 Unfortunately the stochastic averaging does not apply directly to systems (1.3), coming from equations (1.1), since then the coefficients of the ϕ-equation have singularities when some I k vanishes, and since the fast ϕ-equation is rather degenerate if the vector Λ is resonant. Instead we borrow from the theory the Khasminski method [START_REF] Khasminski | On the averaging principle for Ito stochastic differential equations[END_REF] of stochastic averaging and apply it to eq. (1.1), written in the interaction representation, thus arriving at the assertion of Theorem 4.7. Averaging theorem for stationary solutions of eq. (1.3) and for the corresponding stationary measures are known in the stochastic averaging, but (naturally) they control the limiting behaviour only of the I-components of the stationary solutions and measures, while our Theorem 5.5 describes a limit of the whole stationary measure. It seems that no analogy of Theorem 7.4 is known in the stochastic averaging.

At the origin of this paper are lecture notes for an online course which SK was teaching in the Shandong University (PRC) in the autumn term of the year 2020.

u α = sup 0⩽τ <τ ′ ⩽T u(τ ′ ) -u(τ ) E τ ′ -τ α + sup τ ∈[0,T ] u(τ ) E ≤ ∞.
This is a norm in the Hölder space C α ([0, T ], E). We denote the standard C m -norm for C m -smooth functions on E as ⋅ C m (E) . By D(ξ) we denote the law of a random variable ξ, by ⇀ -the weak convergence of measures, and by P(M ) -the space of Borel measures on a metric space M . For a measurable mapping F ∶ M 1 → M 2 and µ ∈ P(M 1 ) we denote by

F ○ µ ∈ P(M 2 ) the image of µ under F ; i.e. F ○ µ(Q) = µ(F -1 (Q)). If m ≥ 0 and L is R n or C n , then Lip m (L, E) is the collection of maps F ∶ L → E such that for any R ⩾ 1 we have (1.5) (1 + R ) -m Lip(F BR (L) ) + sup v∈ BR (L) F (v) E =∶ C m (F ) < ∞,
where Lip(f ) is the Lipschitz constant of a mapping f (note that, in particular,

F (v) E ≤ C m (F )(1 + v L ) m
for any v ∈ L). For a complex matrix A = (A ij ), A * = (A * ji ) stands for its Hermitian conjugate: A * ij = Āji (so for a real matrix B, B * is the transposed matrix). For a set Q we denote by 1 Q its indicator function, and by Q c -its complement. Finally, R + (Z + ) stands for the set of nonnegative real numbers (nonnegative integers), and for real numbers a and b, a ∨ b and a ∧ b indicate their maximum and minimum.

Linear systems and their perturbations

In this section we give the setting of the problem and specify assumptions on the operator A, vector field P and noise √ ε B(v)dW in eq. (1.1). To simplify presentation and explain better the ideas, in the main part of the text we assume that the noise is additive, i.e. B is a constant matrix (possibly degenerate). We will discuss general equations (1.1) in Section 8.

2.1.

Assumptions on A and W (t). We assume that the unperturbed linear system (2.1)

(d dt)v + Av = 0, v ∈ R D ,
is such that all its trajectories are bounded as t → ±∞. Then the eigenvalues of A are pure imaginary, go in pairs ±iλ j and A has no Jordan cells. We also assume that A is invertible. So (1) eigenvalues of A are of the form ±iλ j , where R ∋ λ j ≠ 0;

(2) in the Jordan normal form of A there are no Jordan cells.

By these assumptions D = 2n, and in R 2n there exists a base {e + 1 , e - 1 , . . . , e + n , e - n } in which the linear operator A takes the block-diagonal form:

A = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -λ 1 λ 1 0 0 ⋱ 0 0 -λ n λ n 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
We denote by (x 1 , y 1 , . . . , x n , y n ) the coordinates, corresponding to this base, and for j = 1, . . . , n set z j = x j + iy j . Then R 2n becomes the space of complex vectors (z 1 , . . . , z n ), i.e. R 2n ≃ C n . In the complex coordinates the standard inner product in R 2n reads

(2.2) ⟨z, z ′ ⟩ = Re n j=1 z j z′ j , z, z ′ ∈ C n .
Let us denote by

Λ = (λ 1 , . . . , λ n ) ∈ (R ∖ {0}) n
the frequency vector of the linear system (2.1). Then in the complex coordinates z the operator A reads

Az = diag{iΛ}z,
where diag{iΛ} is the diagonal operator, sending a vector

(z 1 , . . . , z n ) to (iλ 1 z 1 , . . . , iλ n z n ).
Therefore in R 2n , written as the complex space C n , linear equation (2.1) takes the diagonal form,

(d dt)v k + iλ k v k = 0, 1 ⩽ k ⩽ n.
Below we examine the perturbed eq. (1.1), using these complex coordinates. We next discuss the random process W (t), written in the complex coordinates. The standard complex Wiener process has the form

(2.3) β c (t) = β + (t) + iβ -(t) ∈ C,
where β + (t) and β -(t) are independent standard (real) Wiener processes, defined on some probability space (Ω, F, P). Then βc (t) = β + (t)iβ -(t), and any Wiener process W (t) ∈ C n may be conveniently written in the complex from as (2.4)

W k = n 1 l=0 Ψ 1 kl β c l + n 1 l=1 Ψ 2 kl βc l , k = 1, . . . , n,
where Ψ 1 = (Ψ 1 kl ) and Ψ 2 = (Ψ 2 kl ) are complex n × n 1 matrices and {β c l } are independent standard complex Wiener processes. Again, in order to simplify presentation below we suppose that the noise in eq. (1.1) is of the form

W (t) = n 1 l=1 Ψ kl β c l (t), k = 1, . . . , n.
We do not assume that the matrix Ψ is non-degenerate (in particular, it may be zero). Then the perturbed equation (1.1) in the complex coordinates reads as (2.5)

dv k + iλ k v k dt = εP k (v)dt + √ ε Ψ kl dβ c l (t), k = 1, . . . , n , where v = (v 1 , . . . , v n ) ∈ C n and 0 < ε ≤ 1.
The results, obtained below for eq. (2.5), remain true for general equations (1.1) on the price of heavier calculation. The corresponding argument is sketched in Section 8.

2.2.

Assumptions on P and on the perturbed equation. Our first goal is to study equation (2.5) with 0 < ε ≪ 1 on a time interval 0 ≤ t ≤ ε -1 T , where T > 0 is a fixed constant. Introducing the slow time τ = εt we write the equation as

(2.6) dv k (τ ) + iε -1 λ k v k dτ = P k (v)dτ + n 1 l=1 Ψ kl d βc l (τ ), k = 1, . . . , n, 0 ≤ τ ≤ T.
Here { βc l (τ ), l = 1, . . . , n 1 } is another set of independent standard complex Wiener processes, which we now re-denote back to {β c l (τ ), l = 1, . . . , n 1 }. We stress that the equation above is nothing but the original eq. (1.1), where its linear part (2.1) is conservative and nondegenerate in the sense of conditions ( 1) and (2), written in the complex coordinates and slow time. So all results below concerning eq. (2.6) (and eq. (2.11)) may be reformulated for eq. (1.1) on the price of heavier notation.

Everywhere below in this paper we assume the following assumption concerning the wellposedness of eq. (2.6): Assumption 2.1.

(1) The drift P (v) = (P 1 (v), . . . , P n (v)) is a locally Lipschitz vector field, belonging to Lip m 0 (C n , C n ) for some m 0 ≥ 0 (see (1.5)).

(2) For any v 0 ∈ C n , equation (2.6) has a unique strong solution v ε (τ

; v 0 ), τ ∈ [0, T ], equal v 0 at τ = 0. Moreover, there exists m ′ 0 > (m 0 ∨ 1) such that (2.7) E sup 0⩽τ ⩽T v ε (τ ; v 0 ) 2m ′ 0 ⩽ C m ′ 0 ( v 0 , T ) < ∞ ∀ 0 < ε ≤ 1,
where C m ′ 0 (⋅) is a non-negative continuous function on R 2 + , non-decreasing in both arguments.

Our proofs easily generalize to the case when the vector field P is locally Lipschitz, satisfying P (v) ≤ C(1 + v ) m 0 for all v and some C > 0, m 0 ≥ 0. 3 In this case the argument remains essentially the same (but become a bit longer), and the constants in estimates depend not only on m 0 , but also on the locally Lipschitz constant of P , which is the function

R ↦Lip P BR (C n ) .
Below T > 0 is fixed and the dependence of constants on T usually is not indicated. Solutions of eq. (2.6) are assumed to be strong, unless otherwise stated. As usual, strong solutions are understood in the sense of an integral equation. I.e., v ε (τ

; v 0 ) = v(τ ), 0 ≤ τ ≤ T , is a strong solution, equal v 0 at τ = 0, if a.s. v k (τ ) + τ 0 iε -1 λ k v k (s) -P k (v(s)) ds = v 0k + n 1 l=1 Ψ kl β c l (τ ), k = 1, . . . , n, for 0 ≤ τ ≤ T .
2.3. Interaction representation. Now in eq. (2.6) we pass to the interaction representation, which means that we substitute

(2.8) v k (τ ) = e -iτ ε -1 λ k a k (τ ), k = 1, . . . , n.
Then v k (0) = a k (0) and we obtain the following equations for variables a k (τ ):

(2.9)

da k (τ ) = e iτ ε -1 λ k P k (v) + e iτ ε -1 λ k n 1 l=1
Ψ kl dβ c l (τ ), k = 1, . . . , n.

Actions I k = a k 2 2 for solutions of (2.9) are the same as the actions for solutions of (2.6). Compare to (2.6), in eq. (2.9) we have removed from the drift the large term ε -1 diag(iΛ)v on the price that now coefficients of the system are fast oscillating functions of τ .

To rewrite conveniently the equations above we introduce the rotation operators Φ w : for any real vector w = (w 1 , . . . , w n ) ∈ R n we denote

(2.10) Φ w ∶ C n → C n , Φ w = diag{e iw 1 , . . . , e iwn }.
Then

(Φ w ) -1 = Φ -w , Φ w 1 ○ Φ w 2 = Φ w 1 +w 2 , Φ 0 = id, each Φ w is an unitary transformation, so Φ * w = Φ -1 w . Moreover, (Φ w z) j = z j , ∀z, w, j.
In terms of operators Φ we write v(τ ) as Φ τ ε -1 Λ a(τ ), and write system (2.9) as

(2.11) da(τ ) = Φ τ ε -1 Λ P (Φ -τ ε -1 Λ a(τ ))dτ + Φ τ ε -1 Λ Ψdβ c (τ ), a(τ ) ∈ C n ,
where β c (τ ) = (β c 1 (τ ), . . . , β c n 1 (τ )). This is the equation which we are going to study for small ε for 0 ⩽ τ ⩽ T under the initial condition (2.12)

a(0) = v(0) = v 0 .
The solution a ε (τ

; v 0 ) = Φ -τ ε -1 Λ v ε (τ ; v 0 ) of (2.11
), (2.12) also satisfies estimate (2.7), for each

ε ∈ (0, 1]. We recall that a C 1 -diffeomorphism G of C n transforms a vector filed V to the field G * V , where (G * V )(v) = dG(u) V (u) for u = G -1 v. In particular, (Φ τ ε -1 Λ ) * P (v) = Φ τ ε -1 λ k ○ P (Φ -ετ Λ v).
So equation (2.11) may be written as

da(τ ) = (Φ τ ε -1 Λ ) * P (a(τ ))dτ + Φ τ ε -1 Λ Ψdβ c (τ ).
2.4. The compactness. For 0 < ε ⩽ 1 we denote by a ε (τ ; v 0 ) a solution of equation (2.11), equals v 0 at τ = 0. Then a ε (τ (2.6) exists by Assumption 2.1, so a solution a ε (τ ; v 0 ) as well exists and is unique. Our goal is to examine its law

; v 0 ) = Φ τ ε -1 λ k v ε (τ ; v 0 ). A unique solution v ε (τ ; v 0 ) of
Q ε ∶= D(a ε (⋅; v 0 )) ∈ P(C([0, T ], C n ))
as ε → 0. While v 0 is fixed we will usually write a ε (τ ; v 0 ) as a ε (τ ). Lemma 2.2. Under Assumption 2.1, the set of probability measures Q ε , ε ∈ (0, 1]} is precompact in the space P(C([0, T ], C n )) with respect to the weak topology.

Proof. Let us denote the random force in equation (2.11)

by dζ ε (τ ) ∶= Φ τ ε -1 Λ Ψdβ c (τ ), where ζ ε (τ ) = (ζ ε l (τ ), l = 1, . . . , n 1 ).
For any k we have

ζ ε k (τ ) = τ 0 dζ ε k = τ 0 e isε -1 λ k n 1 l=1 Ψ kl dβ c l (s).
So ζ ε (τ ) is a stochastic integral of a non-random vector function. Hence, it is a Gaussian random process with zero mean value, and its increments over non-intersecting time-intervals are independent. For each k,

E ζ ε k (τ ) 2 = τ 0 2 n 1 l=1 Ψ kl 2 ds =∶ 2C ζ k τ, C ζ k = n 1 l=1 Ψ kl 2 ≥ 0,
and

Eζ ε k (τ )ζ ε j (τ ) = E ζε k (τ ) ζε j (τ ) = 0. Therefore ζ ε k (τ ) = C ζ k β c k (τ )
, where by Lévy's theorem (see [14, p. 157]) β c k (τ ) is a standard complex Wiener process. However, the processes ζ ε j and ζ ε k with j ≠ k are not necessarily independent. By the basic properties of Wiener process, a.s. the curve [0, T ] ∋ τ ↦ ζ ε (ω, τ ) ∈ C n is Hölder-continuous with exponent 1 3 , and since C ζ k does not depend on ε, then abbreviating

(C 1 3 ([0, T ], C n ) to C 1 3 we have P ζ ε (⋅) ∈ BR (C 1 3 ) → 1 as R → ∞,
uniformly in ε. Let us write equation (2.11) as

da ε (τ ) = V ε (τ )dτ + dζ ε (τ ).
By Assumption 2.1 and since a ε (τ ) ≡ v ε (τ ) we have that

E sup τ ∈[0,T ] V ε (τ ) ⩽ C m 0 (P )E(1 + sup τ ∈[0,T ] v ε (τ ) ) m 0 ⩽ C( v 0 ) < ∞.
Therefore, by Chebyshev's inequality,

P( sup 0⩽τ ⩽T V ε (τ ) > R) ⩽ C( v 0 )R -1 , uniformly in ε ∈ (0, 1]. Since a ε (τ ) = v 0 + τ 0 V ε (s)ds + ζ ε (τ ),
then we get from the above that (2.13)

P a ε (⋅) 1 3 > R → 0 as R → ∞,
uniformly in ε ∈ (0, 1]. By the Ascoli-Arzela theorem the sets BR (C 1 3 ) are compact in C([0, T ]; C n ), and in view of (2.13) for any δ > 0 there exists R δ such that

Q ε BR δ (C 1 3 ) ⩾ 1 -δ, ∀ε > 0.
So by Prokhorov's theorem the set of measures

{Q ε , 0 < ε ⩽ 1} is pre-compact in P(C[0, T ], C n ).
By this lemma, for any sequence ε l → 0 there exists a subsequence ε ′ l → 0 and a measure

Q 0 ∈ P(C([0, T ], C n )) such that (2.14) Q ε ′ l ⇀ Q 0 as ε ′ l → 0.
3. Averaging of vector fields with respect to the frequency vector

For a vector field P ∈ Lip m 0 (C n , C n ) we denote Y P (a; t) = (Φ tΛ ) * P (a) = Φ tΛ ○ P (Φ -tΛ a), a ∈ C n , t ∈ R,
and for T ′ > 0 define partial averaging ⟪ P ⟫ T ′ of the vector field P with respect to the frequency vector Λ as follows:

(3.1)

⟪ P ⟫ T ′ (a) = 1 T ′ T ′ 0 Y P (a; t)dt = 1 T ′ T ′ 0 Φ tΛ ○ P (Φ -tΛ a)dt. Lemma 3.1. For any T ′ > 0, ⟪ P ⟫ T ′ (a) ∈ Lip m 0 (C n , C n ) and C m 0 (⟪ P ⟫ T ′ ) ≤ C m 0 ( P ) (see (1.5)).
Proof. If a ∈ BR (C n ), then Φ -tΛ a ∈ BR (C n ) for each t. So Y P (a; t) = (Φ tΛ ) * P (a) = P (Φ -tΛ a)) , and from here

⟪ P ⟫ T ′ (a) ⩽ sup 0≤t≤T ′ Y P (a; t) ⩽ C m 0 ( P )(1 + R) m 0 .
Similarly, for any a 1 , a 2 ∈ BR (C n ),

Y P (a 1 ; t) -Y P (a 2 ; t) = P (Φ -tΛ a 1 ) -P (Φ -tΛ a 2 ) ⩽ C m 0 ( P )(1 + R) m 0 a 2 -a 1 ∀t ⩾ 0, so that ⟪ P ⟫ T ′ (a 1 ) -⟪ P ⟫ T ′ (a 1 ) ⩽ C m 0 ( P )(1 + R) m 0 a 1 -a 1 .
This proves the assertion.

Now we define averaging of the vector field P with respect to the frequency vector Λ as

(3.2) ⟪ P ⟫(a) = lim T ′ →∞ ⟪ P ⟫ T ′ (a) = lim T ′ →∞ 1 T ′ T ′ 0 (Φ tΛ ) * P (a)dt.
Lemma 3.2.

(1) The limit (3.2) exists for any a. Moreover, ⟪ P ⟫ belongs to Lip m 0 (C n , C n ) and C m 0 (⟪ P ⟫) ≤ C m 0 ( P ).

(2) If a ∈ BR (C n ), then the rate of convergence in (3.2) depends not on a, but only on R. This is the main lemma of deterministic averaging for vector fields. For its proof see [START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF]Lemma 3.1]. 4 The averaged vector field ⟪ P ⟫ is invariant with respect to transformations Φ θΛ : Lemma 3.3. For all a ∈ C n and θ ∈ R,

Φ θΛ * ⟪ P ⟫(a) ≡ Φ θΛ ○ ⟪ P ⟫ ○ Φ -θΛ (a) = ⟪ P ⟫(a).
Proof. For definiteness let θ > 0. For any T ′ > 0 we have:

⟪ P ⟫ T ′ (Φ -θΛ (a)) = 1 T ′ T ′ 0 Φ tΛ ○ P (Φ -tΛ ○ Φ -θΛ (a))dt = 1 T ′ T ′ 0 Φ tΛ ○ P (Φ -(t+θ)Λ a)dt. Since Φ tΛ = Φ -θΛ ○ Φ (t+θ)Λ , then this equals 1 T ′ Φ -θΛ T ′ 0 Φ (t+θ)Λ ○ P (Φ -(t+θ)Λ a)dt = Φ -θΛ ○ ⟪ P ⟫ T ′ (a) + O 1 T ′ .
Passing to the limit as T ′ → ∞ we obtain the assertion.

The statement below asserts that the averaged vector filed ⟪P ⟫ is as least as smooth as P .

Proposition 3.4. If P ∈ C m (C n ) for some m ∈ N, then ⟪P ⟫ ∈ C m (C n ) and ⟪P ⟫ C m ( BR ) ⩽ P C m ( BR ) , ∀R > 0.
Proof. We first fix any R > 0. Then there exists a sequence of polynomial vector fields {P R,j , j ∈ N} (cf. below item 3) in Section 3.1) such that P R,j -P C m ( BR ) → 0 as j → ∞. An easy calculation shows that 4 In fact, if P is any locally Lipschitz vector field, then ⟪ P ⟫ is well defined and locally Lipschitz with the same locally Lipschitz constant (see discussion after Assumption 2.1) as P . See in [START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF].

(3.3) ⟪P R,j ⟫ T -⟪P R,j ⟫ C m ( BR ) → 0 as T → ∞,
for each j. Since the transformations Φ tΛ are unitary, then differentiating the integral in (3.1) in a we get that

(3.4) ⟪ P ⟫ T C m ( BR ) ⩽ P C m ( BR ) ∀ T > 0,
for any C m -vector field P . Therefore

(3.5) ⟪P R,j ⟫ T -⟪P ⟫ T C m ( BR ) ≤ P R,j -P C m ( BR ) =∶ κ j → 0 as j → ∞, ∀ T > 0. So ⟪P R,j ⟫ T -⟪P R,k ⟫ T C m ( BR ) ≤ 2κ j∧k ∀ T > 0. From this estimate and (3.3) we find that ⟪P R,j ⟫ -⟪P R,k ⟫ C m ( BR ) ≤ 2κ j∧k . Thus {⟪P R,j ⟫} is a Cauchy sequence in C m ( BR ). So it C m -converges to a limiting field ⟪P R,∞ ⟫. As P R,j converges to P in C m ( BR ), then using again (3.4) we find that ⟪P R,∞ ⟫ C m ( BR ) ≤ P C m ( BR ) . But by Lemma 3.2 ⟪P R,∞ ⟫ must equal ⟪P ⟫.
Since R > 0 is arbitrary, the assertion of the proposition follows.

Finally we note that if a vector field P is Hamiltonian, then its averaging ⟪P ⟫ also is. Looking ahead we state the corresponding result here, despite the averaging of functions ⟨⋅⟩ is defined below in Section 3.2. Proposition 3.5. If a locally Lipschitz vector field P is Hamiltonian, i.e.

P (z) = i ∂ ∂ z H(z) for some C 1 -function H, then ⟪P ⟫ also is Hamiltonian and ⟪P ⟫ = i ∂ ∂ z ⟨H⟩. For a proof see [13, Theorem 5.2].
3.1. Calculating the averagings. 1) The frequency vector Λ = (λ 1 , . . . , λ n ) is called completely resonant if its components λ j are proportional to some λ > 0. I.e. if λ j λ ∈ Z for all j.

In this case all trajectories of the original linear system (2.1) are periodic, the operator Φ tΛ is 2π λ-periodic in t and so

(3.6) ⟪ P ⟫(a) = ⟪ P ⟫ 2π λ (a) = λ 2π 2π λ 0 (Φ tΛ ) * P (a)dt.
Completely resonant linear systems (2.1) and their perturbations (1.1) often occur in applications. In particular -in non-equilibrium statistical physics. There the dimension D = 2n is large, all λ j 's are equal, and the Wiener process W (t) in (1.1) may be very degenerate (it may have only two non-zero components). E.g. see [START_REF] Dymov | Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators[END_REF], where more references may be found.

2) Let us consider the case, opposite to the above and assume that frequency vector Λ is non-resonant:

(3.7) n i=1 m j λ j ≠ 0 ∀(m 1 , . . . , m n ) ∈ Z n ∖ {0}
(that is, the real numbers λ j are rationally independent). Then

(3.8) ⟪ P ⟫(a) = 1 (2π) n T n (Φ w ) * P (a) dw, T n = R n (2πZ n ).
Indeed, if P is a polynomial vector field, then (3.8) easily follows from (3.2) by a direct component-wise calculation. The general case is a consequence of this result since any vector field may be approximated by polynomial fields. Details are left to the reader (cf. Lemma 3.5 in [START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF], where P res equals to the r.h.s. of (3.8) if the vector Λ is non-resonant).

The r.h.s. of (3.8) obviously is invariant with respect to all rotations Φ w ′ , so it depends not on the vector a, but only on the corresponding torus

(3.9) {z ∈ C n ∶ I j (z) = I j (a) ∀ j} (see (1.2)) to which a belongs, (3.10) Φ w * ⟪ P ⟫(a) ≡ ⟪ P ⟫(a) ∀w ∈ C n if Λ is non-resonant.
See below Section 6 for discussing equations (1.1) with non-resonant vectors Λ.

3) If the field P in (3.2) is polynomial, i.e.

(3.11)

Pj (a) = α , β ≤N C α,β j a α āβ , j = 1, . . . , n,
for some N ∈ N, where α, β ∈ Z n + , a α = ∏ a α j j and α = ∑ α j , then ⟪ P ⟫ = P res . Here P res is a polynomial vector field such that for each j, P res j (a) is given by the r.h.s. of (3.11), where the summation is taken over all α , β ≤ N , satisfying Λ ⋅ (αβ) = λ j . This easily follows from an explicit calculation of the integral in (3.1) (see [START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF]Lemma 3.5]).

Averaging of functions.

Similarly to definition (3.2), for a locally Lipschitz function f ∈ Lip m (C n , C), m ≥ 0, we define its averaging with respect to a frequency vector Λ as (3.12) ⟨f ⟩(a) = lim

T ′ →∞ 1 T ′ T ′ 0 f (Φ -tΛ a)dt, a ∈ C n .
Then by the same argument as above we obtain:

Lemma 3.6. If f ∈ Lip m (C n , C), then (1)
The limit (3.12) exists for every a, and for a ∈ BR (C n ) the rate of convergence in (3.12) depends not on a, but only on R.

(2) ⟨f ⟩ ∈ Lip m (C n ; C) and C m (⟨f ⟩) ≤ C m (f ). (3) If f is C m -smooth
for some m ∈ N, then ⟨f ⟩ also is, and the C m -norm of the latter is bounded by the C m -norm of the former. (4) The function ⟨f ⟩ commutes with the operators Φ θΛ , θ ∈ R, in the sense that ⟨f ○Φ θΛ ⟩ = ⟨f ⟩ ○ Φ θΛ = ⟨f ⟩.

If the vector Λ is non-resonant, then similar to (3.8) we have

(3.13) ⟨f ⟩(a) = 1 (2π) n T n f (Φ -w a) dw.
The r.h.s. of (3.13) is the averaging of function f in angles. It is constant on the tori (3.9).

Effective equation and the averaging theorem

In this section we show that the limiting measure Q 0 in (2.14) is independent of the choice of the sequence ε ′ l → 0, so D(a ε ) ⇀ Q 0 as ε → 0, and represent Q 0 as the law of a solution of an auxiliary effective equation. The drift in this equation is the averaged drift in eq. (2.6). Now we construct its dispersion.

The diffusion matrix for equation (2.11) is the n × n complex matrix

A ε (τ ) = (Φ τ ε -1 Λ Ψ) ⋅ (Φ τ ε -1 Λ Ψ) * . Denoting (4.1) Φ τ ε -1 Λ Ψ = e iτ ε -1 λ l Ψ lj =∶ ψ ε lj (τ ) = ψ ε (τ ),
we have

A ε kj (τ ) = n 1 l=1 ψ ε kl (τ ) ψε jl (τ ) = e iτ ε -1 (λ k -λ j ) n 1 l=1 Ψ kl Ψjl .
So for any τ > 0,

1 τ τ 0 A ε kj (s)ds = n 1 l=1 Ψ kl Ψjl 1 τ τ 0 e isε -1 (λ k -λ j ) ds,
and we immediately see that

(4.2) 1 τ τ 0 A ε kj (τ )dτ → A kj as ε → 0, where (4.3) A kj = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∑ n 1 l=1 Ψ kl Ψjl , if λ k = λ j , 0, otherwise.
Clearly,

A kj = Ājk , so A is a Hermitian matrix. If λ k ≠ λ j for k ≠ j, then (4.4) A = diag{b 1 , . . . , b n }, b k = n 1 l=1 Ψ kl 2 .
For any vector ξ ∈ C n we get from (4.2) that ⟨Aξ, ξ⟩ ⩾ 0 since obviously ⟨A ε (τ )ξ, ξ⟩ = ψ ε (τ )ξ 2 ⩾ 0 for each ε. Therefore A is a nonnegative Hermitian matrix, and there exists another nonnegative Hermitian matrix B (called the principal square root of A) such that

BB * = B 2 = A. The matrix B is non-degenerate if Ψ is. Example 4.1. If Ψ is a diagonal matrix diag {ψ 1 , . . . , ψ n }, ψ j ∈ R, then A ε (τ ) = Ψ 2 . In this case A = Ψ 2 and B = Ψ =diag { ψ 1 , . . . , ψ n }.
In fact, it is not needed that the matrix B is square and Hermitian, and the argument below remains true if we take for B any complex n × N -matrix (with any N ∈ N), satisfying the equation BB * = A.

Now we define the effective equation for eq. (2.11) as follows:

(4.5) da k -⟪P ⟫ k (a)dτ = n l=1 B kl dβ c l , k = 1, . . . , n.
Here the matrix B is as above and ⟪P ⟫ is the resonant averaging of the vector field P . We will usually consider this equation with the same initial condition as equations (2.6) and (2.11):

(4.6) a(0) = v 0 .
Since the vector field ⟪P ⟫ is locally Lipschitz and the dispersion matrix B is constant, then a strong solution of eq. (4.5), (4.6), if exists, is unique. Note that the effective dispersion B in (4.5) is a square root of an explicit matrix, and due to item 3) of Section 3.1 if the vector field P (v) is polynomial, then the effective drift ⟪P ⟫(a) also is given by an explicit formula. We recall that a measure

Q ∈ P(C([0, T ], C n )) is a weak solution of equation (4.5), (4.6) if Q = D(ã)
, where the random process ã(τ ), 0 ≤ τ ≤ T , is a weak solution of (4.5), (4.6). 5 A proof of this result is preceded by a number of lemmas. Till the end of this section we assume Assumption 2.1. As in Section 3 we denote

(4.7) Y (a; τ ε -1 ) ∶= (Φ τ ε -1 Λ ) * P (a).
Then equation (2.11) for a ε reads as

(4.8) da ε (τ ) -Y (a ε , τ ε -1 )dτ = Φ τ ε -1 Λ Ψdβ c (τ ). Denote (4.9) ỹ(a, τ ε -1 ) = Y (a, τ ε -1 ) -⟪P ⟫(a) = (Φ τ ε -1 Λ ) * P (a) -⟨⟨P ⟩⟩(a).
The following key lemma shows that integrals of ỹ(a ε , τ ε -1 ) with respect to τ become small with ε, uniformly in the segment of integrating.

Lemma 4.3. For a solution a ε (τ ) of equation (2.11), (2.12) we have

E max 0⩽τ ⩽T τ 0 ỹ(a ε (s), sε -1 )ds → 0 as ε → 0.
The lemma is proved in Subsection 4.1. Now let us introduce a natural filtered measurable space for the problem we consider,

(4.10) ( Ω, B, {B τ , 0 ⩽ τ ⩽ T }),
where Ω is the Banach space C([0, T ], C n ) = {a ∶= a(⋅)}, B is its Borel σ-algebra and B τ is the sigma-algebra, generated by the r.v.'s {a(s) ∶ 0 ⩽ s ⩽ τ )}. Consider the process on Ω, defined by the l.h.s. of (4.5)

(4.11) N ⟪P ⟫ (τ ; a) = a(τ ) - τ 0 ⟪P ⟫(a(s))ds, a ∈ Ω, τ ∈ [0, T ].
Note that for any 0

≤ τ ≤ T , N ⟪P ⟫ (τ ; ⋅) is a B τ -measurable continuous functional on C([0, T ]; C n ).
Lemma 4.4. The random process N ⟪P ⟫ (τ ; a) is a martingale on the space (4.10) with respect to the limiting measure Q 0 in (2.14).

Proof. Let us fix a τ ∈ [0, T ] and take a B τ -measurable function f τ ∈ C b ( Ω). We will show that (4.12)

E Q 0 N ⟪P ⟫ (t; a)f τ (a) = E Q 0 N ⟪P ⟫ (τ ; a)f τ (a) for any τ ⩽ t ⩽ T,
which would imply the assertion. To establish this let us first consider the process

N Y,ε (τ ; a ε ) ∶= a ε (τ ) - τ 0 Y (a ε , sε -1 )ds,
who is a martingale in view of (4.8). As

N Y,ε (τ ; a ε ) -N ⟪P ⟫ (τ ; a ε ) = τ 0 ⟪P ⟫(a ε (s)) -Y (a ε (s), sε -1 ) ds,
then by Lemma 4.3, Here and below in this proof o ε (1) stands for a quantity which goes to zero with ε. Since N Y,ε is a martingale, then by this relation

(4.13) max 0≤τ ≤T E N Y,ε (τ ; a ε ) -N ⟪P ⟫ (τ ; a ε ) = o ε (1).
E N ⟪P ⟫ (t; a ε )f τ (a ε ) + o ε (1) = E N Y,ε (t; a ε )f τ (a ε ) = E N Y,ε (τ ; a ε )f τ (a ε ) = E N ⟪P ⟫ (τ ; a ε )f τ (a ε ) + o ε (1). So (4.14) E Qε N ⟪P ⟫ (t; a)f τ (a) -N ⟪P ⟫ (τ ; a)f τ (a) = o ε (1).
To get (4.12) we will pass in this relation to a limit as ε → 0. For this end for M > 0 consider the function

G M (t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ t, if t ≤ M, M sgn t, otherwise.
Since by Assumption 2.1 and Lemma 3.2

E Qε sup τ ∈[0,T ] N ⟪P ⟫ (τ ; a) 2 ⩽ E Qε C P (1 + sup τ ∈[0,T ] a(τ ) 2(m 0 ∨1) ) ⩽ C P,m 0 ( v 0 ),
then for any ε we have

(4.15) E Qε (1 -G M ) ○ N ⟪P ⟫ (t; a)f τ (a) -N ⟪P ⟫ (τ ; a)f τ (a) ≤ CM -1 .
As Q ε ′ l ⇀ Q 0 , then by Fatou lemma this estimate stays true for ε = 0. Relations (4.14) and (4.15) show that

E Qε G M ○ (N ⟪P ⟫ (t; a)f τ (a) -N ⟪P ⟫ (τ ; a)f τ (a)) = o ε (1) + o M -1 (1).
From this and convergence (2.14) we derive the relation

E Q 0 G M ○ (N ⟪P ⟫ (t; a)f τ (a) -N ⟪P ⟫ (τ ; a)f τ (a)) = o M -1 (1),
which jointly with estimate (4.15) ε=0 imply (4.12) when we send M to ∞. The lemma is proved. Definition 4.5. A measure Q on the space (4.10) is called a solution of the martingale problem for effective equation (4.5) with the initial condition (4.6) if a(0) = v 0 Q-a.s. and 1) the process

{N ⟪P ⟫ (τ ; a) ∈ C n , τ ∈ [0, T ]} (see (4.11)
) is a vector-martingale on the filtered space (4.10) with respect to the measure Q;

2) for any k, j = 1, . . . , n the process (4. [START_REF] Khasminski | On the averaging principle for Ito stochastic differential equations[END_REF])

N ⟪P ⟫ k (τ ; a)N ⟪P ⟫ j (τ ; a) -2 τ 0 (BB * ) kj ds, τ ∈ [0, T ]
(where BB * = A) is a martingale on the space (4.10) with respect to the measure Q, as well as the process

N ⟪P ⟫ k (τ ; a)N ⟪P ⟫ j (τ ; a). 6
This is a classical definition, written in complex coordinates. See [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF] and [14, Section 5.4], where we profited from [14, Remark 4.12 ] and the result of [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]Problem 4.13] since by Lemma 3.2 the vector field ⟪P ⟫ in (4.5) is locally Lipschitz.

We have Lemma 4.6. The limiting measure Q 0 in (2.14) is a solution of the martingale problem for effective equation (4.5), (4.6). 6 That is, ⟨N

⟪P ⟫ k (τ ; a), N ⟪P ⟫ j (τ ; a)⟩(τ ) = 2 ∫ τ 0 (BB * ) kj and ⟨N ⟪P ⟫ k (τ ; a), N ⟪P ⟫ j (τ ; a)⟩(τ ) = 0. See Appen- dix B.
Proof. Since condition 1) in Definition 4.5 has been verified in Lemma 4.4, it remains to check condition 2). For the second term in (4.16) we have, as ε → 0, (4.17)

τ 0 ψ ε (s)(ψ ε (s)) * kj ds = τ 0 e iε -1 (λ k -λ j )s (ψψ * ) kj ds → τ A kj ,
where the matrix (A kj ) is given by (4.3). Let us pass to the first term. Since by (2.11) and (4.1)

N Y,ε (τ ) = v 0 + τ 0 ψ ε (s)dβ c (s), ψ ε lj (s) = e isε -1
λ l Ψ lj , then by the complex Itô formula (see Appendix C) and Assumption 2.1, for any k, j ∈ {1, . . . , n} the process

(4.18) N Y,ε k (τ )N Y,ε j (τ ) -2 τ 0 (ψ ε (s)(ψ ε (s)) * ) kj ds,
is a martingale. As when verifying 1), we will compare (4.16) with (4.18). To do this let us consider

N ⟪P ⟫ k (τ ; a ε )N ⟪P ⟫ j (τ ; a ε ) -N Y,ε k (τ ; a ε )N Y,ε j (τ ; a ε ) = a ε k (τ ) - τ 0 ⟪P ⟫ k (a ε (s)) ds āε j (τ ) - τ 0 ⟪P ⟫ j (a ε (s)) ds -a ε k (τ ) - τ 0 Y k (a ε (s), sε -1 ) ds āε j (τ ) - τ 0 Ȳj (a ε (s), sε -1 ) ds =∶ M kj (a ε , τ ).
Closely repeating the proof of (4.13) we get that

sup 0≤τ ≤T E M kj (a ε ; τ ) = o ε (1) as ε → 0.
Since (4.18) is a martingale, then this relation and (4.17) imply that (4. [START_REF] Khasminski | On the averaging principle for Ito stochastic differential equations[END_REF]) is a martingale due to the same reasoning by which relations (4.13) and the fact that N Y,ε (τ ; a ε ) is a martingale imply that N ⟪P ⟫ (τ ; a) also is one. To pass to a limit as ε → 0 the proof uses that the random variables like N Y,ε k (τ ; a ε )N Y,ε j (τ ; a ε ) are integrable uniformly in ε > 0 due to Assumption 2.1, where m ′ 0 > m 0 . Similarly for any k and j, the process

N ⟪P ⟫ k (τ )N ⟪P ⟫ j
(τ ) also is a martingale. The lemma's assertion is established.

Now we can prove Proposition 4.2:

Proof. [of Proposition 4.2] It is well known that a solution of the martingale problem for a stochastic differential equation is its weak solution. Instead of referring to a corresponding theorem (see [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF] or [14, Section 5.4]), again following [START_REF] Khasminski | On the averaging principle for Ito stochastic differential equations[END_REF], we give a short direct proof, based on another strong result from the stochastic calculus. By Lemma 4.6 and the martingale representation theorem for complex processes (see Appendix B), we know that there exists an extension ( Ω, B, P) of the probability space ( Ω, B, Q 0 ) and on it exist standard independent complex Wiener processes β c 1 (τ ), . . . , β c n (τ ) such that

da j (τ ) -⟪P ⟫ j (a)dτ = n l=1 B jl dβ c l (τ ), j = 1, . . . , n,
where the dispersion B is a nonnegative Hermitian matrix, satisfying BB * = A. Therefore the measure Q 0 is a weak solution of effective equation (4.5). We thus proved the assertion of the proposition.

By Lemma 3.2, in effective equation (4.5) the drift term ⟪P ⟫ is locally Lipschitz. So its strong solution (if exists) is unique. By Proposition 4.2 the measure Q 0 is a weak solution of eq. (4.5). Hence, by the Yamada-Watanabe theorem [14, Section 5.3.D], [24, Chapter 8] a strong solution for the effective equation exists, and its weak solution is unique. Therefore the limit Q 0 = lim ε l →0 Q ε l does not depend on the sequence ε l → 0. So the convergence holds as ε → 0 and we thus have established Theorem 4.7. For any v 0 ∈ C n the solution a ε (τ ; v 0 ) of problem (2.11), (2.12) satisfies

(4.19) D(a ε (⋅; v 0 )) ⇀ Q 0 in P(C([0, T ], C n )) as ε → 0,
where Q 0 is the law of a unique weak solution a 0 (τ ; v 0 ) of effective equation (4.5), (4.6).

Remark 4.8. 1) A straightforward analysis of the theorem's proof shows that it goes without changes if a ε (τ ) solves eq. (2.11) with an initial data v ε0 which converges to v 0 when ε → 0. So

(4.20) D(a ε (⋅; v ε0 )) ⇀ Q 0 in P(C([0, T ], C n )) as ε → 0, if v ε0 → v 0 when ε → 0.
2) Denoting as before D(a ε (⋅;

v 0 )) = Q ε ∈ P(C([0, T ]; C n ))
we use the Skorokhod representation theorem (see [START_REF] Billingsley | Convergence of Probability Measures[END_REF]Section 6]) to find a sequence ε j → 0 and processes ξ j (τ ), 0 ≤ τ ≤ T , j = 0, 1, . . . , such that D(ξ 0 ) = Q 0 , D(ξ j ) = Q ε j , and ξ j → ξ 0 in C([0, T ]; C n ), a.s. Then (2.7) and Fatou's lemma imply that

(4.21) E a 0 2m ′ 0 C([0,T ];C n ) = E Q 0 a 2m ′ 0 C([0,T ];C n ) = E ξ 0 2m ′ 0 C([0,T ];C n ) ≤ C m ′ 0 ( v 0 , T
). The result of Theorem 4.7 admits an immediate generalisation to the case when the initial data v 0 in (2.12) is a random variable: Amplification 4.9. Let v 0 be a r.v., independent from the Wiener process β c (τ ). Then still the convergence (4.19) holds.

Proof. It suffices to establish (4.19) when a ε is a weak solution of the problem. Now, let (Ω ′ , F ′ , P ′ ) be another probability space and ξ ω ′ 0 be a r.v. on Ω ′ , distributed as v 0 . Then a εω (τ ; ξ ω ′ 0 ) is a weak solution of (2.11), (2.12), defined on the probability space Ω ′ × Ω. Take f to be a bounded continuous function on C([0, T ], C n ). Then by the theorem above, for each

ω ′ ∈ Ω ′ lim ε→0 E Ω f (a εω (⋅; ξ ω ′ 0 )) = E Ω f (a 0ω (⋅; ξ ω ′ 0 )
). Since f is bounded, then by the Lebesgue dominated convergence theorem we have

lim ε→0 Ef (a ε (⋅; v 0 )) = lim ε→0 E Ω ′ E Ω f (a εω (⋅; ξ ω ′ 0 )) = E Ω ′ E Ω f (a 0ω (⋅; ξ ω ′ 0 )) = Ef (a 0 (⋅; v 0 )).
This implies the required convergence (4.19).

The convergence, stated in the last amplification, holds uniformly in the class of random initial data v 0 , a.s. bounded by a fixed constant. To state the result we have to introduce a distance in the space of measures. Definition 4.10. Let M be a Polish (i.e. a complete and separable) metric space. For any two measures µ 1 , µ 2 ∈ P(M ) we define the dual-Lipschitz distance between them as

µ 1 -µ 2 * L,M ∶= sup f ∈C b (M ), f L ⩽1 ⟨f, µ 1 ⟩ -⟨f, µ 2 ⟩ ≤ 2,
where

f L = f L,M = Lip f + f C(M ) .
In the definition and below we denote

Π t ∶ C([0, T ], C n ) → C n , a(⋅) ↦ a(t), 0 ≤ t ≤ T. Noting that f ○ Π t L,C n ≤ f L,C([0,T ],C n ) for each t we get that (4.23) Π t ○ µ 1 -Π t ○ µ 2 * L,C n ≤ µ 1 -µ 2 * L,C([0,T ],C n ) for all µ 1 , µ 2 ∈ P(C([0, T ], C n ))
and all 0 ≤ t ≤ T (where Π t ○ µ j ∈ P(C n ) stands for the image of µ j under the mapping Π t ).

The dual-Lipschitz distance converts P(M ) to a complete metric space and induces on it a topology, equivalent to the weak convergence of measures, e.g. see [START_REF] Dudley | Cambridge studies in advanced mathematics[END_REF]Section 11.3] and [5, Section 1.7]). Proposition 4.12. Under the assumption of Amplification 4.9 let the r.v. v 0 be such that v 0 ≤ R a.s., for some R > 0. Then the rate of convergence in (4. [START_REF] Kuksin | Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations[END_REF]) with respect to the dual-Lipschitz distance depends only on R.

Proof. The proof of Amplification 4.9 shows that it suffices to verify that for a non-random initial data v 0 ∈ BR (C n ) the rate of convergence in (4. [START_REF] Kuksin | Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations[END_REF]) depends only on R. Assume the opposite. Then there exist a δ > 0, a sequence ε j → 0 and vectors

v j ∈ BR (C n ) such that (4.24) D(a ε j (⋅; v j )) -D(a 0 (⋅; v j )) * L,C([0,T ],C n ) ⩾ δ.
By the same argument as in the proof of Lemma 2.2, we know that the two sets of probability measures {D(a ε j (⋅; v j ))} and {D(a 0 (⋅; v j ))} are pre-compact in C([0, T ], C n ). Therefore, there exists a sequence k j → ∞ such that ε k j → 0, v k j → v 0 and D(a

ε k j (⋅; v k j )) ⇀ Q0 , D(a 0 (⋅; v k j )) ⇀ Q 0 in P(C([0, T ], C n )). Then (4.25) Q0 -Q 0 * L,C([0,T ],C n ) ⩾ δ.
Since in the well-posed eq. (4.5) the drift and dispersion are locally Lipschitz, then the law D(a 0 (⋅; v ′ )) is continuous with respect to the initial condition v ′ . 7 Therefore Q 0 is the unique weak solution of the effective equation (4.5) with initial condition a 0 (0) = v 0 . By (4.20) the measure Q0 also is a weak solution of the problem (4.5), (4.6). Hence, Q 0 = Q0 . This contradicts (4.25) and proves the assertion.

We continue with an obvious application of Theorem 4.7 to solutions v ε (τ ; v 0 ) of the original eq. (2.6). Consider the action-mapping (z 1 , . . . , z n ) ↦ (I 1 , . . . , I n ) =∶ I (see (1.2)). Since the interaction representation (2.8) does not change the actions, then we have from the theorem that Corollary 4.13. For any v 0 , as ε → 0 (4.26)

D(I v ε (⋅; v 0 )) ⇀ I ○ D(a(⋅; v 0 )) in P(C([0, T ], R n + ))
, where a(⋅; v 0 ) is a unique weak solution of effective equation (4.5), (4.6).

Example 4.14. If the drift P in (2.6) is globally Lipschitz, that is, Lip(P ) ⩽ M for some M > 0, then it is not hard to see that Assumption 2.1 holds, so Theorem 4.7 and Corollary 4.13 apply. A more interesting example is discussed below in Section 9.

4.1. Proof of Lemma 4.3. In this subsection we denote by H k (r; c 1 , . . . ), k = 1, 2, . . . , non-negative functions of r > 0 which go to zero with r and depend on parameters c 1 , . . . (dependence of H k 's on T and P is not indicated). Also, for an event Q we denote

E Q f (ξ) = E1 Q f (ξ).
For any M 1 ⩾ 1 we set

E 1 = E 1ε M 1 = {ω ∈ Ω ∶ sup 0⩽τ ⩽T a ε (τ ) ⩽ M 1 }.
By Assumption 2.1 and Chebyshev's inequality,

P(Ω ∖ E 1 ) ⩽ H 1 (M 1 -1 ; v 0 ).
Recalling that ỹ was defined in (4.9), by Lemma 3.2 we have that

ỹ(a ε (s), sε -1 ) ⩽ Y (a ε (s), sε -1 ) + ⟪P ⟫(a ε (s)) ⩽ 2C m 0 (P ) a ε (s) m 0 .
So, abbreviating ỹ(a ε (s), sε -1 ) to ỹ(s), in view of (2.7) we have: (1.4)). Then by (2.13) P(Ω ∖ E 2 ) ⩽ H 3 (M -1 2 ; v 0 ). Therefore,

E Ω∖E 1 max τ ∈[0,T ] τ 0 ỹ(s)ds ⩽ T 0 E 1 Ω∖E 1 ỹ(s) ds ⩽ 2C m 0 (P ) P (Ω ∖ E 1 ) 1 2 ( T 0 E a ε (s) 2m 0 ds) 1 2 ⩽ 2C m 0 (P )(H 1 (M 1 -1 ; v 0 )) 1 2 =∶ H 2 (M 1 -1 ; v 0 ).

Now we should estimate

E E 1 max τ ∈[0,T ] ∫ τ 0 ỹ(s)ds . For an M 2 ⩾ 1 consider the event E 2 = E 2ε M 2 = {ω ∈ Ω ∶ a ε 1 3 ⩽ M 2 } (see
E Ω∖E 2 max τ ∈[0,T ] τ 0 ỹ(s)ds ⩽ P(Ω ∖ E 2 ) 1 2 (C P T 0 E a ε (s) 2m 0 ds) 1 2 ⩽ H 4 (M -1 2 ; v 0 ) . It remains to bound E E 1 ∩E 2 max τ ∈[0,T ] ∫ τ 0 ỹ(s)ds . We set N = [T √ ε] + 1, L = T N. Then C -1 √ ε ⩽ L ⩽ C √ ε and c -1 √ ε ⩽ N ⩽ c √ ε
for some constants C and c. We consider a partition of interval [0, T ] by points τ l = lL, l = 0, . . . , N , and denote

η l = τ l+1 τ l ỹ(s)ds, l = 0, . . . , N -1. For any τ ∈ [0, T ] let us find l = l(τ ) such that τ ∈ [τ l , τ l+1 ]. Then τ 0 ỹ(s)ds ⩽ η 1 + ⋅ ⋅ ⋅ + η l + τ τ l ỹ(s)ds . If ω ∈ E 1 , then ∫ τ τ l ỹ(s)ds ⩽ 2C m 0 (P )M 1 m 0 L. Therefore (4.27) E E 1 ∩E 2 max 0⩽τ ⩽T τ 0 ỹ(s)ds ⩽ 2C m 0 (P )M 1 m 0 L + N -1 l=0 E E 1 ∩E 2 η l ,
and it remains to estimate the integrals E E 1 ∩E 2 η l for l = 0, . . . , N -1. Observe that

η l ⩽ τ l+1 τ l [ỹ(a ε (s), sε -1 ) -ỹ(a ε (τ l ), sε -1 )]ds + τ l+1 τ l ỹ(a ε (τ l ), sε -1 )ds =∶ U 1 l + U 2 l . Since ỹ(a ε , τ ε -1 ) = (Φ τ ε -1 Λ ) * P (a ε )-⟪P ⟫(a ε ) and P, ⟪P ⟫ ∈ Lip m 0 (C n , C n ), then for ω ∈ E 1 ∩E 2 the integrand in U 1 l is bounded by 2C m 0 (P )M 1 m 0 sup τ l ⩽s⩽τ l+1 a ε (s) -a ε (τ l ) ⩽ 2C m 0 (P )M 1 m 0 M 2 L 1 3 . So U 1 l ⩽ 2C m 0 (P )M 1 m 0 M 2 L 4 3 . Now consider integral U 2 l .
By the definition of ỹ(a ε , τ ε -1 ), we have

U 2 l = τ l+1 τ l Y (a ε (τ l ), sε -1 )ds -L⟪P ⟫(a ε (τ l ) =∶ Z 1 + Z 2 .
For integral Z 1 , making the change of variable s = τ l + εx with x ∈ [0, L ε ], we have

Z 1 = ε L ε 0 Y (a ε (τ l ), τ l ε -1 + x)dx. Since Y (a ε (τ l ), τ l ε -1 + x) = Φ τ l ε -1 Λ ○ Φ xΛ P Φ -xΛ (Φ -τ l ε -1 Λ a ε (τ l )) , then Z 1 = LΦ τ l ε -1 Λ ε L L ε 0 Φ xΛ P Φ -xΛ Φ -τ l ε -1 Λ a ε (τ l ) dx = LΦ τ l ε -1 Λ ⟪P ⟫ L ε Φ -τ l ε -1 Λ a ε (τ l ) (see (3.1)). As L ε ∼ ε -1 2 ≫ 1 and Φ -τ l ε -1 Λ a ε (τ l ) = a ε (τ l ) ⩽ M 1 for ω ∈ E 1 ∩ E 2 ,
then by Lemma 3.2 the partial averaging ⟪P ⟫ L ε is close to the complete averaging ⟪P ⟫. Thus

Z 1 -LΦ τ l ε -1 Λ ⟪P ⟫ Φ -τ l ε -1 Λ (a ε (τ l )) = Z 1 -L(Φ τ l ε -1 Λ ) * ⟪P ⟫(a ε (τ l )) = Z 1 -L⟪P ⟫(a ε (τ l )) ≤ LH 5 ( √ ε; M 1 , v 0 ),
where we used Lemma 3.3 to get the second equality. Since Z 2 = -L⟪P ⟫(a ε (τ l )), then

U 2 l = Z 1 + Z 2 ≤ LH 5 ( √ ε; M 1 , v 0 ).
We then have obtained that

E E 1 ∩E 2 η l ⩽ 2C m 0 (P )M 1 m 0 M 2 L 4 3 + LH 5 ( √ ε; M 1 , v 0 ).
Together with (4.27) it gives us that

E E 1 ∩E 2 max 0⩽τ ≤T τ 0 ỹ(s)ds ⩽ 2C m 0 (P )M 1 m 0 L + 2C m 0 (P )M 1 m 0 M 2 L 1 3 + H 5 ( √ ε; M 1 , v 0 ). Therefore E max 0⩽τ ⩽T τ 0 ỹ(s)ds ⩽ H 2 (M 1 -1 ; v 0 ) + H 4 (M -1 2 ; v 0 ) + 2C m 0 (P )M 1 m 0 (M 2 + 1)ε 1 6 + H 5 ( √ ε; M 1 , v 0 ).
Now for any δ > 0, we perform the following procedure:

(1) choose M 1 so large that

H 2 (M 1 -1 ; v 0 ) ⩽ δ; (2) choose M 2 so large that H 4 (M -1 2 ; v 0 ) ⩽ δ; (3) finally, choose ε δ > 0 so small that 2C m 0 (P )M 1 m 0 (M 2 + 1)ε 1 3 + H 5 ( √ ε; M 1 , v 0 ) ⩽ δ if 0 < ε ⩽ ε δ .
We have seen that for any δ > 0,

E max 0⩽τ ⩽T ∫ τ 0 ỹ(a ε (s), sε -1 )ds ≤ 3δ if ε ≤ ε δ . So E max 0⩽τ ⩽T τ 0
[Y (a ε (s), sε -1 ) -⟪P ⟫(a ε (s))]ds → 0 as ε → 0, which completes the proof of Lemma 4.3.

Stationary solutions and mixing

In this section we study relation between stationary solutions of equation (2.6) and of the effective equation. We recall that a solution a(τ ), τ ⩾ 0, of equation (2.6) (or of effective equation (4.5)) is stationary if D(a(τ )) ≡ µ for all τ ⩾ 0 and some measure µ ∈ P(C n ), called a stationary measure for the corresponding equation.

Through this section we assume that equation (2.6) satisfies the following strengthening of Assumption 2.1: Assumption 5.1.

(1) The drift P (v) is a locally Lipschitz vector filed, belonging to Lip m 0 (C n , C n ) for some m 0 ∈ N.

(2) For any v 0 ∈ C n equation (2.6) has a unique strong solution v ε (τ ; v 0 ), τ ≥ 0, equal v 0 at τ = 0. There exists m ′ 0 > (m 0 ∨ 1) such that (5.1) E sup

T ′ ⩽τ ⩽T ′ +1 v ε (τ ; v 0 ) 2m ′ 0 ⩽ C m ′ 0 ( v 0 )
for every T ′ ⩾ 0 and ε ∈ (0, 1], where

C m ′ 0 (⋅) is a continuous non-decreasing function. (3) Equation (2.6) is mixing. That is, it has a stationary solution v ε st (τ ), D(v ε st (τ )) ≡ µ ε ∈ P(C n ), and (5.2) D(v ε (τ ; v 0 )) ⇀ µ ε in P(C n ) as τ → +∞,
for every v 0 .

Under Assumption 5.1 equation (2.6) defines in C n a mixing Markov process with the transition probability Σ τ (v) ∈ P(C n ), τ ≥ 0, v ∈ C n , where Σ τ (v) = Dv ε (τ ; v); e.g. see [14, Section 5.4.C]. Let us denote by X the complete separable metric space X = C([0, ∞), C n ) with the distance

(5.3) dist(a 1 , a 2 ) = ∞ N =1 2 -N a 1 -a 2 C([0,N ],C n ) 1 + a 1 -a 2 C([0,N ],C n ) , a 1 , a 2 ∈ X ,
and consider on X continuous function g(a) = sup 0≤t≤1 a(t) 2m ′ 0 . Denoting µ τ ε = D(v ε (τ ; 0)) we have by the Markov property that (5.4) E sup

T ′ ⩽τ ⩽T ′ +1 v ε (τ ; 0) m = C n Eg(v ε (⋅; v 0 ))µ T ′ ε (dv 0 ), and
(5.5) E sup 0⩽τ ⩽1 v ε st (τ ; 0) m = C n Eg(v ε (⋅; v 0 ))µ ε (dv 0 ).
The l.h.s. of (5.4) is estimated in (5.1). To estimate (5.5) we will pass in the r.h.s. of (5.4) to the limit as T ′ → ∞, using (5.2). To do that we start with a lemma Lemma 5.2. Let n 1 , n 2 ∈ N, B ⊂ R n 1 is a closed convex set which contains more than one point, and F ∶ B → R n 2 is a Lipschitz mapping. Then F may be extended to a map

F ∶ R n 1 → R n 2 in such a way that a) Lip F = Lip F , b) F (R n 1 ) = F (B).
Proof. Let Π ∶ R n 1 → B be the projection, sending any point of R n 1 to a nearest point in B.

Then Lip Π =1, see Appendix D, and obviously F = F ○ Π is a required extension of F .

Since C m 0 (P ) =∶ C * < ∞, then for any M ∈ N the restriction of P to BM (C n ) and its Lipschitz constant both are bounded by (1 + M ) m 0 C * . In view of the lemma above we may extend P BM (C n ) to a Lipschitz mapping

P M ∶ C n → C n such that Lip (P M ) ≤ (1 + M ) m 0 C * , sup P M (v) ≤ (1 + M ) m 0 C * .
Considering for a solution v(τ ) of equation (2.6) the stopping time τ M = inf{t ≥ 0 ∶ v(t) ≥ M } and denoting by v εM the stopped solution v εM (τ ; v 0 ) = v ε (τ ∧τ M ; v 0 ), we note that the process v εM will not change if in (2.6) we replace P (v) by P M (v). So v εM (τ ; v 0 ) is a stopped solution of a stochastic equation with Lipschitz coefficients, and thus a.s. the curve v εM ω (⋅; v 0 ) ∈ X continuously depends on v 0 , for each M ∈ N. As g(v εM ) ≤ g(v ε ) a.s., then in view of (5.2) and (5.1), for every M and any N > 0,

E(N ∧ g)(v εM (⋅; v)) µ ε (dv) = lim T ′ →∞ E(N ∧ g)(v εM (⋅; v)) µ T ′ ε (dv) ≤ C m (0)
(to get the last inequality from (5.1) we used the Markov property). Passing in the l.h.s. to the limit as N → ∞ using the monotone convergence theorem we see that

(5.6) Eg(v εM (⋅; v)) µ ε (dv) ≤ C m (0).
Since for every v a.s. g(v εM (⋅; v)) ↗ g(v ε (⋅; v)) ≤ ∞ as M → ∞, then using again the latter theorem we derive from (5.6) that

Eg(v ε (⋅; v)) µ ε (dv) ≤ C m (0).

Evoking (5.5) we get

Lemma 5.3. The stationary solution v ε st (τ ) satisfies estimate (5.1) with C m ′ 0 ( v 0 ) replaced by C m ′ 0 (0). Let us consider the interaction representation for v ε st , v ε st (τ ) = Φ -τ ε -1 Λ a ε (τ ) (note that a ε is not a stationary process!). Then a ε (τ ) satisfies equation (2.11), so for any N ∈ N the collection of measures {D(a ε [0,N ] ), 0 < ε ⩽ 1} is tight in view of (5.1) (for the same reason as in Section 2.4). We choose a sequence ε l → 0 (depending on N ) such that

D(a ε l [0,N ] ) ⇀ Q 0 in P C([0, N ], C n ) .
Applying the diagonal process and replacing {ε l } by a subsequence, which we still denote {ε l }, we achieve that Da

ε l ⇀ Q 0 in P(X) (see (5.3)). Since a ε (0) = v ε st (0), then µ ε l ⇀ µ 0 ∶= Q 0 τ =0 .
Let a 0 (τ ) be a process in

C n such that D(a 0 ) = Q 0 . Then (5.7) D(a ε l (τ )) ⇀ D(a 0 (τ )) ∀ 0 ≤ τ < ∞.
In particular, D(a 0 (0)) = µ 0 .

Proposition 5.4.

(1) The limiting process a 0 is a stationary weak solution of effective equation (4.5) and D(a 0 (τ )) ≡ µ 0 , τ ∈ [0, ∞). In particular, the limiting points of the collection of stationary measures {µ ε , ε ∈ (0, 1]} as ε → 0 are stationary measures of the effective equation.

(2) Any limiting measure µ 0 is invariant under operators Φ θΛ , θ ∈ R. So D(Φ θΛ a 0 (τ )) = µ 0 for all θ ∈ R and τ ∈ [0, ∞).

Proof.

(1) Using Lemma 5.3 and repeating the argument in the proof of Proposition 4.2 we obtain that a 0 is a weak solution of the effective equation. It remains to prove its stationarity. Take any bounded Lipschitz function f on C n and consider

E 1 0 f (v ε l st (τ ))dτ = E 1 0 f (Φ -τ ε l -1 Λ a ε l (τ ))dτ.
Using the same argument as in the proof of Lemma 4.3 (but applied to averaging of functions rather than that of vector-fields), we obtain that

E 1 0 f v ε l st (τ ) dτ -E 1 0 ⟨f ⟩ a ε l (τ ) dτ = E 1 0 f Φ -τ ε -1 l Λ a ε l (τ )) -⟨f ⟩(a ε l (τ ) dτ → 0 as ε l → 0.
(5.8)

By Lemma 3.6, ⟨f ⟩ a ε l (τ ) = ⟨f ⟩ Φ τ ε -1 l Λ v ε l st (τ ) = ⟨f ⟩ v ε l st (τ ) for every τ . Since the process v ε l st (τ ) is stationary, then Ef (v ε l st (τ )) = Const and E⟨f ⟩ a ε l (τ ) = E⟨f ⟩(v ε l st (τ )) = Const ′ . So we get from (5.8) that (5.9) Ef v ε l st (τ ) -E⟨f ⟩ a ε l (τ ) → 0 as ε l → 0 ∀ τ.
For any τ let us consider fτ = f ○ Φ τ ε -1 l Λ . Then f (a ε l (τ )) = fτ (v ε l st (τ )). Since ⟨f ⟩ = ⟨ fτ ⟩ by Lemma 3.6.( 4), then applying (5.9) to fτ we get:

lim ε l →0 Ef (a ε l (τ )) = lim ε l →0 E fτ (v ε l st (τ )) = lim ε l →0 E⟨ fτ ⟩(a ε l (τ )) = lim ε l →0
E⟨f ⟩(a ε l (τ )).

From this relation, (5.9) and (5.7) we find that Ef (a 0 (τ )) = ∫ f (v) µ 0 (dv) for each τ and every f as above. This implies the first assertion of the lemma.

(2) Passing to the limit in (5.9) using (5.7) we have

f (v)µ 0 (dv) = E⟨f ⟩(a 0 (τ )) ∀ τ.
Using this relation with f ∶= f ○ Φ θΛ and next with f ∶= f we get that

f ○ Φ θΛ (v)µ 0 (dv) = E⟨f ○ Φ θΛ ⟩(a 0 (τ )) = E⟨f ⟩(a 0 (τ )) = f (v)µ 0 (dv),
for any θ ∈ R and any τ , for every bounded Lipschitz function f . This implies the second assertion.

If effective equation (4.5) is mixing, then it has a unique stationary measure. In this case the measure µ 0 in Proposition 5.4 does not depend on a choice of the sequence ε l → 0, and so µ ε ⇀ µ 0 as ε → 0. Therefore, we have Theorem 5.5. If in addition to Assumption 5.1 we assume that the effective equation is mixing and µ 0 is its unique stationary measure, then

µ ε ⇀ µ 0 in P(C n ) as ε → 0.
Moreover, the measure µ 0 is invariant for all operators Φ θΛ , and the law of the stationary solution of equation (2.6), written in the interaction presentation, converges to the law of the stationary solution of effective equation (4.5).

We recall that Theorem 4.7 and Corollary 4.13 only assure that on finite time-intervals τ ∈ [0, T ] the actions of solutions for eq. (2.6), as ε → 0, converge in law to the actions of solutions for the effective equation with the same initial data. In difference, when ε → 0 all of the stationary measure for eq. (2.6) converges to that for the effective equation. This important fact was first observed in [START_REF] Dymov | Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators[END_REF] for a special class of equations (2.6).

Corollary 5.6. Under the assumption of Theorem 5.5 , for any v 0 ∈ C n we have

lim ε→0 lim τ →∞ D(v ε (τ ; v 0 )) = µ 0 .
Proof. Since lim τ →∞ D(v ε (τ )) = µ ε , then the result follows from Theorem 5.5.

Remark 5.7. Let us decomplexify C n to R 2n and write the effective equation in the real coordinates {x = (x 1 , . . . , x 2n )}:

dx j (τ ) -⟪P ⟫ j (x)dτ = 2n l=1 B jl dW l (τ ), j = 1, . . . , 2n,
where W l are independent standard real Wiener processes. Then the stationary measure µ 0 ∈ P(R 2n ) satisfies the stationary Fokker-Plank equation

1 2 2n l=1 2n j=1 ∂ 2 ∂x l ∂x j (B lj µ 0 ) = 2n l=1 ∂ ∂x l (⟪P ⟫ l (x)µ 0 )
in the sense of distributions. If the dispersion matrix Ψ is non-degenerate, then the diffusion B also is, and since the drift ⟪P ⟫(x) is locally Lipschitz, then by the standard theory of the Fokker-Plank equation µ 0 = ϕ(x)dx, where ϕ ∈ C 1 (R 2n ).8

The non-resonant case

Assume that the frequency vector Λ = (λ 1 , . . . , λ n ) is non-resonant (see (3.7)). In Section 3.1.2) we saw that in this case vector field ⟪P ⟫ may be calculated via the averaging (3.8) and commutes with all rotations Φ w , w ∈ R n . For any j ∈ {1, . . . , n} let us denote w j,t ∶= (0, . . . 0, t, 0, . . . , 0) (only the j-th entry is non-zero). Consider ⟪P ⟫ 1 (z) and write it as ⟪P ⟫ 1 (z) = z 1 R 1 (z 1 , . . . , z n ), for some complex function R 1 . Since for w = w 1,t we have Φ w (z) = (e it z 1 , z 2 , . . . , z n ), then now the first component in relation (3.10) reads

e -it z 1 R 1 (e -it z, z 1 , . . . , z n ) = e -it z 1 R 1 (z 1 , . . . , z n ), for every t. So R 1 (e -it z 1 , z 2 , . . . , z n ) ≡ R 1 (z 1 , . . . , z n )
and R 1 (z 1 , . . . , z n ) depends not on z 1 , but only on z 1 . Similarly we verify that R 1 (z 1 , . . . , z n ) depends only on z 2 , . . . , z n . Therefore ⟪P ⟫ 1 (z) = z 1 R 1 ( z 1 , . . . , z n ). Same is true for any ⟪P ⟫ j (z). We then obtain the following statement: Proposition 6.1. If Assumption 2.1 holds and the frequency vector Λ is non-resonant, then ⟪P ⟫ satisfies (3.8), and

(1) ⟪P ⟫ j (a) = a j R j ( a 1 , . . . , a n ), j = 1, . . . , n.

(2) The effective equation reads (6.1) da j (τ )a j R j ( a 1 , . . . , a n )dτ = b j dβ c j (τ ), j = 1, . . . , n, where b j = (∑ n l=1 Ψ jl 2 ) 1 2 (and a ↦ (a 1 R 1 , . . . , a n R n ) is a locally Lipschitz vectorfield).

(3) If a(τ ) is a solution of (6.1), then the vector of its actions

I(τ ) = (I 1 , . . . , I n )(τ ) ∈ R n + is a weak solution of equation (6.2) dI j (τ ) -2I j (RR j ) 2I 1 , . . . , 2I n dτ -b 2 j dτ = b j 2I j dW j (τ ), I j (0) = 1 2 v 0j 2 , j = 1, . . . , n
, where {W j } are independent standard real Wiener processes. (4) If in addition the assumptions of Theorem 5.5 are met and the matrix Ψ is nondegenerate, then the stationary measure µ 0 reads dµ 0 = p(I)dI dϕ, where p is a continuous function on R n + , C 1 -smooth outside the boundary ∂R n + . Proof. (1) is proved above, and (2) follows from it and (4.4).

(3) Writing the diffusion in effective equation (4.5) as in eq. ( 6.1) and applying Itô's formula (see Appendix C) to I j = 1 2 a j 2 we get that

(6.3) dI j (τ ) -1 2 āj ⟨⟨P ⟩⟩ j + a j ⟨⟨P ⟩⟩ j dτ -b 2 j dτ = b j ⟨a j (τ ), dβ j (τ )⟩ =∶ b j a j dξ j (τ )
, where dξ j (τ ) = ⟨a j a j , dβ j (τ )⟩ (see (2.2)) and for a j = 0 we define a j a j as 1. Using [START_REF] Arnold | Mathematical Aspects of Classical and Celestial Mechanics[END_REF] we see that the l.h.s. in (6.3) is the same as that in (6.2). Since a j a j (τ ) ≡ 1 for each j, then by Lévy's theorem (e.g. see [14, p. 157]), ξ(τ ) = (ξ 1 , . . . , ξ n )(τ ) is a standard n-dimensional Wiener process and (3) follows.

(4) By Theorem 5.5 the stationary measure µ 0 is invariant for all operators Φ θΛ , θ ∈ R. Since the curve θ ↦ θΛ ∈ T n is dense in T n , then µ 0 is invariant for all operators Φ w , w ∈ T n . As the matrix Ψ is non-degenerate, then by Remark 5.7 we have dµ 0 = p(z)dz, where p is a C 1 -function (dz stands for the volume element in C n ≃ R 2n ). Let us write z j = 2I j e iϕ j . Then dµ 0 = p(I, ϕ)dz. In the coordinates (I, ϕ) the operators Φ w reads as (I, ϕ) ↦ (I, ϕ+w). Since µ 0 is invariant for all of them, then p does not depend on ϕ. So dµ 0 = p(I)dz = p(I) dI dϕ and (4) holds.

By assertion (3) of the proposition, in the non-resonant case eq. ( 6.2) describes asymptotic as ε → 0 behaviour of actions I j of solutions for eq. (2.6). But how regular is that equation? Let us denote by r j = a j = 2I j , 1 ≤ j ≤ n, the radii of components of a vector a ∈ C n , consider the smooth polar-coordinates mapping R n + × T n → C n , (r, ϕ) ↦ (r 1 e iϕ 1 , . . . , r n e iϕn ), and extend it to a mapping Φ ∶ R n × T n → C n , defined by the same formula. A j-th component of the drift in eq. (6.2), written in the form (6.3) without the Itô term b 2 j dτ , is R a j ⟪P ⟫ j (a) . Due to (3.8), in the polar coordinates we write this as 1 (2π) n T n R r j e iϕ j e iw j P j (r, ϕ -

w) dw = 1 (2π) n T n R r j e iθ j Pj (r, θ) dθ =∶ F j (r), r ∈ R n ,
where F j (r) is a continuous function, vanishing with r j . Since the integrand in the second integral will not change if for some l = 1, . . . , n we replace r l by -r l and replace θ l by θ l + π, then F j (r) is even in each its argument r l . So it may be written as

F j (r 1 , . . . , r n ) = f j (r 2 1 , . . . , r 2 n ), f j ∈ C(R n + )
, where f j (x 1 , . . . , x n ) vanishes with x j . Now assume that the vector field P is C 2 -smooth. In this case the integrand in the integral for

F j is C 2 -smooth in (r, θ) ∈ R n × T n and F j is a C 2 - smooth function of r.
Then by a result of H. Whitney (see in [START_REF] Whitney | Differentiable even functions[END_REF] Theorem 1 with s = 1 and the remark, concluding the paper), f j extends to R n in such a way that f j (x) is C 1 -smooth in each its argument x l and (∂ ∂x l )f j (x) is continuous on R n . So f j is C 1 -smooth. Since r 2 j = 2I j , then we have established the following result.

Proposition 6.2. If the frequency vector Λ is non-resonant and P is C 2 smooth, then equation (6.2) may be written as

(6.4) dI j (τ ) -G j (I 1 , . . . , I n )dτ -b 2 j dτ = b j 2I j dW j (τ ), 1 ≤ j ≤ n,
where G is a C 1 -smooth vector-field such that G j (I) vanishes with I j , for each j.

We stress that although due to the square-root singularity in the dispersion the averaged I-equation (6.4) is a stochastic equation without uniqueness of a solution, still the limiting law D(I(⋅)) for actions of solutions for eq. (2.6) is uniquely defined by Corollary 4.13.

Uniform in time convergence.

In this section we study the uniform in time convergence in distribution of solutions for eq. (2.11) to those for effective equation (4.5), with respect to the dual-Lipschitz metric (see Definition 4.10). These results are finite-dimensional versions of those, obtained in [START_REF] Huang | On averaging and mixing for stochastic PDEs[END_REF] for stochastic PDEs. Throughout the section we assume Assumption 7.1. The first two items of Assumption 5.1 hold, and

(3 ′ ) Effective equation (4.5) is mixing with a stationary measure µ 0 . For any its solution a(τ ), τ ≥ 0, such that D(a(0)) =∶ µ and ⟨ z 2m ′ 0 , µ(dz)⟩ = E a(0) 2m ′ 0 ≤ M ′ for some M ′ > 0 (we recall notation (4.22)) we have

(7.1) D(a(τ )) -µ 0 * L,C n ⩽ g M ′ (τ, d) ∀τ ⩾ 0, if µ -µ 0 * L,C n ⩽ d ⩽ 2. Here the function g ∶ R 3 + → R + , (τ, d, M ) ↦ g M (τ, d)
, is continuous, vanishes with d, converges to zero when τ → ∞ and is such that for each fixed M ≥ 0 the function

(τ, d) ↦ g M (τ, d) is uniformly continuous in d for (τ, d) ∈ [0, ∞) × [0, 2]. 9
We emphasize that now we assume mixing for the effective equation, but not for the original equation (2.6). Since Assumptions 7.1 imply Assumptions 2.1, then the assertions of Section 4 with any T > 0 hold to solutions of equations (2.11) which we analyse in this section.

Proposition 7.2. Assume that the first two items of Assumption 5.1 hold, eq. (4.5) is mixing with a stationary measure µ 0 , and for each M > 0 and any v 1 , v 2 ∈ BM (C n ) we have

(7.2) Da(τ ; v 1 ) -Da(τ ; v 2 ) * L,C n ≤ g M (τ )
, where g is a continuous function of (M, τ ) which goes to zero when τ → ∞ and is a nondecreasing function of M . Then condition (3 ′ ) holds with some function g.

The proposition is proved below in Subsection 7.1. Note that (7.2) holds (with g replaced by 2g) if

(7.3) Da(τ ; v 1 ) -µ 0 * L,C n ≤ g M (τ ) ∀ v 1 ∈ BM (C n ).
Usually a proof of mixing for eq. (4.5) in fact establishes (7.3). So condition (3 ′ ) is a rather mild restriction.

v 0 ∈ C n lim ε→0 sup τ ⩾0 D(a ε (τ ; v 0 )) -D(a 0 (τ ; v 0 )) * L,C n = 0,
where a ε (τ ; v 0 ) and a 0 (τ ; v 0 ) solve respectively (2.11) and (4.5) with the same initial condition

a ε (0; v 0 ) = a 0 (0; v 0 ) = v 0 .
Proof. Since v 0 is fixed, we abbreviate a ε (τ ; v 0 ) to a ε (τ ). Due to (5.1)

(7.4) E a ε (τ ) 2m ′ 0 ⩽ C m ′ 0 ( v 0 ) =∶ M * ∀ τ ≥ 0. By (7.4) and (4.19) 10 (7.5) E a 0 (τ ; v 0 ) 2m ′ 0 = ⟨ a 2m ′ 0 , Da 0 (τ ; v 0 )⟩ ≤ M * ∀ τ ≥ 0.
Since Da 0 (τ ; 0) ⇀ µ 0 as τ → ∞, then from the estimate above with v 0 = 0 we get that

(7.6) ⟨ a 2m ′ 0 , µ 0 ⟩ ≤ C m ′ 0 (0) =∶ C m ′ 0 .
For later usage we note that since to derive estimates (7.5) and (7.6) we only used Assumptions 5.1.(1), 5.1.(2) and the fact that eq. (4.5) is mixing, then the two estimates hold under the assumptions of Proposition 7.2.

The constants in estimates below depend on M * , but usually this dependence is not indicated. For any T ≥ 0 we denote by a 0 T (τ ) a weak solution of effective equation (4.5) such that Da 0 T (0) = Da ε (T ). Note that a 0 T (τ ) depends on ε and that a 0 0 (τ ) = a 0 (τ ; v 0 ). Lemma 7.5. Take any δ > 0. Then (1) for any T > 0 there exists

ε 1 = ε 1 (δ, T ) > 0 such that if ε ≤ ε 1 , then (7.7) sup τ ∈[0,T ] D(a ε (T ′ + τ )) -D(a 0 T ′ (τ )) * L,C n ⩽ δ 2 ∀ T ′ ⩾ 0.
(2) Let us choose a T * = T * (δ) > 0, satisfying g M * (T, 2) ≤ δ 4 for any T ≥ T * . Then there exists ε 2 = ε 2 (δ) > 0 such that if ε ≤ ε 2 and D(a ε (T ′ ))µ 0 * L,C n ⩽ δ for some T ′ ≥ 0, then also

(7.8) D(a ε (T ′ + T * )) -µ 0 * L,C n ⩽ δ,
10 Indeed, for any N > 0 the estimate with a 2m ′ 0 replaced by a 2m ′ 0 ∧ N follows from the convergence Da ε (τ ; v0) ⇀ Da 0 (τ ; v0). Then the required estimate follows from Fatou's lemma as N → ∞. and (7.9) sup

τ ∈[T ′ ,T ′ +T * ] D(a ε (τ )) -µ 0 * L,C n ⩽ δ 2 + sup τ ≥0 g M * (τ, δ).
Below we abbreviate ⋅ * L,C n to ⋅ * L , for a measure ν ∈ P(C n ) denote by a ε (τ ; ν) a weak solution of eq. ( 2.11) such that D(a ε (0)) = ν, and define a 0 (τ ; ν) similarly. Since eq. (2.11) defines a Markov process in C n (e.g. see [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]Section 5.4.C] and [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]Section 3.3]), then

Da ε (τ ; ν) = C n Da ε (τ ; v) ν(dv),
and a similar relation holds for Da 0 (τ ; ν).

Proof. Denote ν ε = D(a ε (T ′ )). Then (7.10) D(a ε (T ′ + τ )) = D(a ε (τ ; ν ε )), D(a 0 T ′ (τ )) = D(a 0 (τ ; ν ε ))
. By (7.4), for any δ > 0 there exists K δ > 0 such that for each ε, ν ε (C n ∖ BK δ ) ≤ δ 8, where BK δ ∶= BK δ (C n ). So

ν ε = A ε ν ε δ + Āε νε δ , A ε = ν ε ( BK δ ), Āε = ν ε (C n ∖ BK δ )
, where ν ε δ and νε δ are the conditional probabilities ν ε (⋅ BK δ ) and ν ε (⋅ C n ∖ BK δ ). Accordingly, (7.11)

D(a κ (τ ; ν ε )) = A ε D(a κ (τ ; ν ε δ )) + Āε D(a κ (τ ; νε δ )), where κ = ε or κ = 0. Therefore, D(a ε (τ ; ν ε ))-D(a 0 (τ ; ν ε )) * L ≤ A ε D(a ε (τ ; ν ε δ ))-D(a 0 (τ ; ν ε δ )) * L + Āε D(a ε (τ ; νε δ ))-D(a 0 (τ ; νε δ )) * L .
The second term on the r.h.s obviously is bounded by 2 Āε ⩽ δ 4 . While by Proposition 4.12 and (4.23) there exists ε 1 > 0, depending only on K δ and T , such that for 0 ≤ τ ≤ T and ε ∈ (0, ε 1 ] the first term in the r.h.s. is ⩽ δ 4 . Due to (7.10) this proves the first assertion. To prove the second assertion let us choose ε 2 = ε 1 (δ 2, T * (δ)). Then we get from (7.7), (7.4), (7.1) and the definition of T * that for ε ≤ ε 2 ,

D(a ε (T ′ + T * )) -µ 0 * L ⩽ D(a ε (T ′ + T * )) -D(a 0 T ′ (T * )) * L + D(a 0 T ′ (T * )) -µ 0 * L ⩽ δ.
This proves (7.8). Next, in view of (7.7) and (7.1), (7.5),

sup θ∈[0,T * ] D(a ε (T ′ + θ)) -µ 0 * L ⩽ sup θ∈[0,T * ] D(a ε (T ′ + θ)) -D(a 0 T ′ (θ)) * L + sup θ∈[0,T * ] D(a 0 T ′ (θ)) -µ 0 * L ⩽ δ 2 + max θ∈[0,T * ] g M * (θ, δ).
This implies (7.9). Now we continue to prove the theorem. Let us fix arbitrary δ > 0 and take some 0 < δ 1 ≤ δ 4. Below in the proof the functions ε 1 , ε 2 and T * are as in Lemma 7.5. i) By the definition of T * , (7.1) and (7.4), (7.12)

D a 0 T ′ (τ )) -µ 0 * L ≤ δ 1 ∀ τ ≥ T * (δ 1
), for any T ′ ≥ 0. We will abbreviate T * (δ 1 ) to T * .

ii) By (7.7), if ε ≤ ε 1 = ε 1 (δ 1 , T * ) > 0, then (7.13) sup

0≤τ ≤T * D a ε (τ )) -D a 0 (τ ; v 0 ) * L ≤ δ 1 2 .
In particular, in view of (7.12) with T ′ = 0, (7.14) D a ε (T * ))µ 0 * L < 2δ 1 . iii) By (7.14) and (7.8) with δ ∶= 2δ 1 and with T ′ = nT * , n = 1, 2, . . . we get inductively that

(7.15) D a ε (nT * )) -µ 0 * L ≤ 2δ 1 ∀ n ∈ N, if ε ≤ ε 2 = ε 2 (2δ 1 ).
iv) Now by (7.15) and (7.9) with δ ∶= 2δ 1 , for any n ∈ N and 0 ≤ θ ≤ T * , (7.16)

D a ε (nT * + θ)) -µ 0 * L ≤ δ 1 + sup θ≥0 g M * (θ, 2δ 1 ), if ε ≤ ε 2 (2δ 1 ). v) Finally, if ε ≤ ε # (δ 1 ) = min ε 1 (δ 1 , T * ), ε 2 (2δ 1 )
, then by (7.13) if τ ≤ T * and by (7.12)+(7.16) if τ ≥ T * we have that

D a ε (τ )) -D a 0 (τ ; v 0 ) * L ≤ 2δ 1 + sup θ≥0 g M * (θ, 2δ 1 ) ∀ τ ≥ 0.
By the assumption, imposed in (3 ′ ) on function g M , g M (t, d) is uniformly continuous in d and vanishes at d = 0. So there exists δ * = δ * (δ), which we may assume to be ≤ δ 4, such that if δ 1 = δ * , then g M * (θ, 2δ 1 ) ≤ δ 2 for every θ ⩾ 0. Then by the estimate above,

D a ε (τ )) -D a 0 (τ ; v 0 ) * L ≤ δ if ε ≤ ε * (δ) ∶= ε # (δ * δ) > 0,
for every positive δ. This proves the theorem's assertion.

Since the interaction representation does not change actions, then for the action variables of solutions for the original equations (2.6) we have In [9, Theorem 2.9] the assertion of the corollary is proved for a class of systems (2.6). The proof in [START_REF] Dymov | Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators[END_REF] is based on the observation that the mixing rate in the corresponding equation (2.6) is uniform in 0 < ε ≤ 1. This is a delicate property which is harder to establish than (3 ′ ) in Assumption 7.1. We also note that Theorem 7.4 immediately implies that if equations (2.11) are mixing with stationary measures µ ε , then µ ε ⇀ µ 0 . Cf. Theorem 5.5. 7.1. Proof of Proposition 7.2. In this subsection we write solutions a 0 (τ ; v) of effective equation (4.5) as a(τ ; v). We will prove the assertion of Proposition 7.2 in four steps. i) At this step, for any non-random v 1 , v 2 ∈ BM (C n ) we denote by a j (τ ) ∶= a(τ ; v j ), j = 1, 2, and examine the distance D(a 1 (τ )) -D(a 2 (τ )) * L as a function of τ and v 1v 2 . Let us set w(τ ) = a 1 (τ )a 2 (τ ) and assume that v 1v 2 ≤ d for some d ≥ 0. Then we have ẇ = ⟪P ⟫(a 1 ) -⟪P ⟫(a 2 ).

Since by Lemma 3.2 and Assumption 2.1.(1)

⟪P ⟫(a 1 (τ )) -⟪P ⟫(a 2 (τ )) ≤ C w(τ ) X(τ ), X(τ ) = 1 + a 1 (τ ) m 0 ∨ a 2 (τ ) m 0 , then (d dτ ) w 2 ≤ 2CX(τ ) w 2 with w(0) ≤ d. So (7.17) w(τ ) ≤ d exp C τ 0 X(l) dl .
Denote Y (T ) = sup 0≤τ ≤T X(τ ) . By (5.1) estimate (2.7) holds with C m ′ 0 ( v 0 , T ) = C m ′ 0 (M )(T + 1). So by Remark 4.8.2) EY (T ) ≤ (C m ′ 0 (M ) + 1)(T + 1) (since m ′ 0 > (m 0 ∨ 1)). For K > 0 denote by Ω K (T ) the event {Y (T ) ≥ K}. Then P(Ω K (T )) ≤ (C m ′ 0 (M ) + 1)(T + 1)K -1 , and ∫ τ 0 X(l) dl ≤ τ K for ω ∉ Ω K (T ). From here and (7.17) we see that if f is such that f ≤ 1 and Lip f ≤ 1, then

E f (a 1 (τ )) -f (a 2 (τ )) ≤ 2P(Ω K (τ )) + d exp Cτ K) = 2(C m ′ 0 (M ) + 1)(τ + 1)K -1 + d exp Cτ K) ∀ K > 0.
; v 2 )) * L = D(a 1 (τ )) -D(a 2 (τ )) * L ≤ g M (τ ) ∧ g 1 M (τ, d) ∧ 2 =∶ g 2 M (τ, d) if v 1 -v 2 ≤ d. (7.19)
The function g 2 M is continuous in the variables (τ, d, M ), vanishes with d and goes to zero when τ → ∞ since g M (τ ) does.

ii) At this step we consider a solution a 0 (τ ; µ) =∶ a(τ ; µ) of (4.5) with D(a(0)) = µ as in Assumption 7.1.(3 ′ ) and examine the l.h.s of (7.1) as a function of τ . For any M > 0 consider the conditional probabilities µ

M = P(⋅ BM (C n )) and μM = P(⋅ C n ∖ BM (C n )). Then (7.20) D(a(τ ; µ)) = A M D(a(τ ; µ M )) + ĀM D(a(τ ; μM )),
where

A M = µ( BM (C n )) and ĀM = µ(C n ∖ BM (C n )) (cf. (7.11)). As E a(0) 2m ′ 0 ≤ M ′ , then ĀM = P{a(0) > M } ⩽ M ′ M 2m ′ 0 .
Since eq. (4.5) is assumed to be mixing, then D(a(τ ; 0))µ 0 * L ≤ ḡ(τ ), where ḡ ≥ 0 is a continuous function, going to 0 as τ → ∞. So in view of (7.2),

D(a(τ ; v)) -µ 0 * L ≤ g M (τ ) + ḡ(τ ) =∶ gM (τ ), ∀ v ∈ BM (C n ).
From here,

D(a(τ ; µ M )) -µ 0 * L = D(a(τ ; v)) µ M (dv) -µ 0 * L ≤ D(a(τ ; v)) -µ 0 * L µ M (dv) ≤ gM (τ ).
Therefore due to (7.20),

D(a(τ ; µ))-µ 0 * L ⩽ A M D(a(τ ; µ M )) -µ 0 * L + ĀM D(a(τ ; μM )) -µ 0 * L ≤ D(a(τ ; µ M )) -µ 0 * L + 2 ĀM ≤ gM (τ ) + 2 M ′ M 2m ′ 0
for any M > 0 and τ ≥ 0.

Let M 1 (τ ) > 0 be a continuous non-decreasing function, growing to infinity with τ , and such that gM 1 (τ ) (τ ) → 0 as τ → ∞ (it exists since gM (τ ) is a continuous function of (M, τ ), going to 0 as τ → ∞ for each fixed M ). Then

(7.21) D(a(τ ; µ)) -µ 0 * L ⩽ 2 M ′ M 1 (τ ) 2m ′ 0 + gM 1 (τ ) (τ ) =∶ ĝM ′ (τ ).
Obviously ĝM ′ (τ ) ≥ 0 is a continuous function on R 2 + , converging to 0 as τ → ∞. iii) Now we examine the l.h.s of (7.1) as a function of τ and d. Recall that the Kantorovich distance between measures ν 1 , ν 2 on C n is

ν 1 -ν 2 K = sup Lip f ≤1 ⟨f, ν 1 ⟩ -⟨f, ν 2 ⟩ ≤ ∞.

Averaging for systems with general noises

In this section we sketch a proof of Theorem 4.7 for equations (1.1) with general stochastic terms √ ε B(v) dW. The proof follows the argument in Section 4 with an extra difficulty which appears in the case of equations with non-additive degenerate noises.

Let us consider v-equation (2.6) with a general noise (possibly non-additive) and decomplexify it by writing the components v k (τ ) as (ṽ 2k-1 (τ ), ṽ2k (τ )) ∈ R 2 , k = 1, . . . , n. Now a solution v(τ ) is a vector in R 2n and the equation reads (8.1) dv(τ

) + ε -1 Av(τ ) dτ = P (v(τ )) dτ + B(v(τ )) dβ(τ ), v(0) = v 0 ∈ R 2n .
Here A is the block-diagonal matrix as in Section 2, B(v) is a real 2n × n 2 -matrix and β(τ ) = (β 1 (τ ), . . . , β n 2 (τ )), where {β j (τ )} are independent standard real Wiener processes. Note that in the real coordinates in R 2n ≃ C n the operator Φ w in (2.10) with some vector w ∈ R n is given by a block-diagonal matrix, where for j = 1, . . . , n its j-th diagonal block is the 2 × 2-matrix of rotation by the angle w j .

In this section we assume that Assumption 8.1. The dirft P ∈ Lip m 0 (R 2n , R 2n ), the matrix-function B(v) belongs to Lip m 0 R 2n , M (2n × n 2 ) , equation (8.1) is well-posed and its solutions satisfy (2.7).

Passing to the interaction representation v(τ ) = Φ τ ε -1 Λ a(τ ) we rewrite the equation as

(8.2) da(τ ) = Φ τ ε -1 Λ P (v(τ )) dτ + Φ τ ε -1 Λ B(v(τ )) dβ(τ ), a(0) = v 0 .
As in Section 4 we will see that as ε → 0, the asymptotic behaviour of distributions of the equation's solutions is described by an effective equation. As before, the effective drift is ⟪P ⟫(a). To calculate the effective dispersion, as in the proof of Lemma 4.6 we consider the martingale

N Y,ε ∶= a ε (τ ) - τ 0 Y (a ε (s), sε -1 )ds = v 0 + τ 0 B Λ (a ε (s); sε -1 )dβ(s),
where Y is defined in (4.7) and B Λ (a; t) = Φ tΛ B(Φ -tΛ a). By Itô's formula, for i, j = 1, . . . , n the process

N Y,ε i (τ )N Y,ε j (τ ) - τ 0 A Λ ij (a ε (s); sε -1 )ds, (A Λ ij (a; t)) = B Λ (a; t)B Λ * (a; t),
where B Λ * is the transpose of B Λ , also is a martingale. By a straightforward analogy of Lemma 3.2, the limit

A 0 (a) ∶= lim T →∞ 1 T T 0 A Λ (a; t)dt
exists and belongs to Lip 2m 0 (R 2n , M (2n × 2n)). Then, by analogy with Lemma 4.3,

E τ 0 A Λ (a ε (s); sε -1 )ds - τ 0 A 0 (a ε (s))ds → 0, as ε → 0,
for any τ ⩾ 0. From here, as in Section 4, we conclude that now for the effective diffusion we should take A 0 (a), which is a non-negative symmetric matrix. Denoting by B 0 (a) = A 0 (a)

B(v)B * (v)ξ ⩾ α ξ , ∀ v, ∀ ξ ∈ R 2n ,
for some α > 0, then the matrix B Λ (a, τ ) also satisfies (8.4) for all a and τ , i.e. ⟨A Λ (a; sε -1 )ξ, ξ⟩ ⩾ α ξ 2 . Thus A 0 (a) ⩾ αI, and so B 0 (a) = A 0 (a) To treat equations (8.1) with degenerate non-additive noises we write the matrix A 0 (a) as

A 0 (a) = lim T →∞ 1 T T 0 Φ tΛ B(Φ -tΛ a) ⋅ Φ tΛ B(Φ -tΛ a) * dt.
By the same reason as in Proposition 3.4 we find that

A 0 C 2 (B R ) ⩽ C B 2 C 2 (B R ) , ∀R > 0. Now using [24, Theorem 5.2.3], we get that (8.5) Lip(B 0 (a) BR ) ⩽ C A 0 1 2 C 2 (B R+1 ) ⩽ C 1 B C 2 (B R+1
) , ∀R > 0. So the matrix-function B 0 (a) is locally Lipschitz continuous, eq. ( 8.3) has a unique solution and the assertion of Theorem 4.7 remains true for eq. (8.1). We have obtained Theorem 8.2. Suppose that Assumption 8.1 holds and for the matrix function B(v) in (8.1) one of the following three options is true:

(1) it is v-independent;

(2) it satisfies the non-degeneracy condition (8.4);

(3) it is a C 2 -smooth matrix-function of v. Then for any v 0 ∈ R 2n a solution a ε (τ ; v 0 ) of eq. (8.2) satisfies

D(a ε (⋅; v 0 )) ⇀ Q 0 in P(C([0, T ], C n )) as ε → 0,
where Q 0 is the law of a unique weak solution of the effective equation (8.3).

An obvious analogy of Corollary 4.13 holds for solutions of eq. (8.1). 9. A sufficient condition for Assumptions 2.1, 5.1 and 7.1

In this section we derive a condition which implies Assumptions 2.1, 5.1 and 7.1. So when it is met, all theorems in Sections 4, 5 and 7 apply to eq. (2.6).

Consider a stochastic differential equation on R l , (9.1)

dx = b(x)dτ + σ(x)dβ(τ ), x ∈ R l , τ ⩾ 0,
where σ(x) is an l × k-matrix and β(τ ) is a standard Wiener processes in R k . We assume The diffusion a(x) = σ(x) ⋅ σ t (x) is a nonnegative symmetric l × l-matrix. Consider the differential operator

L v(x) = l j=1 b j (x) ∂v ∂x j + 1 2 l i=1 l j=1 a ij (x) ∂ 2 v ∂x i ∂x j .
We have the following result concerning the well-posedness of eq. (9.1) from [17, Theorem 3.5]:

Theorem 9.2. Assume Assumption 9.1 and suppose that there exists a nonnegative function V (x) ∈ C 2 (R l ) such that for some constant c > 0 we have L (V (x)) ⩽ cV (x) ∀τ ⩾ 0, x ∈ R l , and inf

x >R V (x) → ∞ as R → ∞.

Then for any x 0 ∈ R l equation (9.1) has a unique strong solution X(τ ) with the initial condition X(0) = x 0 . Furthermore, the process X(τ ) satisfies EV (X(τ )) ⩽ e cτ V (x 0 ) ∀τ ⩾ 0.

The function V is called a Lyapunov function for equation (9.1). In its terms a sufficient condition for the mixing in equation (9.1) is given by the following statement: for some constants α 1 > 0 and α 2 ⩾ 0, where ⟨⋅, ⋅⟩ is the standard inner product in R l . (2) The diffusion matrix a(x) = σ(x)σ(x) t is uniformly non-degenerate, that is

(9.3) γ 2 I ⩽ a(x) ⩽ γ 1 I ∀x ∈ R l ,
for some γ 1 ⩾ γ 2 > 0. Then for any c ′ > 0 equation (9.1) has a smooth Lyapunov function V (x), equal exp(c ′ x ) for x ≥ 1, estimate (5.1) holds true for its solutions for every m ∈ N, and the equation is mixing.

In Appendix A it is shown how to derive the proposition from abstract results in [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]. Moreover, it can be proved that under the assumptions of the proposition the equation is exponentially mixing and (7.3) holds, see Example 7.3.

Let us decomplexify C n to R 2n and identify equation (2.6) with a real equation (9.1), where l = 2n (and x = v). Then b(v) ≅ (b j (v) = (-iε -1 λ j v j + P j (v), j = 1, . . . , n), where b j ∈ C ≅ R 2 ⊂ R 2n . Since in complex terms the real inner product reads as ⟨v, w⟩ = Re ∑ v j wj , then ⟨b(v), v⟩ = ⟨P (v), v⟩. So for equation (2.6) condition (9.2) is equivalent to uniformly in X. Then (A.3) dF X ⩽ -α 1 2 η ′ F X + C 0 dτ + η ′ F (X)⟨∇f (x), σ t (X)dW ⟩,

where the constant C 0 > 0 depends on k 1 , k 2 , α 1 , η ′ and α 2 . Taking expectation and applying Gronwall's lemma we obtain that (A.4) Ee η ′ f (X(τ )) ⩽ e -α 1 2 η ′ τ e η ′ f (x 0 ) + C 1 , τ ⩾ 0, where C 1 > 0 depends on the same parameters as C 0 . Now let us take any T ⩾ 0 and for τ ∈ [T, T + 1] consider relation (A.3), where F (X) is replaced by F (X) = e ηf (X) with 0 < η ≤ To prove the proposition it remains to show that under the imposed assumptions equation (9.1) is mixing. Due to [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]Theorem 4.3] we just need to verify that there exists an absorbing ball B R = { x ⩽ R} such that for any compact set K ⊂ R l ∖ B R (A.6) sup

x 0 ∈K Eτ (x 0 ) < ∞,
where τ (x 0 ) is the hitting time for B R of trajectory X(τ ; x 0 ). Indeed, let x 0 ∈ K ⊂ R l ∖ B R for some R > 0 to be determined later. We set τ M ∶= min{τ (x 0 ), M }, M > 0. Applying Itô's formula to the process F (τ, X(τ )) = e Here v(t) and g(t) are adapted processes in C n , M 1 and M 2 are adapted processes in the space of complex n × N -matrices, and B(t) = (β c 1 (t), . . . , β c N (t)), B(t) = ( βc 1 (t), . . . , βc N (t)), where {β c j } are independent standard complex Wiener processes. We recall that for a C 1function f on C n , ∂f ∂z j = 1 2 ∂f ∂x j -i∂f ∂y j and ∂f ∂ zj = 1 2 ∂f ∂x j + i∂f ∂y j . If f is a polynomial of z j and zj , then ∂f ∂z j and ∂f ∂ zj may be calculated as if z j and zj are independent variables.

The processes g, M 1 , M 2 and the function f (t, v) in the theorem below are assumed to satisfy the usual conditions, needed for the applicability of Itô's formula (e.g. see in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]), which we do not repeat here.

Theorem C.1. Let f (t, v) be a C 2 -smooth complex function. Then

df (t, v(t)) = ∂f ∂t + d v f (t, v)g + d vf (t, v)ḡ + Trace M 1 (M 2 ) t + M 2 (M 1 ) t ∂ 2 f ∂v∂v + M 1 ( M 2 ) t + M 2 ( M 1 ) t ∂ 2 f ∂v∂v + 2 M 1 ( M 1 ) t + M 2 ( M 2 ) t ∂ 2 f ∂v∂v dt + d v f (M 1 dB + M 2 d B) + d vf ( M 1 d B + M 2 dB) . (C.2)
Here d v f (t, v)g = ∑ ∂f ∂v j g j , d vf (t, v)ḡ = ∑ ∂f ∂v j ḡj , ∂ 2 f ∂v∂v stands for the matrix with entries ∂ 2 f ∂v j ∂v k , etc. If the function f is real valued, then d vf (v) = d v f (v), and the Itô term, given by the second and third lines of (C.2), reeds 2Re Trace (M 1 (M 2 ) t + M 2 (M 1 ) t ) ∂ 2 f ∂v∂v + (M 1 ( M 1 ) t + M 2 ( M 2 ) t ) ∂ 2 f ∂v∂v .

To prove the result one may re-write v(t) as an Itô process in R 2d in terms of the real Wiener processes Re W j (t), Im W j (t), apply the usual Itô's formula to f (t, v(t)) and then re-write the result back in terms of the complex Wiener processes. Corresponding straightforward calculation is rather heavy and it is not easy to make it without mistake. Below we suggest a better way to get the formula.

Proof. The linear part of the formula (C.2), given by its first and forth lines, follows from the real case by linearity. It remains to prove that the Itô term has the form, given by the second and third lines. From the real formula we see that the Itô term is an expression, linear in ∂ 2 f ∂v∂v, ∂ 2 f ∂v∂v and ∂ 2 f ∂v∂v, with the coefficients, quadratic in the matrices M 1 and M 2 . So it may be written as

(C.3) Trace Q 1 ∂ 2 f ∂v∂v + Trace Q 2 ∂ 2 f ∂v∂v + Trace Q 3 ∂ 2 f ∂v∂v dt,
where Q j are complex n × n-matrices, quadratic in M 1 , M 2 . We should show that they have the form, specified in (C.2). To do that we note that since the processes β c j are independent and have the form (2.3), then for all j, l their brackets have the following form:

Notation.

  For a Banach space E and R > 0, B R (E) stands for the open R-ball {e ∈ E ∶ e E < R}, and BR (E) -for its closure { e E ≤ R}; C b (E) stands for the space of bounded continuous function on E, and C([0, T ], E) -for the space of continuous curves [0, T ] → E, given the sup-norm. For any 0 < α ⩽ 1 and u ∈ C([0, T ], E), (1.4)

Proposition 4 . 2 .

 42 The limiting probability measure Q 0 in (2.14) is a weak solution of effective equation (4.5), (4.6).

(4. 22 )Example 4 . 11 .

 22411 ⟨f, µ⟩ ∶= M f (m)µ(dm). Consider the Polish spaces C([0, T ], C n ) and C n and the mappings

Example 7 . 3 .

 73 If the assumption of Proposition 9.4 below is fulfilled, then (7.2) is satisfied since in this case (7.3) holds with g M (τ ) = V (M )e -cτ . Here c > 0 is a constant and V (M ) = max{V (x) ∶ x ∈ BM (C n )}, where V (x) is the Lyapunov function as in Proposition 9.3. See e.g. [22, Theorem 2.5] and [20, Section 3.3]. Theorem 7.4. Under Assumption 7.1, for any

Corollary 7 . 6 .

 76 Under the assumptions of Theorem 7.4 the actions of a solution v ε (τ ; v 0 ) for eq. (2.6) which equals v 0 at τ = 0 satisfy lim ε→0 sup τ ⩾0 D(I v ε (τ ; v 0 )) -D((I a 0 (τ ; v 0 )) * L,C n = 0.

(7. 18 )

 18 Let us denote by g 1 M (τ, d) the function in the r.h.s. with K = ln ln d-1 ∨ 3 . This is a continuous function of (τ, d, M ) ∈ R 3 + , vanishing when d = 0. Due to (7.2) and (7.18), D(a(τ ; v 1 )) -D(a(τ

Assumption 9 . 1 .

 91 The drift b(x) and dispersion σ(x) are locally Lipschitz in x, and C m (b), C m (σ) ≤ C < ∞ for some m ≥ 0.

Proposition 9 . 3 .

 93 Assume that in addition to Assumption 9.1 we have (1) the drift b satisfies(9.2) ⟨b(x), x⟩ ⩽ -α 1 x + α 2 ∀x ∈ R l ,

(9. 4 )

 4 ⟨P (v), v⟩ ⩽ -α 1 v + α 2 , ∀v ∈ C n ,for some constants α 1 > 0 and α 2 ⩾ 0. Now consider effective equation (4.5). Since in eq. (2.11) the drift isY (a, τ ε -1 ) = (Φ τ ε -1 Λ ) * P (a),then under the assumption (9.4) we have⟨Y (a, τ ε -1 ), a⟩ = ⟨P (Φ τ ε -1 Λ a), Φ τ ε -1 Λ a⟩ ⩽ -α 1 Φ -τ ε -1 Λ a + α 2 = -α 1 a + α 2 ,for all ε. Therefore ⟪P ⟫ satisfies ⟨⟪P ⟫(a), a⟩ = limT →∞ 1 T T 0⟨Y (a, τ ε -1 ), a⟩dτ ⩽ -α 1 a + α 2 .

1 2 ηF

 2 ′ , and integrate it from T to τ :F (X(τ )) ≤ F (X(T )) + C 0 + η τ T (X)⟨∇f (x), σ t (s, X)dW ⟩ =∶ F (X(T )) + C 0 + M(τ ). (A.5)Due to estimate (A.4), M(τ ) is a continuous square-integrable martingale. Therefore by Doob's inequality,E sup T ≤τ ≤T +1 M(τ ) 2 ≤ 4E M(T + 1) 2 ≤ C T +1 T E F 2 (X(s))ds ≤ C T +1 T EF (X(s))ds ≤ C ′ ,where C ′ depends on k 1 , k 2 , α 1 , η ′ , α 2 and x 0 . From here, (A.5) and (A.4) it follows thatE sup T ≤τ ≤T +1 e ηf (X(τ )) ≤ C ′′ ,where C′′ depends on the same parameters as C ′ . This bound implies that solutions X(τ ) satisfy estimate (5.1) in Assumption 5.1 for every m ≥ 0.

1 4 η′ α 1 4 F′ α 1 4 0 eη ′ α 1 s 4 (2α 1 X 3 τ M 0 eη ′ α 1 s 4 ds

 441041304 ′ α 1 τ X(τ ) 2 and using (A.4) we find thatdF (τ, X(τ )) = η (τ, X(τ )) + L F (τ, X(τ )) dτ + dM(τ ),where M(τ ) is a corresponding stochastic integral. By (A.1), (A.4) and (9.2), we haveEe η τ M X(τ M ) 2 + E τ M (τ ) -C 3 )ds ⩽ x 0 2 + 2e η ′ f (x 0 ) =∶ γ(x 0 ),whereC 3 > 0 depends on α 1 , α 2 , k 1 and k 2 . Since X(s) ⩾ R for 0 ⩽ s ⩽ τ M , then E C ⩽ γ(x 0 ), if R ≥ C 3 α 1 . Therefore Eτ M ⩽ γ(x 0 ) C 3 . Letting M → ∞ we verify (A.6) for R ≥ C 3 α 1 .This completes the proof of Proposition 9.3.

(C. 4 )

 4 ⟨β c j , β c l ⟩ = ⟨ βc j , βc l ⟩ = 0, ⟨β c j , βc l ⟩ = ⟨ βc j , β c l ⟩ = 2δ j,l t.Now let in (C.1) g = 0, v(0) = 0 and M 1 , M 2 are constant matrices. Thenv(t) = M 1 B(t) + M 2 B(t).

  .3) remains true for solutions of eq. (8.2). In particular, the theorem applies to equation (2.5) with general additive random forces (2.4) (but then the effective dispersion matrix is given by a more complicated formula than in Section 4).Similarly, if the diffusion in (8.1) is non-degenerate in the sense that

	(8.4)
	1 2
	its principal square root, as in Section 4 we verify that any limiting measure Q 0 as in (2.14)
	is a solution for the martingale problem for the effective equation
	(8.3)

da(τ ) -⟪P ⟫(a(τ ))dτ = B 0 (a(τ ))dβ(τ ), a(0) = v 0 , and so is a weak solution of the equation. If the noise in equation (8.1) is additive, then B 0 is a constant matrix, eq. (

8

.3) has a unique solution and Theorem 4.7 with (modified) effective equation (

8

  1 2 is a locally Lipschitz matrix function of a (e.g. see[START_REF] Stroock | Multidimensional Diffusion Processes[END_REF] Theorem 5.2.2]). So again eq. (8.3) has a unique solution and Theorem 4.7 remains true for eq. (8.1) (with the effective equation of the form (8.3)).

See[START_REF] Jian | The Bogolyubov-Krylov averaging[END_REF] for averaging in deterministic perturbation of eq. (2.1) by locally Lipschitz vector fields.

This is well known, and can be easily proved using the estimate in Remark 4.8.2).

E.g. firstly Theorem 1.6.8 from[START_REF] Bogachev | Fokker-Plank-Kolmogorov equations[END_REF] implies that µ 0 = ϕ(x)dx, where ϕ is a Hölder function, and then by the usual elliptic regularity ϕ ∈ C 1 .

So g M extends to a continuous function on [0, ∞] × [0, 2] which vanishes when τ = ∞ or d = 0.

Acknowledgement GH was supported by National Natural Science Foundation of China (No. 20221300605). Both authors were supported by the Ministry of Science and Higher Education of the Russian Federation (megagrant No. 075-15-2022-1115).

Appendix C. Itô's formula for complex processes Appendix D. Projections to convex sets Acknowledgement References Obviously ν 1ν 2 * L ≤ ν 1ν 2 K . Due to (7.6) and the assumption on µ, the 2m ′ 0 -moments of µ and µ 0 are bounded by M ′ ∨ C m ′ 0 . Thus (7.22) µ - Section 11.4] and [START_REF] Villani | Optimal Transport[END_REF]Chapter 7]. So by the Kantorovich-Rubinstein theorem (see [START_REF] Villani | Optimal Transport[END_REF][START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]) there exist r.v.'s ξ and ξ 0 , defined on a new probability space (Ω ′ , F ′ , P ′ ), such that D(ξ) = µ, D(ξ 0 ) = µ 0 and (7. [START_REF] Skorokhod | Asymptotic Methods in the Theory of Stochastic Differential Equations[END_REF] E ξ 1ξ 0 = µµ 0 K . Then using (7.19) and denoting by a st (τ ) a stationary solution of eq. (4.5), D(a st (τ )) ≡ µ 0 , we have:

) by (7.5) and the assumption on µ, then denoting

Then by (7.23) and (7.22),

iv) End of the proof. Let g 0 (s) be a positive continuous function on R + such that g 0 (s) → ∞ as s → +∞ and C m ′ 0 g 0 (s) (ln ln s) -1 2 ⩽ 2C m ′ 0 (0) for s ⩾ 3. Taking r = D 1 2 and choosing in the r.h.s. of (7.24) K = g 0 (r -1 ), we denote the r.h.s as g 3 M ′ (τ, r) (so we substitute in (7.24) D = r 2 and K = g 0 (r -1 )). By (7.24) and the definition of g 2 M (see (7.18) and (7.19)), we have

By the choice of g 0 , as r → 0 the first, the second and the fourth terms converge 0. The third term is ⩽ 4(C m ′ 0 (0) + 1)(ln ln(r -1 )) -1 2 for r ⩽ 1 3 , so it also converges to zero with r. Hence g 3 M ′ (τ, r) defines a continuous function on R 3 + , vanishing with r. Using the expression for D in (7.22)

Finally, evoking (7.21) we arrive at (7.1) with g = g 5 , where

The function g 5 is continuous, vanishes with d and converges to zero as τ → ∞. For any fixed M ′ > 0 this convergence is uniform in d due to the term ĝM ′ (τ ). So for a fixed

, where it vanishes when τ = ∞. Thus g 5 M ′ is uniformly continuous in d, and the assertion of the proposition is proved.

We saw that assumption (9.4) implies the validity of condition (9.2) also for the effective equation.

As it was pointed out, if the dispersion matrix Ψ is non-degenerate, then the dispersion B in the effective equation is non-degenerate as well. The corresponding diffusion matrix also is non-degenerate and condition (9.3) holds for it. Thus we obtained the following statement: Proposition 9.4. If in equation (2.6) the dispersion matrix Ψ is non-degenerate, the drift P ∈ Lip m 0 for some m 0 ∈ N and (9.4) holds for some constants α 1 > 0 and α 2 ⩾ 0, then the assumption of Theorem 5.5 holds true, and so does the assumption of Theorems 4.7 and 7.4.

Appendix A. Proof of Proposition 9.3 By condition (9.3) the diffusion a is uniformly bounded. So there exist constants

, where c ′ > 0 is a positive constant and f (x) is a non-negative smooth function, equal x for x ≥ 1, and such that its first and second derivatives are bounded by 3. Then ∂V (x)

Therefore we have that

where

From the condition (9.2) and (A.1) it is evident that

where C > 0 is a constant depending on k 1 , k 2 and sup x ⩽1 b(x) . Then we achieve that

is a Lyapunov function for equation (9.1). Then by Theorem 9.2 for any x 0 ∈ R l the equation has a unique solution X(τ ) = X(τ ; x 0 ), equal x 0 at τ = 0, which satisfies

Let us apply Itô's formula to the process F X(τ ) = e η ′ f (X(τ )) , where 0 < η ′ ⩽ 1 2 c ′ is a constant to be determined later. Then

Appendix B. Representation of martingales

Let {M k (t), t ∈ [0, T ]}, k = 1, . . . , d, be continuous square-integrable martingales on a filtered probability space (Ω, F, P, {F t }). We recall that their brackets (or their crossvariational process) is an {F t }-adapted continuous matrix-valued process of bounded variation ⟨M k , M j ⟩(t), 1 ≤ k, j ≤ d, a.s. vanishing at t = 0, and such that for all k, j the process M k (t)M j (t) -⟨M k , M j ⟩(t) is an {F t }-martingale; see [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF], Definition 1.5.5 and Theorem 1.5.13.

Theorem B.1 ([14], Theorem 3.4.2). Let (M k (t), 1 ≤ k ≤ d) be a vector of martingale as above. Then there exists an extension ( Ω, F, P, Ft ) of the probability space on which are defined independent standard Wiener processes W 1 (t), . . . , W d (t), and a measurable adapted matrix X = (X kj (t)) k,j=1,...,d , t ∈ [0, T ] such that E ∫ T 0 X(s) 2 ds < ∞, and P-a.s. we have the representations

and

is a vector of real continuous martingales. The brackets ⟨N i , N j ⟩ and ⟨N i , Nj ⟩ are defined by linearity. For example, ⟨N i , 11 Equivalently ⟨N i , N j ⟩ may be be defined as the only adapted continuous complex process of bounded variation, vanishing at zero, and such that N i N j -⟨N i , N j ⟩ is a martingale. The brackets ⟨N i , Nj ⟩ may be defined similarly. The result above implies a representation theorem for complex continuous martingales. Below we give its special case, relevant for our work.

Corollary B.2. Suppose that the brackets ⟨N i , N j ⟩(t) and ⟨ Ni , Nj ⟩(t) all vanish, while the brackets ⟨N i , Nj ⟩(t), 1 ≤ i, j ≤ d, a.s. are absolutely continuous complex processes. Then there exist an adapted process Ψ(t), valued in d × d complex matrices, satisfying E ∫ T 0 Ψ(t) 2 dt < ∞, and independent standard complex Wiener processes β c 1 (t), . . . , β c d (t), all defined on an extension of the original probability space, such that a.s.

Moreover, ⟨N i , N j ⟩(t) ≡ 0 and

Appendix C. Itô's formula for complex processes Consider a complex Itô process v(t) ∈ C n , defined on a filtered probability space:

11 There is no need to define the brackets ⟨ Ni, Nj⟩ and ⟨ Ni, Nj⟩ since these are just the processes, complexconjugated to ⟨Ni, Nj⟩ and ⟨Ni, Nj⟩, respectively. Note that an analogy of the lemma's statement for a Banach space X in general is wrong.