Sergei Kuksin
email: kuksin@math.jussieu.fr

Alberto Maiocchi
email: alberto.maiocchi@unimi.it

Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations

Keywords: stochastic PDEs, NLS equation, weak turbulence, averaging theorem, effective equation 2010 Mathematics subject classification: Primary 35R60

We consider the free linear Schrödinger equation on a torus T d , perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping:

Here u = u(t, x), x ∈ T d , 0 < ν 1, q * ∈ N, f is a positive continuous function, ρ is a positive parameter and β k (t) are standard independent complex Wiener processes. We are interested in limiting, as ν → 0, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time τ = νt, we prove that the limiting behaviour of them both is described by the effective equation

where the nonlinearity F (u) is made out of the resonant terms of the monomial |u| 2q * u.

Introduction

Equations

The nonlinear Schrödinger equation on the torus with small nonlinearity u t (t, x) -i∆u(t, x) = -iε 2q * |u| 2q * u, u = u(t, x), x ∈ T d L = R d /(2πLZ d ), (1.1) where q * ∈ N and 0 < ε 1, is a popular model in various branches of science. The nonlinearity in (1.3) is Hamiltonian and may be written as

-iε 2q * |u| 2q * u = ε 2q * i∇H(u), H(u) = H 2q * +2 (u) = - 1 2q * + 2 |u(x)| 2q * +2 dx, (1.2)
so the equation describes a conservative system. To describe systems interacting with the 'environment', at some stage physicists often add (to this and to other similar equations) terms that describe the pumping of energy into the system and its dissipation (see, for example, [28, § 2.2.3]). A way to describe the pumping of energy is to add to the equation a small random force, which is usually Gaussian, smooth in x, and often white in time t, while to describe the dissipation a suitable function of the Laplacian is usually used. In this way we arrive at (see [27, (5)] and [3, (1.2), § 1.2])

u t -i∆u = -iε 2q * |u| 2q * u -νf (-∆)u + √ ν d dt k∈Z d L b k β k (t)e ik•x , x ∈ T d L . (1.3)
Here 0 < ν 1 and Z d L denotes the set of vectors of the form k = l/L with l ∈ Z d . The damping -f (-∆) is the self-adjoint linear operator in L 2 (T d L ), which acts on the exponentials e ik•x , k ∈ Z d L , according to f (-∆)e ik•x = γ k e ik•x , γ k = f (λ k ), where λ k = |k| 2 .

(1.4)

The real-valued smooth function f (t), t 0, is positive and f > 0. To avoid technicalities not relevant for this work, we assume that f (t) C 1 |t| + C 2 for all t, for suitable positive constants C 1 , C 2 (for example, f (-∆)u = -∆u + u). The processes β k , k ∈ Z d L , are standard independent complex Wiener processes, i.e. β k (t) = β k + (t) + iβ k -(t), where β k ± (t) are standard independent real Wiener processes. The real numbers b k are all non-zero and decay fast when |k| → ∞. The factor in front of the random force is chosen to be √ ν to guarantee that solutions of (1.3) stay of order 1 when t 1 and 0 < ν 1. We assume that (1.3) with sufficiently smooth initial data u(0, x) = u 0 (x) is well posed. It is well known that this assumption holds (at least) under some restriction on d, q * and the growth of f (t) at infinity; see § 2. [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF] The parameters ν and ε measure, respectively, the inverse time-scale of the forced oscillations and their amplitude. Physicists consider different regimes, where the two parameters are tied in various ways. 1 To do this they assume some relations between ε and ν, explicitly or implicitly. In our work we choose

ε 2q * = ρν,
where ρ > 0 is a constant. This assumption is within the usually imposed bounds (see [START_REF] Nazarenko | Wave turbulence[END_REF]). Passing to the slow time τ = νt, we get the rescaled equation

u + iν -1 (-∆u) = -f (-∆)u -iρ|u| 2q * u + b k βk (τ )e ik•x , ( 1.5)
where u = u(τ, x), x ∈ T d L and the upper dot stands for d/dτ . If we write u(τ, x) as a Fourier series, u(τ, x) = k v k (τ )e ik•x , then in view of (1.2), (1.5) may be written as the system vk + iν -1

λ k v k = -γ k v k + 2ρi ∂H(v) ∂v k + b k βk (τ ), k ∈ Z d L .
(1.6)

Here H(v) is the Hamiltonian H expressed in terms of the Fourier coefficients

v = (v k , k ∈ Z d L ), H(v) = - 1 2q * + 2 k1,...,k2q * +2∈Z d L v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +2 δ 1•••q * +1 q * +2•••2q * +2 , (1.7)
and we use the standard notation (see [START_REF] Nazarenko | Wave turbulence[END_REF])

δ 1•••q * +1 q * +2•••2q * +2 = 1 if k 1 + • • • + k q * +1 -k q * +2 -• • • -k 2q * +2 = 0, 0 otherwise. (1.8)
As before, we are interested in the limit ν → 0. 2We note that our method also applies to (1.6) with Hamiltonians H of the form (1.2), where the density of the Hamiltonian is a real-valued polynomial of u and ū (not necessarily a polynomial of |u| 2 ). For instance, we could work with the cubic Hamiltonians H 3 = |u| 2 (u + ū) dx or H 3 = (u 3 + ū3 ) dx.

Discrete turbulence

In physics, (1.1) and (1.5) with ν, ε 1 are treated by the theory of weak turbulence (WT); see, for instance, [START_REF] Nazarenko | Wave turbulence[END_REF][START_REF] Zakharov | Kolmogorov spectra of turbulence[END_REF]. That theory either deals with (1.5), where L = ∞ by formally replacing Fourier series for L-periodic functions with Fourier integrals and making them bold transformations, or considers the limit ν, ε → 0 simultaneously with the limit L → ∞. That is, WT theory considers the iterated limit

L → ∞, ε,ν → 0, (1.9)
and treats it in an equally bold way. Concerning this limit, the WT theory makes a number of remarkable predictions based on tools and ideas developed in the community that can be traced back to the work [START_REF] Peierls | On the kinetic theory of thermal conduction in crystals[END_REF]. The relation between the parameters in (1.9) is not quite clear, and it may be better to talk about the WT limits (rather than about a single case).

In order to understand the double limit above, it is natural to first study the limit ν → 0 (with L fixed). Its deterministic version recently received attention in the physical literature as the 'discrete turbulence (DT) limit' (see [START_REF] Kartashova | Nonlinear resonance analysis: theory, computation, applications[END_REF] and [21, § 10]). Similar limits were considered by mathematicians interested in related problems (see [START_REF] Gérard | Effective integrable dynamics for a certain nonlinear wave eqaution[END_REF]), and were used by them for intermediate arguments (see, for example, [START_REF] Faou | The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation[END_REF]).

Our work is dedicated to rigorous justification of the DT limit for the dampeddriven equation (1.3) ν∼ε 2q * =(1.5). Namely, we show that when ν → 0, statistical characteristics of actions of solution u ν for (1.5) have limits of order 1, described by actions of solutions for a certain effective equation that is a nonlinear stochastic equation with coefficients of order 1 and with a Hamiltonian nonlinearity that is made out of the resonant terms of the nonlinearity |u| 2q * u.

The effective equation above is a natural stochastic version of similar equations from the deterministic (physical) DT (see [START_REF] Kartashova | Nonlinear resonance analysis: theory, computation, applications[END_REF][START_REF] Nazarenko | Wave turbulence[END_REF]). So, in a sense, our results justify the physical DT in the stochastic setting. But in the stochastic case we do more than that since we also treat the stationary regime for (1.5) and show that it converges to that for the effective equation. So solutions of the latter approximate (in distribution) solutions of the former as t → ∞ and ν → 0. Remarkably, in the stationary regime the effective equation approximates not only the actions of solutions with ν 1, but also their angles (see § 1.4). As the title of the paper suggests, our argument is a form of averaging. The latter is a tool that is used by the WT community on a regular basis, either explicitly (see, for example, [START_REF] Nazarenko | Wave turbulence[END_REF]), or implicitly.

Inviscid limits for damped-driven Hamiltonian PDE, effective equations and interaction representation

Equation (1.3) is the linear Hamiltonian partial differential equation (HPDE) (1.1) ε=0 , driven by a random force, subject to the linear damping -νf (-∆u) and perturbed by the Hamiltonian nonlinearity -ε 2q * iρ|u| 2q * u. Damped-driven HPDEs and the inviscid limits in these equations when the random force and the damping go to zero are very important for physics. In particular, since the d-dimensional Navier-Stokes equation (NSE) with a random force can be regarded as a dampeddriven Euler equation (which is an HPDE), the inviscid limit for the NSE describes the d-dimensional turbulence. The NSE with a random force, especially when d = 2, has been intensively studied in the past years, but the corresponding inviscid limit turned out to be very complicated even for d = 2 (see [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF]). The problem of this limit becomes feasible when the underlying HPDE is integrable or linear. The most famous integrable PDE is the Korteweg-de Vries (KdV) equation. Its dampeddriven perturbations and the corresponding inviscid limits were studied in [START_REF] Kuksin | Damped-driven KdV and effective equations for long-time behaviour of its solutions[END_REF][START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF]. In [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF] the method of those works was applied to the situation when the unperturbed HPDE is the Schrödinger equation

u t + i(-∆u + V (x)u) = 0, x ∈ T d L , (1.10)
where the potential V (x) is in general a function of position. Crucial for the previously mentioned works is that the unperturbed equation is free from strong resonances: for [START_REF] Kuksin | Damped-driven KdV and effective equations for long-time behaviour of its solutions[END_REF][START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF] it means that all solutions of the KdV equation are almostperiodic functions of time such that for a typical solution the corresponding frequency vector is free from resonances; for [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF] it means that, for the typical potentials V (x) considered there, the spectrum of the linear operator in (1.10) is nonresonant.

In contrast, now the linear operator in the unperturbed equation (1.1) ε=0 has the eigenvalues

λ k ∈ k -2 Z, k ∈ Z d L (see (1. 4
)), which are highly resonant (accordingly, all solutions for (1.1) ε=0 are periodic with the same period 2πL -2 ). This gives rise to an additional difficulty. To explain it, we rewrite (1.5)=(1.6) as a fast-slow system, defining

I k = 1 2 |v k | 2 , ϕ k = Arg v k (
these are the action angles for the linear Hamiltonian system (1.1) ε=0 ). In the new variables (1.5) reads

İk (τ ) = v k • P k (v) + b 2 k + b k (v k • βk ), (1.11) φk (τ ) = -ν -1 λ k + I -1 k • • • , (1.12)
where k ∈ Z d L , • indicates the real scalar product in C R 2 , P (v) is the vector field on the right-hand side of the v-equation (1.6) and • • • abbreviates a factor of order 1 (as ν → 0). If the frequencies {λ k } are resonant, then the equations for some linear combinations of the phases ϕ k are slow, which makes it more difficult to analyse the system. The method of resonant averaging treats this problem in finite dimension; see [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF] and § 2.2. In the situation at hand, we have an additional problem: the ϕ-equations (1.12) have singularities at the locus = {I : I k = 0 for some k}, (1.13) which is dense in the space of sequences (I k , k ∈ Z d L ), and the averaged I-equations

İk (τ ) = v k • P k (I) + b 2 k + b k 2I k βk (τ ), k ∈ Z d L , (1.14)
where • signifies the average in ϕ ∈ T ∞ , have weak singularities there. A way to overcome these difficulties is to find for (1.11), (1.12) an effective equation, which is a system of regular equations

vk = R k (v) + b k βk (τ ), k ∈ Z d L , (1.15)
such that under the natural projection

v k → I k = 1 2 |v k | 2 , k ∈ Z d L ,
solutions of (1.15) transform to solutions of (1.14). In [START_REF] Kuksin | Damped-driven KdV and effective equations for long-time behaviour of its solutions[END_REF] this approach was used to study the perturbed KdV equation when written as a fast-slow system, similar to (1.11), (1.12). That system has strongly nonlinear behaviour, and in [START_REF] Kuksin | Damped-driven KdV and effective equations for long-time behaviour of its solutions[END_REF] the effective equation was constructed as a kind of averaging of the corresponding I-equations. In [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF] an effective equation for the damped-driven non-resonant equation (1.10) was derived in a similar way. If the introduced damping is linear and the nonlinearity is Hamiltonian, as in (1.3), then the effective equation in [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF] is linear. When the unperturbed Hamiltonian system is linear, an alternative way to find an effective equation is to use the interaction representation, i.e. to pass from the complex variables v k (τ ) (which diagonalize the linear system) to the fast rotating variables

a k (τ ) = e iν -1 λ k τ v k (τ ), k ∈ Z d L . (1.16) Since |a k | = |v k |,
the limiting dynamics of the a-variables controls the limiting behaviour of the actions I k . So a regular system of equations, describing the limiting a-dynamics, is the effective equation. Bogolyubov used this approach for finitedimensional deterministic averaging, calling it averaging in quasilinear systems (see [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF]). The interaction representation is systematically used in WT theory. Now consider the fast-slow equations (1.11), (1.12), which come from (1.6), where the fast motion (1.12) is highly resonant. Repeating the construction of the effective equation from [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF], but replacing there the usual averaging by the resonant averaging, we find an effective equation corresponding to (1.6). It turns out to be another damped-driven Hamiltonian system with a Hamiltonian H res obtained via the resonant averaging of H(v); see § 3.2. As we said above, an alternative way to derive the effective equation is through the interaction representation, i.e. by transition from the v-variables to the a-variables (1.16). In view of (1.6), the a-variables

satisfy the system of equations ȧk = -γ k a k + e iν -1 λ k τ b k βk (τ ) -ρi k1,...,k2q * +1∈Z d L a k1 • • • a kq * +1 ākq * +2 • • • āk2q * +1 δ 1•••q * +1 q * +2•••2q * +1k × exp(-iν -1 τ (λ k1 + • • • + λ kq * +1 -λ kq * +2 -• • • -λ k2q * +1 -λ k )), (1.17) for k ∈ Z d L .
The terms constituting the nonlinearity oscillate rapidly as ν goes to zero unless the sum of the eigenvalues in the exponent in the final two lines vanishes. The processes {e iν -1 λ k τ βk (τ ), k ∈ Z d L } make another set of standard independent complex white noises. This leads to the right guess that only the terms for which this sum equals zero (i.e. the resonant terms) contribute to the limiting dynamics, and that the effective equation is the following damped-driven Hamiltonian system:

vk = -γ k v k + 2ρi ∂H res (v) ∂v k + b k βk (τ ), k ∈ Z d L . (1.18)
Here the Hamiltonian H res (v) is given by the sum

- 1 2q * + 2 k1,...,k2q * +2∈Z d L v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +2 δ 1•••q * +1 q * +2•••2q * +2 δ(λ 1•••q * +1 q * +2•••2q * +2 ), (1.19) so that 2ρi(∂H res /∂v k )(v) is -ρi k1,...,k2q * +2∈Z d L v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +1 δ 1•••q * +1 q * +2•••2q * +1k δ(λ 1•••q * +1 q * +2•••2q * +1k ),
(1.20) where we use another abbreviation commonly used in physics

δ(λ 1•••q * +1 q * +2•••2q * +2 ) = 1 if λ k1 + • • • + λ kq * +1 -λ kq * +2 -• • • -λ k2q * +2 = 0, 0 otherwise.
(1.21) This representation for H res is different from that given by the resonant averaging. Its advantage is the natural relation with the a-variables, with which it is convenient to study the limit ν → 0. The representation for H res by means of the resonant averaging turns out to be more useful in the study of properties of H res and of the corresponding Hamiltonian vector field.

We saw that the effective equation can be obtained from system (1.6) by a simple procedure: drop the fast rotations and replace the Hamiltonian H by its resonant average H res . In contrast with the non-resonant case, this is a nonlinear system. The corresponding Hamiltonian system

vk = 2ρi ∂H res (v) ∂v k , k ∈ Z d L , (1.22)
has a vector field, locally Lipschitz in sufficiently smooth spaces, so (1.18) is well posed locally in time. In fact, it is globally well posed. We get this result in § 5.1 as a simple consequence of our main theorems.

The Hamiltonian H res has two convex quadratic integrals

H 0 (v) = 1 2 |v k | 2 , H 1 = 1 2 λ k |v k | 2 ,
which are similar to the energy and the enstrophy integrals for the two-dimensional Euler equation on T 2 (see (3.33)), and the vector integral of moments M (u) =

1 2 k|u k | 2 ∈ R d ,
which can be compared with the extra integrals of the twodimensional Euler equation. In addition, the vector field (1.20) is nonlinear homogeneous and Hamiltonian, as is that of the Euler equation. This makes the effective equation (1.18) similar to the two-dimensional Navier-Stokes system on T 2 . Fortunately the former is significantly simpler than the latter.

Results

The main results of our work are stated and proved in § 5, and are based on properties of the effective equation, established earlier. They imply that the longtime behaviour of solutions for (1.5), when ν → 0, is controlled in distribution by solutions for the effective equation. We start with the results on the Cauchy problem. So, let v ν (τ ) be a solution of (1.6)

such that v ν (0) = v 0 , where v 0 = (v 0k , k ∈ Z d L )
corresponds to a sufficiently smooth function u 0 (x). Let us fix any T > 0.

Consider the vector of actions

I(v ν (τ )) = {I k (v ν (τ )), k ∈ Z d L }.
Theorem A. When ν → 0, we have the weak convergence of measures

D(I(v ν (τ ))) D(I(v 0 (τ ))), (1.23)
where v 0 (τ ), 0 τ T , is a unique solution of (1.18) such that v 0 (0) = v 0 .

For any ξ ∈ Z ∞ 0 , where Z ∞ 0 is the set of integer vectors (ξ k , k ∈ Z d L ) of finite length, we define

Φ ξ (v ν (τ )) := ξ k ϕ k (v ν (τ )) ∈ S 1 = R/2πZ.
Then, in addition to (1.23), for a resonant vector ξ ∈ Z ∞ 0 (i.e. for a vector ξ such that k ξ k λ k = 0) the distribution of Φ ξ (v ν (τ )), mollified in τ , converges as ν → 0 to that of Φ ξ (v 0 (τ )). Conversely, if ξ is non-resonant, then the measure D(Φ ξ (v ν (τ ))), mollified in τ , converges to the Lebesgue measure on S 1 . All of this is proved in § 5.1 using the interaction representation (1.17) for (1.3).

The limiting behaviour of solutions v ν (τ ) can be described without evoking the effective equation (see proposition 5.5). Now consider a stationary measure µ ν for (1.5) (it always exists). We have the following theorem.

Theorem B. Every sequence ν j → 0 has a subsequence ν j → 0 such that

I • µ νj I • m 0 , Φ (ξ) • µ νj Φ (ξ) • m 0
for any resonant vector ξ ∈ Z ∞ 0 , where m 0 is a stationary measure for (1.18). If a vector ξ is non-resonant, then the measure Φ ξ • µ ν converges, as ν → 0, to the Lebesgue measure on S 1 .

If the effective equation has a unique stationary measure m 0 , then the limits in theorem 1.4 do not depend on the sequence ν j → 0, so the convergences hold as ν → 0. Remarkably, in this case the measure m 0 controls not only the slow, but also the fast components of the measures µ ν .

Theorem C. If the effective equation has a unique stationary measure m 0 , then µ ν m 0 as ν → 0.

In particular, if the effective equation has a unique stationary measure m 0 and (1.3) is mixing,3 then m 0 describes asymptotic behaviour of distributions of solutions u(t) for (1.3) as t → ∞ and ν → 0:

lim ν→0 lim t→∞ D(u(t)) = m 0 .
In view of the last theorem, it is important to understand when the effective equation has a unique stationary measure and is mixing. This is discussed in § 5.3. In particular, the mixing holds if q * = 1, f (t) = t + 1 and d 3.

Other equations. Our approach applies to other equations usually considered in WT theory. In particular, in [START_REF] Kuksin | The limit of small Rossby numbers for randomly forced quasigeostrophic equation on β-plane[END_REF] we applied it to the two-dimensional quasigeostrophic equation on the β-plane with random force:

(-∆ + K)ψ -ρJ(ψ, ∆ψ) -βψ x = random force -κ∆ 2 ψ + ∆ψ. (1.24)
Here ψ is the stream function, ψ = ψ(t, x, y), where x ∈ R/LZ and y ∈ R/Z, ∆ψ is the Ekman damping, -κ∆ 2 ψ is the kinematic viscosity and the random force is similar to that in (1.3) and (1.5). The equation has the same structure as (1.5), and our approach is to prove that for typical values of the horizontal period L (when the structure of resonances is relatively simple) the limiting, as β → ∞, behaviour of solutions for (1.24) exists, is uniform in κ ∈ (0, 1], and is described by an effective equation that is an infinite system of stochastic equations. This system splits into invariant systems of complex dimension less than or equal to 3; each of them is an integrable Hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits lim L=ρ→∞ lim β→∞ and lim κ→0 lim β→∞ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.

Weak turbulence

The most famous prediction of WT theory (see § 1.2) deals with the distribution of the energy of solutions for (1.1) and (1.3) between the frequencies. To describe the corresponding claims, consider the quantity E|v k (τ )| 2 , average it in time 4 τ and in wave vectors k ∈ Z d L such that |k| ≈ r > 0; next, properly scale this and denote the result E r . The function r → E r is called the energy spectrum. It is predicted by WT theory that, in a certain inertial range r ∈ [r 1 , r 2 ], which is contained in the spectral zone where the random force is negligible (i.e. where |b k | ≪ (E|v k | 2 ) 1/2 ), the energy spectrum has an algebraic behaviour

E r ∼ r -α for r ∈ [r 1 , r 2 ], (1.25)
for a suitable α > 0. If the WT is stated in terms of the iterated limits (1.9), then only the limits that lead to the algebraic energy spectra (1.25) are relevant.

In our companion work [START_REF] Kuksin | Derivation of a wave kinetic equation from the resonantaveraged stochastic NLS equation[END_REF] we study the effective equation for (1.5) under the limit L → ∞, evoking the heuristic tools from WT theory as presented in [START_REF] Zakharov | Kolmogorov spectra of turbulence[END_REF] and mimicking the logic of that book. There we show (heuristically) that a suitable choice of the function ρ(L) leads, in the limit of L → ∞, to a wave kinetic equation for the averaged actions

n k (t) = 1 2 E|v k (t)| 2 .
That equation is different from the heuristic kinetic equations obtained by WT methods (see [3, § 1.2] and [28, § 2.2.3]), but is closely related to them so that the Zakharov ansatz applies and allows one to obtain stationary solutions of the equation, algebraic in r = |k| and corresponding to energy spectra of the desired form (1.25). 5The rigorous and heuristic results obtained in this work and in [START_REF] Kuksin | Derivation of a wave kinetic equation from the resonantaveraged stochastic NLS equation[END_REF] encourage us to pursue our program to study the WT in the model, given by (1.3), which brings to WT theory the advantage of a rigorous foundation based on the recent results of stochastic calculus. We believe that some predictions of WT theory (including the fact that the limiting behaviour of the averaged actions n k (t) is described by a certain wave kinetic equation that admits stationary solutions of the algebraical form (1.25)) may be obtained if not under the iterated limit 'first ν → 0, next L → ∞', then under its suitable modification (for example, 'ν → 0 and L → ∞ in such a way that νL → 0').

Notation and agreement.

The stochastic terminology that we use agrees with [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]. All filtered probability spaces we work with satisfy the usual condition (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]). 6Sometimes we neglect to mention that a certain relation holds almost surely (a.s.) Spaces of integer vectors. We denote by Z ∞ 0 the set of vectors in Z ∞ of finite length, and define Z ∞ +0 = {s ∈ Z ∞ 0 : s k 0 ∀k}. See also (2.15) and (3.9).

Infinite vectors. For an infinite vector ξ = (ξ 1 , ξ 2 , . . . ) (integer, real or complex) and N ∈ N we denote by ξ N the vector (ξ 1 , . . . , ξ N ) or the vector (ξ 1 , . . . , ξ N , 0, . . . ), depending on the context. For a complex vector ξ and s ∈ Z ∞ +0 we define ξ s = j ξ sj j .

Norms. We use

| • | to denote the Euclidean norm in R d and in C R 2 , as well as the 1 -norm in Z ∞ 0 . For the norms | • | h m and | • | h m I see (2.13) onwards.
Scalar products. The notation '•' stands for the scalar product in Z ∞ 0 , the pairing of Z ∞ 0 with Z ∞ , the Euclidean scalar product in R d and in C. The latter means that if u, v ∈ C, then u • v = Re(ūv). The L 2 -product is denoted by •, • , and we also denote by f, µ = µ, f the integral of a function f against a measure µ.

Preliminaries

Since in this work we are not interested in the dependence of the results on L, from now on it will be kept fixed and equal to 1, apart from in § 4. There we make explicit calculations, controlling how the results depend on L.

A priori estimates

In this section we discuss preliminary properties of solutions for (1.5). We found it convenient to parametrize the vectors from the trigonometric basis {e ik•x } with natural numbers and to normalize them. That is, to use the basis {e j (x), j 1}, where

e j (x) = (2π) -d/2 e ik•x , k = k(j). (2.1)
The functions e j (x) are eigenfunctions of the Laplacian, -∆e j = λ j e j , so ordered

that 0 = λ 1 < λ 2 • • • . Accordingly, (1.5) reads u + iν -1 (-∆u) = -f (-∆)u -iρ|u| 2q * u + d dτ ∞ j=1 b j β j (τ )e j (x), (2.2)
u = u(τ, x)
, where f (-∆)e j = γ j e j with γ j = f (λ j ). The processes β j = β j + iβ -j , j 1, are standard independent complex Wiener processes. The real numbers b j are such that for a suitable sufficiently large even integer r (defined in (3.13)) we have

B r := 2 ∞ j=1 λ r j b 2 j < ∞.
By H p , p ∈ R, we denote the Sobolev space H p = H p (T d , C), regarded as a real Hilbert space, and by •, • we denote the real L 2 -scalar product on T d . We provide H p with the norm • p ,

u 2 p = ∞ j=1 |u j | 2 (λ j ∨ 1) p for u(x) = ∞ j=1
u j e j (x).

Let u(t, x) be a solution of (2.2) such that u(0, x) = u 0 ; it satisfies standard a priori estimates that we now discuss, following [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF]. Firstly, for a suitable ε 0 > 0, uniformly in ν > 0 one has

Ee ε0 u(τ ) 2 0 C(B 0 , u 0 0 ) ∀τ 0. (2.3) Assume that q * < ⎧ ⎨ ⎩ ∞ if d = 1, 2, 2 d -2 if d 3.
Then the following bounds on the Sobolev norms of the solution hold for each 2m r and every n:

E sup 0 τ T u(τ ) 2n 2m + T 0 u(s) 2 2m+1 u(s) 2n-2 2m ds u 0 2n 2m + C(m, n, T )(1 + u 0 cm,n 0 ), (2.5)
E u(τ ) 2n 2m C(m, n) ∀τ 0, (2.6)
where C(m, n, T ) and C(m, n) also depend on B 2m . Estimates (2.5), (2.6) are assumed everywhere in our work. As we have explained, they are fulfilled under assumption (2.4), but if the function f (t) grows superlinearly, then restriction (2.4) may be weakened.

Relations (2.5) imply in the usual way (see [START_REF] Hairer | Exponential mixing properties of stochastic PDE's through asymptotic coupling[END_REF][START_REF] Kuksin | Randomly forced CGL equation: stationary measures and the inviscid limit[END_REF][START_REF] Odasso | Ergodicity for the stochastic complex Ginzburg-Landau equations[END_REF][START_REF] Shirikyan | Ergodicity for a class of Markov processes and applications to randomly forced PDEs[END_REF]) that (2.2) is regular in the space H r in the sense that for any u 0 ∈ H r it has a unique strong solution u(t, x), equal to u 0 at t = 0, and satisfying estimates (2.3), (2.5) for any n. By the Bogolyubov-Krylov argument applied to a solution of (2.2), starting from the origin at t = 0, this equation has a stationary measure µ ν , supported by the space H r , and a corresponding stationary solution u ν (τ ), Du ν (τ ) ≡ µ ν , also satisfies (2.3) and (2.6).

Resonant averaging

Let W ∈ Z n , n 1, be a non-zero integer vector such that its components are relatively prime (so if W = mV , where m ∈ Z and V ∈ Z n , then m = ±1). We call the set

A = A(W ) := {s ∈ Z n : W • s = 0} (2.7)
the set of resonances for W . This is a Z-module. Denote its rank by r. Here and everywhere below the finite-dimensional vectors are regarded as column vectors.

Lemma 2.1. The rank r equals n -1. There exists a system ζ 1 , . . . , ζ n of integer vectors in Z n such that span Z {ζ 1 , . . . , ζ n-1 } = A, and the n

× n matrix R = (ζ 1 ζ 2 • • • ζ n ) is unimodular (i.e. det R = ±1).
That is to say, the vectors (ζ 1 , . . . , ζ n-1 ) make an integer basis of the hyperspace

W ⊥ ⊂ R n .
Proof. We restrict ourselves to the case in which some component of the vector W equals 1 since this is the result we need below. For the general case and for a more general statement, see, for example, [2, § 7].

Without loss of generality we assume that W n = 1. Consider the matrix such that its nth column is W and for j < n the jth column is the vector e j = (e j 1 , . . . , e j n ) T , where e j l = δ j,l . It is unimodular and transforms the basis vector e n to W . Its inverse is a unimodular matrix B such that BW = e n . Let s be any vector in A. where the m j s are some integers. This proves the lemma if we choose ζ j = B T e j , j = 1, . . . , n. Note that the matrix R equals B T .

Since W • s = 0 ⇐⇒ BW • (B T ) -1 s = 0 ⇐⇒ e n • (B T ) -1 s = 0, we have (B T ) -1 s = n-1 j=1 m j e
Since R T W = BW = e n , the automorphism of the torus T n → T n , ϕ → y = R T ϕ, 'resolves the resonances' in the differential equation φ = W in the sense that it transforms it into the equation ẏ = R T W = (0, . . . , 0, 1) T .

(2.8)

Let us consider a mapping L = L A : T n → T n-1 that is 'dual to the module A':

L : T n ϕ → (ϕ • Re 1 , . . . , ϕ • Re n-1 ) T ∈ T n-1 .
(2.9)

The basis {η j = (R T ) -1 e j , 1 j n} is dual to the basis {ζ j = Re j , 1 j n}, since

η j • ζ l = (R T ) -1 e j • Re l = δ j,l . Therefore, if we decompose ϕ ∈ T n in the η-basis, ϕ = k y k η k = (R T ) -1 y, then Lϕ = (y 1 , . . . , y n-1 ) T . That is, L • (R T ) -1 (y 1 , . . . , y n ) T = (y 1 , . . . , y n-1 ) T . (2.10)
In particular, the fibres of the mapping L are the circles R({y} × S 1 ), where y = (y 1 , . . . , y n-1 ) T ∈ T n-1 .

For a continuous function f on T n we define its resonant average with respect to the integer vector W as the function

f W (ϕ) := 2π 0 f (ϕ + tW ) dt, (2.11)
where we have set dt := dt/2π.

Lemma 2.2. Let f be a C ∞ -function on T n , f (ϕ) = f s e is•ϕ . Then f W (ϕ) = f s δ 0,s•W e is•ϕ = s∈A(W ) f s e is•ϕ . (2.12)
Proof. It is immediate that (2.12) holds for a trigonometric polynomial. Since for C ∞ -functions the series in (2.12) converges well, by continuity the result holds for smooth functions f .

Resonant averaging in a Hilbert space

Consider the Fourier transform for complex functions on T d , which we write as the mapping

F : H u(x) → v = (v 1 , v 2 , . . . ) ∈ C ∞ ,
defined by the relation u(x) = v k e k (x). In the space of complex sequences we introduce the norms

|v| 2 h p = k 1 |v k | 2 (λ k ∨ 1) p , p ∈ R, (2.13)
and set h p = {v | |v| h p < ∞}. Then

|Fu| h p = u p ∀p.
For k 1 let us define

I k = I(v k ) = 1 2 |v k | 2 and ϕ k = ϕ(v k ), where, for v ∈ C, ϕ(v) = Arg v ∈ S 1 if v = 0,
and ϕ(0) = 0 ∈ S 1 . For any r 0 consider the mappings

Π I : h r v → I = (I 1 , I 2 , . . . ) ∈ h r I+ , Π ϕ : h r v → ϕ = (ϕ 1 , ϕ 2 , . . . ) ∈ T ∞ .
(2.14) Here h r I+ is the positive octant {I : I k 0 ∀k} in the space h r I , where

h r I = I |I| h r I = 2 k (λ k ∨ 1) r |I k | < ∞ .
Abusing notation slightly, we will write

Π I (F(u)) = I(u), Π ϕ (F(u)) = ϕ(u).
The mapping I : H r → h r I is 2-homogeneous continuous, while the mapping ϕ : H r → T ∞ is Borel measurable (the torus T ∞ is given the Tikhonov topology and the corresponding Borel σ-algebra).

For infinite integer vectors s = (s 1 , s 2 , . . . ) (and only for them) we will write the

l 1 -norm of s as |s|, |s| = j |s j |.
We define Z ∞ 0 = {s ∈ Z ∞ : |s| < ∞}, and for a vector s = (s 1 , s 2 , . . .

) ∈ Z ∞ 0 write Λ • s = k λ k s k , supp s = {k : s k = 0}, s = max{k : s k = 0}. (2.15) Similarly, for ϕ ∈ T ∞ and s ∈ Z ∞ 0 we write ϕ • s = s • ϕ = k ϕ k s k ∈ S 1 .
Let us fix some m ∈ N ∪ ∞ and define the set of resonances of order m for the (integer) frequency vector Λ = (λ 1 , λ 2 , . . . ) as

A(Λ, m) = {s ∈ Z ∞ 0 : |s| m, Λ • s = 0}.
(2.16)

We will abbreviate

A(Λ) = A(Λ, ∞) = {s ∈ Z ∞ 0 : Λ • s = 0}. Let us define Z ∞ +0 = {s ∈ Z ∞ 0 : s k 0 ∀k}
, and consider a series on some space h r , r 0,

F (v) = p,q,l∈Z ∞ +0 C pql (2I) p v q vl , ( 2.17)
where

I = I(v), C pql = 0 if supp q ∩ supp l = ∅, and for v ∈ h r , q ∈ Z ∞ +0 we write v q = v qj j .
We assume that the series converges normally in h r in the sense that for each R > 0 we have

p,q,l∈Z ∞ +0 |C pql | sup |v| h r ,|w| h r R |v p w p v q w l | < ∞. (2.18) Clearly, F (v) = F (v, v)
, where F is a (complex) analytic function on h r × h r . With some abuse of language and following a tradition in physics we will say that F is analytic in v and v. In particular, F (v) is a real-analytic (so continuous) function of v, and the series (2.17) converges absolutely.

The resonant averaging of F can be conveniently defined by introducing, for any θ ∈ T ∞ , the rotation operator Ψ θ , which is a linear operator in h 0 :

Ψ θ (v) = v , v k = e iθ k v k .
Clearly, this is a unitary isomorphism of every space h r . Note that (I × ϕ)(Ψ θ v) ≡ (I(v), ϕ(v) + θ). Using that Λ is an integer vector and based on definition (2.11), we give the following.

Definition. If a function F ∈ C(h r ) is given by a normally converging series (2.17), then its resonant average with respect to Λ is the function

F Λ (v) := 2π 0 F (Ψ tΛ (v)) dt, dt = dt/2π. (2.19)
Defining a function F (I, ϕ) by the relation

F (v) = F (I(v), ϕ(v)), we see that F Λ (v) = 2π 0
F (I, ϕ + tΛ) dt. So this definition agrees well with (2.11). Consider a monomial F = (2I) p v q vl . By lemma 2.2 we have (2I) p v q vl Λ = (2I) p v q vl δ 0,(q-l)•Λ .

Now assume that F is given by a normally convergent series (2.17) and has degree m ∞ in the sense that C pql = 0 unless |q| + |l| m. Then

F Λ (v) = q-l∈A(Λ,m) C pql (2I) p v q vl = (q-l)•Λ=0 C pql (2I) p v q vl . (2.20)
If the series (2.17

Averaging for (2.2)

Everywhere below T is a fixed positive number.

Equation (2.2) in v-variables, resonant monomials and combinations of phases

Let us pass in (2.2), with u ∈ H r , r > d/2, to the v-variables, v = F(u) ∈ h r :

dv k + iν -1 λ k v k dτ = P k (v) dτ + b k dβ k (τ ), k 1, v(0) = F(u 0 ) =: v 0 . (3.1)
Here

P k = P 1 k + P 0 k , ( 3.2)
where P 1 and P 0 are, correspondingly, the linear and nonlinear Hamiltonian parts of the perturbation. So P 1 k is the Fourier image of -f (-∆), i.e. P 1 k = diag{-γ k , k 1}, while the operator P 0 is the mapping u → -iρ|u| 2q * u, written in the v-variables. That is,

P 0 (v) = -iρF(|u| 2q * u), u = F -1 (v).
Every component P 0 k of it is a sum of monomials

P 0 k (v) = p,q,l∈Z ∞ +0 C pql k (2I) p v q vl = p,q,l∈Z ∞ +0 P 0pql k (v), k 1, (3.3)
where

C pql k = 0 unless 2|p| + |q| + |l| = 2q * + 1 and |q| = |l| + 1. It is straightforward that P 0 k (I, ϕ) (see (2.14)
) is a function of ϕ = (ϕ j , j 1) of order 2q * + 1, and that the mapping P 0 is analytic and of polynomial growth.

Lemma 3.1. The nonlinearity P 0 defines a real-analytic transformation of h r if r > d/2. The mapping P 0 (v) and its differential dP 0 (v) both have polynomial growth in |v| h r .

We will refer to (3.1) as the v-equations. For any s ∈ Z ∞ 0 consider the linear combination of phases

Φ s : h 0 → S 1 , v → s • ϕ(v).
We fix m = 2q * + 2 and find the corresponding set A = A(Λ, m) of resonances of order m (see (2.16)). We order vectors in the set A, that is, we write it as A = {s (1) , s (2) , . . . } in such a way that s (j1) s (j2) if j 1 j 2 , and for N 1 define

J(N ) = max{j : s (j) N }. ( 3.4)
For any s (j) ∈ A consider the corresponding resonant combination of phases ϕ(v), Φ j (v) = Φ sj (v), and introduce the Borel measurable mappings

h r v → Φ = (Φ 1 , Φ 2 , . . . ) ∈ S 1 × S 1 × • • • =: T ∞ , h r v → (I × Φ) ∈ h r I+ × T ∞ .
Note that the system Φ of resonant combinations is highly overdetermined: there are many linear relations between its components Φ j . Let us pass in (3.1) from the complex variables v k to the action-angle variables

I, ϕ dI k (τ ) = (v k • P k )(v) dτ + b 2 k dτ + b k (v k • dβ k ) (3.5)
(here • indicates the real scalar product in C R 2 ), and

dϕ k (τ ) = (-ν -1 λ k + |v k | -2 (iv k • P k (v))) dτ + |v k | -2 b k (iv k • dβ k ). (3.6)
The equations for the actions are slow, while equations for the angles are fast since dϕ k ∼ ν -1 . But the resonant combinations Φ j of angles satisfy slow equations:

dΦ j (τ ) = k 1 s (j) k (|v k | -2 (iv k • P k ) dτ + |v k | -2 b k (iv k • dβ k )), j 1. (3.7)
Repeating for (3.1) and (3.5) the argument of [17, § 7] (see also [12, § 6.2]), we get lower bounds for the norms of the components v k (τ ) of v(τ ). Lemma 3.2. Let v ν (τ ) be a solution of (3.1) and let I ν (τ ) = I(v ν (τ )). Then for any k 1 the convergence

T 0 P{I ν k (τ ) δ} dτ → 0 as δ → 0 (3.8)
holds uniformly in ν > 0 (the rate of the convergence depends on k).

Now we define and study corresponding resonant monomials of v. For any s ∈ Z ∞ 0 , vectors s + , s -∈ Z ∞ +0 such that s = s +s -and supp s = supp s + ∪ supp s -, supp s + ∩ supp s -= ∅ are uniquely defined. Denote by V s the monomial

V s (v) = v s + vs - = l v s + l l l vs - l l . (3.9)
This is a real-analytic function on every space h l , and ϕ(V s (v)) = Φ s (v). Resonant monomials are the functions7

V j (v) = V s (j) (v), j = 1, 2, . . . .

Clearly, they satisfy

I(V j (v)) = (2I) |s (j) |/2 := l (2I l ) |s (j) l |/2 , ϕ(V j (v)) = Φ j (v). (3.10)
Now consider the mapping

V : h l v → (V 1 , V 2 , . . . ) ∈ C ∞ , ( 3.11)
where C ∞ is given the Tikhonov topology. It is continuous for any l. For N 1 define

V (N ) (v) = (V 1 , . . . , V J (v)) ∈ C J ,
where J = J(N ) (see (3.4)).

For any s ∈ Z ∞ 0 , applying the Itô formula to the process V s (v(τ )), we get that

dV s = V s -iν -1 (Λ • s) dτ + j∈ supp s + s + j v -1 j (P j (v) dτ + b j dβ j ) + j∈ supp s - s - j v-1 j ( Pj (v) dτ + b j d βj ) . (3.12) If s = s ∈ Z ∞
0 is perpendicular to Λ, then the first term on the right-hand side vanishes. So V s(τ ) is a slow process, dV s ∼ 1. In particular, the processes dV j , j 1, are slow.

Estimates (2.5) and equation (3.12) readily imply the following lemma.

Lemma 3.3. For any j

1 we have E|V j (v(•))| C 1/3 [0,T ] C j (T ) < ∞, uniformly in 0 < ν 1.
Let us provide the space C([0, T ]; C ∞ ) with the Tikhonov topology, identifying it with the space C([0, T ]; C) ∞ . This topology is metrizable by way of the Tikhonov distance. From now on we fix an even integer r,

r d 2 + 1, (3.13)
and write

h r = h, h r I = h I , C([0, T ], h I+ ) × C([0, T ], C ∞ ) =: H I,V .
We provide H I,V with Tikhonov's distance, the corresponding Borel σ-algebra and the natural filtration of the σ-algebras {F t , 0 t T }.

Let us consider a solution u ν (τ ) of (2.2) satisfying u(0) = u 0 . Define v ν (τ ) = F(u ν (τ )) and write

I(v ν (τ )) = I ν (τ ), V(v ν (τ )) = V ν (τ ) ∈ C ∞ .
Lemma 3.4.

(1) Assume that u 0 ∈ H r . Then the set of laws

D(I ν (•), V ν (•)), 0 < ν 1, is tight in H I,V .
(2) Any limiting measure Q for the set of laws in ( 1) satisfies

E Q |I| n C([0,T ],h r I ) C n ∀n ∈ N, E Q T 0 |I(τ )| h r+1 I dτ C , (3.14)
E Q e ε0|I(τ )| h 0 I C ∀τ ∈ [0, T ]. (3.15)
Proof.

(1) Due to lemma 3.3 and the Arzelà theorem, the laws of processes V j (v ν (•)), 0 < ν 1, are tight in C([0, T ], C) for all j. Due to (2.5) with n = 1, and since the actions I ν k satisfy the slow equations (3.5), the laws of processes I ν (τ ) are tight in C([0, T ], h I+ ) (see, for example, [START_REF] Vishik | Mathematical problems in statistical hydromechanics[END_REF]). Therefore, for every N , any sequence ν → 0 contains a subsequence such that the laws D(I ν (•), V (N ) (v ν (•))) converge along it to a limit. Applying the diagonal process we get another subsequence ν such that the convergence holds for each N . The corresponding limit is a measure m N on the space C([0, T ], h I+ ) × C([0, T ], C) J(N ) . Different measures m N agree, so by Kolmogorov's theorem they correspond to some measure m on the σ-algebra, generated by cylindric subsets of the space C([0, T ], h I+ ) × C([0, T ], C) ∞ , which coincides with the Borel σ-algebra for that space. It is not hard to check that

D(I ν (•), V ν (•))
m as ν = ν → 0. This proves the first assertion.

(2) Estimates (3.14) follow from (2.3), (2.5), the weak convergence to Q and the Fatou lemma; see [19, lemma 1.2.17].

Averaged equations, effective equation, interaction representation

Fix u 0 ∈ H r and consider any limiting measure Q 0 , which exists by lemma 3.4, for the laws

D(I ν (•), V ν (•)) Q 0 as ν → 0. (3.16)
Our goal is to show that the limit Q 0 does not depend on the sequence ν → 0 and to develop tools for its study. We begin with writing down averaged equations for the slow components I and Φ of the process v(τ ), using the rules of stochastic calculus (see [START_REF] Freidlin | Averaging principle for stochastic perturbations of multifrequency systems[END_REF][START_REF] Khasminski | On the avaraging principle for Ito stochastic differential equations[END_REF]), and formally replacing there the usual averaging in ϕ by the resonant averaging • Λ . Let us first consider the I-equations (3.5). The drift in the kth equation is b

2 k + v k • P k = b 2 k + v k • P 1 k + v k • P 0 k , where v k • P 1 k = -2γ k I k and v k • P 0 k (v) = p,q,l∈Z ∞ +0 v k • P 0pql k (v)
(see (3.3)). From § 4, the sum converges normally, so the resonant averaging of the drift is well defined. The dispersion matrix for (3.5) with respect to the real Wiener processes (β

1 , β -1 , β 2 , . . . ) is diag{b k (Re v k Im v k ), k 1} (it is formed by (1 × 2)- blocks).
The diffusion matrix equals the dispersion matrix times its conjugate and equals diag{b 2 k |v k | 2 , k 1}. It is independent from the angles, so the averaging does not change it. For its square root we take diag{b k √ 2I k }, and accordingly write the Λ-averaged I-equations as

dI k (τ ) = v k • P k Λ (I, V ) dτ + b 2 k dτ + b k 2I k dβ k (τ ), k 1 (3.
D jk = -s (j) k b k (2I k ) -1 (Im v k -Re v k ).
Again the diffusion matrix does not depend on the angles and equals M = (M j1j2 ),

M j1j2 = k s (j1) k s (j2) k b 2 k (2I k ) -1 .
The matrix D new with the entries

D new jk = s (j) k b k (2I k ) -1/2
satisfies |D new | 2 = M , and we write the averaged equations for the Φ j s as

dΦ j (τ ) = k 1 s (j) k iv k • P k Λ (I, V ) 2I k dτ + b k √ 2I k dβ -k (τ ) , j 1 (3.18)
(we use here Wiener processes that are independent from those in (3.17) since the differentials v k • dβ k and iv k • dβ k , corresponding to the noises in (3.5) and (3.6), are independent). Equations (3.17) and (3.18) constitute a system of stochastic differential equations for the process (I, V )(τ ) since each Φ j is a function of I and V j . It is overdetermined since there are linear relations between various Φ j s. In addition, (3.17) has a weak singularity at the locus (h) = k {v ∈ h : v k = 0}, while (3.18) has a strong singularity there.

Consider a component v k • P 0 k Λ (v) of the averaged drift in the equation for I k . It may be written as

v k • P 0 k Λ (v) = 2π 0 v k • (e -itλ k P 0 k (Ψ tΛ (v))) dt = v k • R 0 k (v), (3.19)
where we set

R 0 k (v) = 2π 0 e -itλ k P 0 k (Ψ tΛ (v)) dt. That is, R 0 (v) = 2π 0 Ψ -tΛ P 0 (Ψ tΛ v) dt. (3.20)
Repeating the derivation of (2.20) and using that |q| + |l| m -1, we see that

R 0 k (v) = p,q,l∈Z ∞ +0 , q-l∈A(Λ,m)+e k , |q|+|l|+1 m C pql k (2I) p v q vl . (3.21)
Relation (3.21) interprets R 0 (v) as a sum of resonant terms of the mapping P 0 (v), very much in the spirit of WT theory, while (3.20) interprets it as a result of the resonant averaging of P 0 .

The vector field R 0 defines locally Lipschitz operators in the spaces h p , p > d/2:

|R 0 (v) -R 0 (w)| h p C p (|v| h p ∨ |w| h p ) 2q * |v -w| h p . (3.22)
Indeed, in view of (3.20), for any v, w such that |v| h p , |w| h p R we have

|(R 0 (v) -R 0 (w))| h p 2π 0 |Ψ -tΛ (P 0 (Ψ tΛ v) -P 0 (Ψ tΛ w))| h p dt. ( 3.23)
Since P 0 (v) = -iρF(|v| 2q * v), where v = F -1 v, defining Ψ tΛ v = v t , defining w t similarly, and using that the operators Ψ θ define isometries of h p , we bound the right-hand side of (3.23) by

2π 0 |P 0 (v t ) -P 0 (w t )| h p dt = ρ 2π 0 | v t | 2q * v t -| w t | 2q * w t p dt ρC p R 2q * 2π 0 v t -w t p dt ρC p R 2q * |v -w| h p .
Finally, we set

R = R 0 + R 1 , where R 1 k (v) = P 1 k (v) = -γ k v k . Since v k • P 1 k Λ = -2γ k I k Λ = v k • P 1 k = v k • R 1 k , in view of (3.19) we have v k • P k Λ (v) = v k • R k (v). (3.24)
For later use, by the same argument we note that iv k

• P 0 k Λ = iv k • R 0 k and iv k • P 1 k Λ = 0 = iv k • R 1 k . So also iv k • P k Λ (v) = iv k • R k (v). (3.25)
Motivated by the averaging theory for equations without resonances in [START_REF] Kuksin | Damped-driven KdV and effective equations for long-time behaviour of its solutions[END_REF][START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF], we now consider the following effective equation for the slow dynamics in (3.5):

dv k = R k (v) dτ + b k dβ k , k 1. (3.26)
In contrast to the averaged equations (3.17) and (3.18), the effective equation is regular, i.e. it does not have singularities at the locus (h). Since R 0 : h → h is locally Lipschitz, strong solutions for (3.26) exist locally in time and are unique.

Lemma 3.5. A strong solution of (3.26) with a specified initial datum v(0

) = v 0 ∈ h is unique a.s.
The relevance of the effective equation for the study of the long-time dynamics in (2.2)=(3.1) is clear from the next lemma.

Lemma 3.6. Let a continuous process v(τ ) ∈ h be a weak solution of (3.26) such that all moments of the random variable max 0 τ T |v(τ )| h are finite. Then I(v(τ )) is a weak solution of (3.17). Let stopping times 0 τ 1 < τ 2 T and numbers δ * > 0, N ∈ N be such that

I k (v(τ )) δ * for τ 1 τ τ 2 and k N . (3.27)
Then the process (I(v(τ )), Φ j (v(τ )), j J(N )) is a weak solution of the system of averaged equations (3.17), (3.18) j J . 8Proof. Let v(τ ) satisfy (3.26). Applying Itô's formula to I k (v(τ )) and Φ j (v(τ )), j J, we get that

dI k = v k • R k dτ + b 2 k dτ + b k v k • dβ k (3.28)
and

dΦ j = k∈ supp s (j) s (j) k iv k • R k |v k | 2 dτ + b k |v k | 2 iv k • dβ k .
Using (3.24) and (3.25) we see that (3.28) has the same drift and diffusion as (3.17). So I(v(τ )) is a weak solution of (3.17) (see [START_REF] Mikulevicius | Martingale problems for stochastic PDEs[END_REF][START_REF] Yor | Existence et unicité de diffusion à valeurs dans un espace de Hilbert[END_REF]). Similarly, for τ ∈ [τ 1 , τ 2 ], in view of (3.25), the process (I, Φ j , j J), is a weak solution of the system (3.17), (3.18) j J . Now we show that the effective equation describes the limiting (as ν → 0) dynamics for the equations of motions, written in the a-variables of the interaction representation (1.16). Indeed, let u ν (τ ) be a solution of (2.2) satisfying u(0) = u 0 . Define v ν (τ ) = F(u ν (τ )) and consider the vector of a-variables a

ν (τ ) = (a ν k (τ ) = e iν -1 λ k τ v ν k (τ ), k 1) (cf. (1. 16
)). Notice that we obviously have (3.11)). From (3.1), for the vector a ν (τ ) we obtain the system of equations

|v ν (τ )| h m ≡ |a ν (τ )| h m ∀m, I(v ν (τ )) ≡ I(a ν (τ )), V(v ν (τ )) ≡ V (a ν (τ )) (3.29) (see
da ν k = (R k (a ν ) + R k (a ν , ν -1 τ )) dτ + b k e iν -1 λ k τ dβ k (τ ), k 1,
where we have defined

R k (a, ν -1 τ ) = p,q,l∈Z ∞ +0 , q-l-e k ∈A(Λ,m), |q|+|l|+1 m P 0pql k (a) exp(-iν -1 τ (Λ • (q -l -e k ))). (3.30)
This is the non-resonant, fast oscillating part of the nonlinearity (because |Λ • (qle k )| 1). Since { βk (τ ) := e iν -1 λ k τ dβ k (τ ), k 1} is another set of standard independent complex Wiener processes, the process a ν (τ ) is a weak solution of the system of equations

da ν k = (R k (a ν ) + R k (a ν , ν -1 τ )) dτ + b k dβ k (τ ), k 1. (3.31)
We will refer to (3.31) as the a-equations. It is crucial that they are identical to the effective equation (3.26), apart from terms that oscillate fast as ν → 0.

Properties of resonant Hamiltonian H res and the effective equation

Lemma 3.7. The vector field R ,

R 0 = iρ∇H res (v) ∀v ∈ h p , p > d/2, (3.32) is Hamiltonian, where H res (v) = H Λ (v) and H is the Hamiltonian (1.2). Proof. Indeed, since P 0 (v) = iρ∇H(v), R 0 (v) = 2π 0 Ψ -tΛ (iρ∇H(Ψ tΛ (v))) dt = iρ∇ v 2π 0 H(Ψ tΛ (v)) dt = iρ∇ v H res (v),
as Ψ * θ ≡ Ψ -θ , and where we have used (3.20).

Clearly, H res (0) = 0. Since H(u) -C u 2q * +2 0 by the Hölder inequality and since the transformations Ψ tΛ preserve u 0 , we have that

H res (u) -C u 2q * +2 0 ∀u.
The resonant Hamiltonian H res has symmetries, given by some rotations Ψ m , m ∈ R ∞ . Lemma 3.8. The following hold:

(i) letting 1 = (1, 1, . . . ), we have H res (Ψ t1 v) = const. (i.

e. it does not depend on t);

(ii) letting M l be the lth component of the sequence

(k(1), k(2), . . . ), l = 1, . . . , d (see (2.1)), we have H res (Ψ tM l v) = const. for each l; (iii) H res (Ψ tΛ v) = const.
Proof. (i) By (2.19) we have

H res (Ψ t1 v) = 2π 0 H(Ψ t Λ (Ψ t1 v)) dt = 2π 0 H(Ψ t1 (Ψ t Λ v)) dt .
Let us define Ψ t1 (Ψ t Λ v) = v(t; t ). Then (d/dt)v(t; t ) = iv. The flow of this Hamiltonian equation commutes with that of the equation with the Hamiltonian H. 9 So H(v(t; t )) is independent from t for each t , and (i) follows since H res (Ψ t1 v) = H(v(t; t )) dt .

(ii) The proof is the same since the transformations Ψ tM l , t ∈ R, are the flow of the momentum Hamiltonian M l (u) = 1 2 ∞ j=1 k l (j)|u j | 2 , which commutes with H. (iii) This is a straightforward consequence of the definition of the resonant averaging (2.19).

Since the transformations Ψ t1 form the flow of the Hamiltonian

H 0 (v) = 1 2 |v j | 2 = 1 2 |v| 2 h 0 , the transformations Ψ tΛ form the flow of H 1 (v) = 1 2 λ j |v j | 2
, and the transformations Ψ tM l , t ∈ R, form the flow of the momentum Hamiltonian, we may recast the assertions of the last lemma as

{H res , H 0 } = 0, {H res , H 1 } = 0, {H res , M l } = 0 ∀l. (3.33)
Here {•, •} signifies the Poisson bracket. As the transformations Ψ m , m ∈ R ∞ , are symplectic, the symmetries in the lemma above preserve the Hamiltonian vector field R 0 and commute with it. In particular, since Ψ tΛ = e -it∆ , the spectral spaces E λ of the operator -∆, E λ = span{e j : λ j = λ}, are invariant for the flow maps of R 0 . Since the transformations Ψ m , m ∈ R ∞ , obviously preserve the vector field R 1 as well as the random force law in (3.26) (see the proof of lemma 3.9), those Ψ m that are symmetries of R 0 (equivalently, are symmetries of the Hamiltonian H res ) preserve weak solutions of (3.26). So we have the following lemma. Lemma 3.9. If v(τ ) is a solution of (3.26) and m ∈ R ∞ is either a vector m = t1, t ∈ R, or a vector m = tΛ, or m = tM l , l = 1, . . . , d, then also Ψ m v(τ ) is a weak solution.

Proof. Define Ψ m v(τ ) = v (τ ). Applying Ψ m to (3.26), using lemma 3.8 and exploiting the invariance of the operator R 1 with respect to Ψ m , we get

dv k = (Ψ m R(v(τ ))) k dτ + e im k b k dβ k = (R(v (τ ))) k + b k (e im k dβ k ).
Since {e im k β k (τ ), k 1} is another set of standard independent Wiener processes, v (τ ) is a weak solution of (3.26).

Corollary 3.10. If µ is a stationary measure for (3.26) and a vector m is as in lemma 3.9, then the measure Ψ m • µ is also stationary. 9 This follows from the fact that the functional 1 2 |v| 2 h 0 is an integral of motion for the Hamiltonian H, which becomes obvious if we note that in the u-representation H has the form (1.2) and

1 2 |v| 2 h 0 is 1 2 |u| 2 (x) dx.
The next lemma characterizes the increments of R 0 (v) in the space h 0 . It will be needed below to study the ergodic properties of the effective equation. Lemma 3.11. Let p > d/2. Then for any v, w ∈ h p we have

|R 0 (v) -R 0 (w)| h 0 C(|v| h p + |w| h p ) 2q * |v -w| h 0 .
Proof. Repeating the proof of the Lipschitz property of R 0 in the space h (see (3.22)) and using the notation of that proof, i.e. defining Ψ tΛ v = v t , v = F -1 v, and similarly for the vector w, we get that

|R 0 (v) -R 0 (w)| h 0 2π 0 |Ψ -tΛ (P 0 (Ψ tΛ v) -P 0 (Ψ tΛ w))| h 0 dt = 2π 0 |P 0 (v t ) -P 0 (w t )| h 0 dt = 2π 0 | v t | 2q * v t -| w t | 2q * w t 0 dt C 2π 0 (| v t | L ∞ + | w t | L ∞ ) 2q * v t -w t 0 dt C 1 (|v| h p + |w| h p ) 2q * |v -w| h 0 .

Explicit calculation

Here we intend to explicitly calculate the effective equation (3.26), keeping track of the dependence on the size L of the torus. To do that, it is convenient to use the natural parametrization of the exponential basis by vectors k ∈ Z d L ; that is, to decompose functions u(x) as Fourier series u(x) = k∈Z d L v k e ik•x . We modify the norms | • | h p accordingly:

u 2 p = (2πL) d k∈Z d L |k| ∨ 1 L 2p |v k | 2 =: |v| 2 h p .
Now, as in the introduction, the eigenvalues of the negative Laplacian are λ k = |k| 2 and the damping coefficients are γ k = f (λ k ).

In the v-coordinates the nonlinearity becomes the mapping v → P 0 (v), whose kth component is

P 0 k (v) = -iρ k1,...,k2q * +1∈Z d L v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +1 δ 1•••q * +1 q * +2•••2q * +1k
(see (1.8)). Accordingly,

v k • P 0 k = ρ k1,...,k2q * +1∈Z d L Im(v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +1 vk )δ 1•••q * +1 q * +2•••2q * +1k . (4.1)
In order to calculate the resonant average, we first notice that v k •P 0 k can be written as a series (2.17), where |C pql | 1 and |q| + |p| + |l| = 2q * + 2. In this case the sum on the left-hand side of (2.18) is bounded by

C k∈Z d L |v k | 2q * +2 C 1 (L)|v| q * +1 p k∈Z d L |k| -2p q * +1 . So condition (2.18) is met if 2p > d.
Since the order of the resonance is

m = 2q * + 2, v k • P 0 k Λ (v) equals ρ k1,...,k2q * +1∈Z d L Im(v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +1 vk )δ 1•••q * +1 q * +2•••2q * +1k δ(λ 1•••q * +1 q * +2•••2q * +1k ),
(see (1.21)). This follows from (4.1) and (2.19) if one notes that the restriction (ql) • Λ = 0 appearing there is now replaced by the factor δ(λ

1•••q * +1 q * +2•••2q * +1k
). In a similar way, we see that the quantity R 0 k , entering (3.26), takes the form

R 0 k (v) = -iρ k1,...,k2q * +1∈Z d L v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +1 × δ 1•••q * +1 q * +2•••2q * +1k δ(λ 1•••q * +1 q * +2•••2q * +1k ).
Taking into account that R 1 k = -γ k v k , we finally arrive at an explicit formula for the effective equation (3.26):

dv k = -γ k v k -iρ k1,...,k2q * +1∈Z d L v k1 • • • v kq * +1 vkq * +2 • • • vk2q * +1 × δ 1•••q * +1 q * +2•••2q * +1k δ(λ 1•••q * +1 q * +2•••2q * +1k ) dτ + b k dβ k (4.2) with k ∈ Z d L . Due to (3.32), R 0 k (v) = iρ∇ v k H res (v) = 2iρ ∂ ∂v k H res (v).
Therefore, (4.2) can be written as the damped-driven Hamiltonian system (1.18).

Examples. (a) If q * = 1, then (4.2) reads

dv k = -γ k v k -iρ k,k ,k ∈Z d L v k v k vk δ k+k ,k +r δ λ k +λ k ,λ k +λ k dτ + b k dβ k ,
where k ∈ Z d L . If f (t) = t + 1, then this equation looks similar to the complex Ginzburg-Landau equation

u -∆u + u = i|u| 2 u + d dτ b k β k (τ )e ik•x ,
written in the Fourier coefficients. The latter equation possesses nice analytical properties; for example, its stationary measure is unique for any d (see [START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF]).

(b) Our results remain true if the Hamiltonian H, corresponding to the nonlinearity in (1.5), has variable coefficients. In particular, let d = 1 and let the nonlinearity in (1.5) be replaced by -ip(x)|u| 2 u, where p(x) is a sufficiently smooth function.

Then the effective equation is

dv k = -γ k v k -i k1,k2,k3,k4∈Z L v k1 v k2 vk3 p k4 δ k1+k2+k4,k3+k δ k 2 1 +k 2 2 ,k 2 3 +k 2 dτ + b k dβ k ,
where k L ∈ Z d and the p k s are the Fourier coefficients of p(x).

Main results

Averaging theorem for the initial-value problem

We recall that r is a fixed even integer such that r d/2 + 1, and write

h r = h, C([0, T ], h) = H a .
We provide H a with the Borel σ-algebra and the natural filtration of the σ-algebras {F t , 0 t T }. Let v ν (τ ) be a solution of (3.1) such that v ν (0) = v 0 = F(u 0 ) ∈ h r , and consider the corresponding process a ν (τ ). Due to (3.29), the process a ν satisfies obvious analogies with estimates (2.3), (2.5) and (2.6). Since (R + R)(a) is the nonlinearity P (v), written in the a-variables, The proof of the theorem is presented at the end of this section. Let Q 0 be a measure in H I,V , as in (3.16). Since (I, V )(v ν (•)) = (I, V )(a ν (•)) for any ν > 0, rewriting a(τ ) as v(τ ) we derive a corollary of the previous theorem.

|(R + R)(a)(τ )| h = |P (v)(τ )| h C|v(τ )| q * +1 h = C|a(τ )| q * +1
Theorem 5.2. There exists a unique weak solution v(τ ) of effective equation (3.26) satisfying v(0) = v 0 a.s. The law of (I, V )(v(•)) in the space H I,V coincides with Q 0 , and the convergence (3.16) holds as ν → 0. Moreover, for any vectors s1 , . . . , sm ∈ Z ∞ 0 , perpendicular to Λ, we have the convergence

D(I, V s1 , . . . , V sm )(v ν (•)) D(I, V s1 , . . . , V sm )(v(•)).
By this result the Cauchy problem for the effective equation has a weak solution. Using lemma 3.5 and the Yamada-Watanabe argument (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF][START_REF] Mikulevicius | Martingale problems for stochastic PDEs[END_REF][START_REF] Yor | Existence et unicité de diffusion à valeurs dans un espace de Hilbert[END_REF]) we get that the equation is well posed.

Corollary 5.3. For any v 0 ∈ h r , (3.26) has a unique strong and a unique weak solution v(τ ) such that v(0) = v 0 . Its law satisfies (3.14). Now, let s ∈ Z ∞ 0 be any non-zero vector, orthogonal to Λ, and consider ϕ(v ν (τ )) • s = ϕ(V s(v ν (τ ))) ∈ S 1 . Since ϕ(V s) is a discontinuous function of V s ∈ C, in order to pass to a limit as ν → 0 we do the following. We identify S 1 with {v ∈ R 2 : |v| = 1}, define s = N , and approximate the discontinuous function

V N = (V 1 , . . . , V N ) → ϕ(V s) by continuous functions V N → f δ ([I(V N )])ϕ(V s) ∈ R 2 , [I] = min 1 k N I k , 0 < δ 1,
where f δ is continuous, 0 f δ 1, f δ (t) = 0 for t δ/2 and f δ = 1 for t δ.

For any measure µ τ in a complete metric space, which weakly continuously depends on τ , and any τ 1 < τ 2 we will define

µ τ τ2 τ1 = 1 τ 2 -τ 1 τ2 τ1 µ τ dτ.
Then the argument above, jointly with lemma 3.2, implies the following corollary.

Corollary 5.4. Let s ∈ Z ∞ 0 be any non-zero vector, orthogonal to Λ, and let 0 τ 1 < τ 2 T . Then

D(ϕ(v ν (τ )) • s) τ2 τ1 D(ϕ(v(τ )) • s) τ2 τ1 as ν → 0.
On the contrary, if s • Λ = 0, then by proposition 5.10 we get that

D(ϕ(v ν (τ )) • s) τ2 τ1 dϕ.
More generally, if vectors s1 , . . . , sM from Z ∞ 0 are perpendicular to Λ and a vector s is not, then

D(I, ϕ • s1 , . . . , ϕ • sM , ϕ • s)(v ν (τ )) τ2 τ1 D(I, ϕ • s1 , . . . , ϕ • sM )(v(τ )) τ2 τ1 × dϕ.
We do not know of an equivalent description of the measure Q 0 only in terms of the slow variables (I, V ) of (3.1), but the following result holds true. Proposition 5.5. Consider the natural process on the space H I,V with the measure Q 0 . If, for some N ∈ N and δ * > 0, stopping times 0 τ 1 < τ 2 T satisfy (3.27), then for τ ∈ [τ 1 , τ 2 ] the process (I, Φ (N ) )((I, V )(τ )) is a weak solution of the averaged equations (3.17) and (3.18)| j J . Here Φ (N ) = (Φ 1 , . . . , Φ J(N ) ).

Since the averaged quantities v k • P k Λ and iv k • P k Λ are functions of I and Φ (see (2.21)), (3.17) and (3.18)| j J form an underdetermined system of equations for the variables (I, Φ).

Proof of theorem 5.1. The proof follows the Khasminski scheme (see [START_REF] Freidlin | Averaging principle for stochastic perturbations of multifrequency systems[END_REF][START_REF] Khasminski | On the avaraging principle for Ito stochastic differential equations[END_REF][START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF]). Its crucial step is given by the following lemma.

Lemma 5.6. For any k 1 one has

A ν k := E max 0 τ T τ 0 R k (a ν (s), ν -1 s) ds → 0 as ν → 0. (5.2)
The lemma is proved in § 5.4, following the arguments in [START_REF] Kuksin | Weakly nonlinear stochastic CGL equations[END_REF][START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF]. We now derive from it the theorem.

For τ ∈ [0, T ] consider the processes

N ν l k = a ν l k (τ ) - τ 0 R k (a ν l (s)) ds, k 1.
Due to (3.31) we can write N ν l k as

N ν l k (τ ) = Ñ ν l k (τ ) + N ν l k (τ ),
where

Ñ ν l k (τ ) = a ν l (τ ) - τ 0 (R k (a ν l (s)) + R k (a ν l (s), ν -1 l s)) ds is a Q 0 a martingale and the disparity N ν l k is N ν l k (τ ) = τ 0 R k (a ν l (s), ν -1 l s) ds.
The convergence D(a ν l ) Q 0 a and lemma 5.6 imply that the processes

N k (τ ) = a k (τ ) - τ 0 R k (a) ds, k 1,
are Q 0 a martingales (for details, see [17, proposition 6.3]). Similarly to (5.2), we find that

E max 0 τ T τ 0 R k (a ν (s), ν -1 s) ds 2 → 0 as ν → 0.
Then, using the same arguments as before, we see that the processes

N k1 (τ )N k2 (τ ) - τ 0 A k1k2 ds are Q 0
a martingales, where A k1k2 denotes the diffusion matrix for system (3.26). That is, Q 0 a is a solution of the martingale problem with drift R k and diffusion A. Hence, Q 0 a is a law of a weak solution of (3.26). Such a solution exists for any v 0 ∈ h. So by lemma 3.5 and the Yamada-Watanabe argument (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF][START_REF] Mikulevicius | Martingale problems for stochastic PDEs[END_REF][START_REF] Yor | Existence et unicité de diffusion à valeurs dans un espace de Hilbert[END_REF]), weak and strong solutions for (3.26) both exist and are unique. Hence, the limit in (3.16) does not depend on the sequence ν l → 0, the convergence holds as ν → 0, and the theorem is proved.

That is (see [4, § 10.2]), write them as

N νn = N νn I n ,y (dy n )p νn (dI n dy), N 0n = N 0n I n ,y (dy n )p 0n (dI n dy),
where p νn = (id ×π 1 ) • N νn and p 0n = (id ×π 1 ) • N 0n . Since y = L(ϕ n ), we have

p νn = D(I n × (L • ϕ n ))(v νn (τ )).
As each vector ζ j in (5.8) is perpendicular to Λ n , in view of (5.5) we have .11) To calculate N 0n , it remains to find the fibre measures N 0n I n ,y . To do this let us take any bounded continuous function

p 0n = lim ν l →0 D(I n × (L • ϕ n ))(v ν l n (τ )) = (I n × (L • ϕ n )) • m 0n . ( 5
f on R n + ×T n-1 ×S 1 and consider N νn , f = Ef (I n , y, y n )(v ν (τ )). Since y(v) = L(ϕ n ) and y n (v) = v • η n ,
where the vector η n is not perpendicular to Λ, by (5.6),

N νn , f → f (I n , y, y n )((I n × (L • ϕ n ))m 0n )(dI n dy) dy n .
On the other hand, by (5.4),

N ν l n , f → N 0n , f = f (I n , y, y n )N 0n
I n ,y (dy n )p 0n (dI n dy).

Since

p 0n = (I n × (L • ϕ n ))
• m 0n , we get from the two convergences above that for p 0n -almost all pairs (I n , y) we have N 0n I n ,y = dy n . Accordingly, N 0n = dy n × p 0n (dI n dy).

(iv) Consider the measure M 0n . Due to (5.11) its disintegration with respect to the mapping id ×π 1 may be written as M 0n = M 0n I n ,y (dy n )p 0n (dI n dy) (5.12)

with some unknown fibre measures M 0n I n ,y . Now consider the rotated measure Ψ tΛ • m 0 , t 0, and its n-dimensional projection. By (5.9),

R T • Π n • Ψ tΛ • m 0 = (id ×l t ) • R T • m 0n ,
where l t (y, y n ) = (y, y n + t). Due to (5.10) and (5.12), the measure on the righthand side equals M 0n I n ,y (dy n + t)p 0n (dI n dy).

But by corollary 3.10 the measure on the left-hand side does not depend on t. So M 0n I n ,y (dy n ) ≡ M 0n I n ,y (dy n + t) is a translation-invariant measure on S 1 , and it must be equal to dy n . Accordingly,

M 0n = dy n × p 0n (dI n dy n ) = N 0n . (v) We have established that N ν l n
M 0n as ν l → 0. So ν ν l n m 0n , which completes the proof.

Mixing in the effective equations

We start with the case in which the function f (λ) has a linear growth. For simplicity of notation we suppose that f (λ) = λ + 1. We are also forced to assume that q * = 1.

The effective equation (3.26)=(1.18) with q * = 1 looks similar to (1.6) ν=∞, q * =1 , studied in [START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF]. It turns out that the two equations are indeed similar, at least for d 3, and that the proof of the mixing in [16, § 4], based on an abstract theorem from [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF], applies to (3.26) with minimal changes. Indeed, the crucial step in [START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF] in order to apply the result from [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF] is to establish for solutions of the equation the exponential estimate of the form

P sup t 0 t 0 |u(s)| 2 L∞ ds -Kt σ C exp(c 1 |u 0 | 2 L∞ -c 2 σ) ∀σ > 0, ( 5
[v] 2 1 = H 0 (v) + H 1 (v) (see (3.33)), since due to (3.33) we have that d[v(τ )] 2 1 + 2 τ 0 [v(s)] 2 2 ds = 4τ B + 2 ∞ j=1 (λ j + 1)(v j (τ ) • dβ j (τ )),
where we define [v] 2 2 = (λ j + 1) 2 |v j | 2 and B = (λ j + 1)b 2 j . Applying to this relation the supermartingale inequality in the standard way (see, for example, [START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF][START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF]), we get that

P sup τ 0 τ 0 [v(s)] 2 2 ds -2Bt σ C exp(c 1 |v 0 | 2 1 -c 2 σ) ∀σ > 0.
If d 3, then by lemma 3.11 the divergence of two solutions for (3.26) with the same ω satisfies

|v 1 (τ ) -v 2 (τ )| h 0 |v 1 (0) -v 2 (0)| h 0 exp C τ 0 ([v 1 (s)] 2 2 + [v 2 (s)] 2
2 ) ds .

These last two estimates allow us to repeat literally for (3.26) the reduction to theorem 3.1.3 from [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF], made in [START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF], and prove the following theorem.

Theorem 5.9. Let q * = 1, f (λ) = λ + 1 and d 3. Then the effective equation (3.26) has a unique stationary measure µ and is mixing. That is, every solution v(τ ) satisfies D(v(τ ))

µ as τ → ∞.

The presented proof uses that the nonlinearity in the effective equation is at most cubic. It also applies to the effective equations for (1.6), where the Hamiltonian H is one of the two functions H 3 with cubic densities as at the end of § 1.1 (in this case the argument works if d 6). The proof without changes applies to (1.6), where q * = 1, d 3 and f (λ) grows superlinearly. The argument may also be adjusted to the case in which q * = 1, d is any integer number and f (λ) = c 1 + λ c d , where c d is sufficiently big. Based on the similarity with (1.6) ν=∞, q * =1 , studied in [START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF] for any space dimension, we conjecture that for q * = 1 and f (λ) = λ + 1 the effective equation is well posed and mixing for any d. But it is unknown how to prove the mixing for equations with q * 2 (in any space dimension).

Proof of lemma 5.6

For this proof we adopt notation from [START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF]. Namely, we denote by κ(t) various functions of t such that κ → 0 as t → ∞, and denote by κ ∞ (t) functions satisfying κ(t) = o(t -N ) for each N . We write κ(t, M ) to indicate that κ(t) depends on a parameter M . In addition, for events Q and O and a random variable f we write where P 1 (M ) is a suitable polynomial. It is not hard to verify using the Doob inequality that for a suitable choice of P 1 the probability of P(F l ) is less than κ ∞ (L -1 ; M ) (see [START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF]). One gets κ ∞ (M ) + κ(ν -1/2 ; M ) + κ ∞ (ν -1 ; M ) + κ(ν -1/6 ; M ) + κ(ν -1/2 ; M ).

Choosing first M large and then ν small, we make the right-hand side above arbitrarily small. This proves the lemma.

An argument similar to the previous one (see appendix A) implies the following assertion. so that we have (here each τn is regarded as a function of τ 0 = τ0 , given by (A 1)), we can choose τ 0 ∈ [0, L) in such a way that

|η n | τn+1 τn ( Ĝ(s • ϕ ν (l); l) -Ĝ(s • ϕ ν (τ n ) + ν -1 (s • Λ)(l -τ n ); τ n )) dl + τn+1 τn Ĝ(s • ϕ ν (τ n ) + ν -1 (s • Λ)(l -τ n ); τ n ) -
K -1 K-1 n=0 P(E n ) = κ(ε -1 ; R, N ).
For any n consider the event

Q n = sup τn τ τn+1
|I ν (τ ) -I ν (τ n )| h I P 1 (R)L1/3 , where P 1 (R) is a suitable polynomial. It is not hard to verify using the Doob inequality that the probability satisfies P(Q n ) κ ∞ (L -1 ) (see [START_REF] Kuksin | Khasminskii-Whitham averaging for randomly perturbed KdV equation[END_REF]). Setting F n = E n ∪ Q n , n = 0, . . . , K -1, we have that

  Maximum and minimum. We define a ∨ b = max(a, b) and a ∧ b = min(a, b).

  ) converges normally, then the series on the right-hand side also does. It defines an analytic-in-(v, v) function. Note that, in view of (2.20), F Λ is a function of I 1 , I 2 , . . . and the variables {s • ϕ, s ∈ A(Λ, m)}.(2.21)

  17)(see(2.21)). Now consider (3.7) for resonant combinations Φ j of the angles. The corresponding dispersion matrix D = (D jk ) is formed by (1 × 2)-blocks

h.. 1 ) 5 . 1 .

 151 Therefore, all moments of |(R + R)(a)| Ha are finite, and we get from (3.31) thatE|a ν | C 1/3 ([0,T ],h)C, uniformly in ν. Now, arguing as when proving lemma 3.4, we get that the set of laws D(a ν (•)), 0 < ν 1, is tight in H a . Consider any limiting measure corresponding to the laws D(a ν (•)):D(a ν (•)) Q 0 a as ν → 0. (5Theorem There exists a unique weak solution a(τ ) of effective equation(3.26), satisfying a(0) = v 0 a.s. The law of a(•) in the space H a coincides with Q 0 a . The convergence (5.1) holds as ν → 0.

0 R

 0 P O (Q) = P(O ∩ Q) and E O (f ) = E(χ O f ).Here and below, M stands for a suitable function of ν such that M (ν) → ∞ as ν → 0, but νM n → 0 as ν → 0 ∀n.Denote byΩ M = Ω ν M the event Ω M = {sup 0 τ T |a ν (τ )| h r M }. Then, by (2.6), P(Ω c M ) κ ∞ (M ) uniformly in ν, so that one has A ν k κ ∞ (M ) + A ν k,M, where we have definedA ν k,M := E Ω M max 0 τ T τ k (a ν (s), ν -1 s) ds . (5.15)So it remains to estimate A ν k,M . Consider a partition of [0, T ] by the pointsτ n = nL, 0 n K ∼ T /L, where τ K is the last point τ n in [0, T ). The diameter L of the partition is L = √ ν. Defining η l = τ l+1 τ l R k (a ν (s), ν -1 s) ds, 0 l Kω ∈ Ω M the integrand in(5.16) is smaller than a suitable C(M ) (see lemma 3.1 and (3.22)). For any l let us consider the eventF l = sup τ l τ τ l+1 |a ν (τ )a ν (τ l )| h P 1 (M )L 1/3 ,

K- 1 l=0 1 Λ

 11 |E Ω M |η l | -E Ω M \F l |η l || C(M )L K-1 l=0 P(F l ) C(M )κ ∞ (L -1 ; M ),(5.18) so that it remains to estimateE Ω M \F l |η l |. We have |η l | τ l+1 τ l (R k (a ν (s), ν -1 s) -R k (a ν (τ l ), ν -1 s)) ds + τ l+1 τ l (R k (a ν (τ l ), ν -1 s)) ds =: Υ 1 l + Υ 2 l .By the regularity of the integrand and the definition ofF l , l E Ω M \F l Υ 1 l κ(L -1/3 ; M ) = κ(ν -1/6 ; M ). (5.19)So it remains to estimate the expectation of Υ 2 l . Defining t = ντ and making use of (3.30), we write Υ 2 l as exp(-it(Λ • (qle k ))) dtLC(M ) ν L sup p,q,l∈Z ∞ +0 , q-l-e k ∈A(Λ,m), |q|+|l|+1 m • (qle k ) Lκ(ν -1 L; M ),because the supremum in the second line is bounded by 1, since both Λ and q-l-e k are integer vectors. Therefore,l E Ω M \F l Υ 2 l κ(ν -1/2 ; M ). (5.20) Now (5.15) and (5.17)-(5.20) imply that A ν k

S 1 Ĝl+1 τ l S 1 Ĝ 1 Ĝ 3 n,

 1113 (θ; τ n ) dθ dl + τ (θ; τ n ) dθ -S (θ; l) dθ dl =: Υ 1 n + Υ 2 n + Υ 3 n .To estimate the quantities Υ 1,2,we first optimize the choice of the phase τ 0 . A crucial point here is that if we set N := M ∨ s , the function G depends only on v N . So we consider the events E n , 1 n K,E n = {I ν k (τ n ) ε for some k N }, where ε ν a , a = 1/10. (A 4)Since for each k M , by lemma 3.2, we haveL 0 K n=0 P(I ν k (τ n ) ε) dτ 0 = T 0 P(I ν k (τ ) ε) dτ = κ(ε -1 ; R, N )

5 )

 5 n ) κ(ε -1 ; R, N ) + κ ∞ (ν -1/2 ; N ) =: κ. n ) C κ := κ1 , j = 1, 2, 3. (A If ω ∈ Ω R \F n , then for τ ∈ [τ n , τ n+1 ] we have that I ν k (τ ) ε -P 1 (R)L 1/3

  .13) with suitable constants K, C , c 1 and c 2 . This estimate is important in the study of the mixing since it allows one to control divergence of trajectories u 1 (t) and u 2 (t), corresponding to the same realization of the random force, through the inequality 11|u 1 (t)-u 2 (t)| L2 |u 1 (0)-u 2 (0)| L2 exp C

	t
	(|u 1 (s)| 2 L∞ +|u 2 (s)| 2
	0

L∞ ) ds .

(5.14)

For (3.26) an analogy of (5.13) follows by applying the Itô formula to

https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec

at 14:15:29, subject to the Cambridge Core terms of use, available at

When forcing and dissipation are not present, a parameter T is introduced that measures the time-scale on which averaging is performed (see[START_REF] Nazarenko | Wave turbulence[END_REF]): to heuristically compare with the present case, we put ν = 1/T . https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec

at 14:15:29, subject to the Cambridge Core terms of use, available at

See[START_REF] Kuksin | Stochastic CGL equations without linear dispersion in any space dimension[END_REF] for a theory of (1.5) for the case in which f (t) = t + 1 and ν = ∞. https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

Both these conditions hold, for example, if q * = 1 and f (λ) = c 1 + λ c d , where c d is sufficiently big in terms of d.

Certainly this is not needed if we consider stationary solutions of the equation. https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

The work[START_REF] Cardy | Non-equilibrium statistical mechanics and turbulence[END_REF] mainly treats systems in which the sources and sinks of energy are well separated, so that there is a large spectral region where the forcing and the dissipation are practically absent and where the kinetic equation does not depend explicitly on the form of the forcing and dissipation. For this reason, the stationary spectra, found in[START_REF] Cardy | Non-equilibrium statistical mechanics and turbulence[END_REF] for(1.3), are equal to those obtained by WT methods for the HPDE (1.1). This explains the difference between the kinetic equation in[START_REF] Cardy | Non-equilibrium statistical mechanics and turbulence[END_REF] and that in[START_REF] Kuksin | Derivation of a wave kinetic equation from the resonantaveraged stochastic NLS equation[END_REF] (which deals with the case in which the dissipation is present on the whole spectral range). Contrary to[START_REF] Cardy | Non-equilibrium statistical mechanics and turbulence[END_REF], the method of[START_REF] Zakharov | Statistical description of nonlinear wave fields[END_REF] applies, at least formally, to all kinds of forcing and damping, so the kinetic equation of that work may depend on the forcing and dissipation.

[START_REF] Freidlin | Averaging principle for stochastic perturbations of multifrequency systems[END_REF] That is, the corresponding filtrations {Ft} are continuous from the right, and each Ft contains all negligible sets. https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

(2.4) https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

It may be better to call V j (v) a minimal resonant monomial since for any l ∈ Z ∞ +0 the monomial I l V j (v) is also resonant and corresponds to the same resonance.https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

This system is heavily underdetermined. https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

Under certain restrictions on the equation it is known that its law (i.e. the stationary measure of the equation) is unique; see, for example,[START_REF] Shirikyan | Ergodicity for a class of Markov processes and applications to randomly forced PDEs[END_REF]. We will not discuss this now.https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

To match (5.13) and(5.14) we use, crucially, that q * 1.https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

ε. On the other hand, by lemma

3.1, for any positive δ we have the estimate||v k | -2 (iv k • P k (v))χ {|v k |>δ} | δ -1 Q k (|v| h r ), https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0308210517000233 Downloaded from https://www.cambridge.org/core. Facolta Medicina Veterinaria, on 04 Dec 2017 at 14:15:29, subject to the Cambridge Core terms of use, available at

Acknowledgements

We thank Sergey Nazarenko, Anatoli Neishtadt and Vladimir Zeitlin for discussions and advice. This work was supported by l'Agence Nationale de la Recherche through the grant STOSYMAP (ANR 2011BS0101501).

Averaging theorem for stationary solutions

Let v ν (τ ) be a stationary solution of (3.1) as at the end of § 2.1. 10 Solutions v ν inherit the a priori estimates (2.3), (2.5), (2.6), and so the set of laws D(I(v ν (•)), V (v ν (•))), 0 < ν 1, is still tight in H I,V (see lemma 3.4). Consider any limit

Q as ν → 0.

(5.3)

As before, the measure Q satisfies (3.14) (with the constants C n , C , C , corresponding to v 0 = 0). Moreover, it is stationary in τ .

Theorem 5.7. There exists a stationary solution v(τ ) of the effective equation (3.26)

Proof. Define µ ν = Dv ν (τ ). Estimate (2.5) with 2m = r and n = 1 implies that |v| 2 h r+1 µ ν (dv) C for all ν. So the set of measures µ ν is tight in H r . Replacing, if necessary, the sequence {ν l } by a subsequence, we obtain that

Clearly, (I, V ) • µ 0 is the marginal distribution for Q for τ = const., which we will denote by q (i.e. q = Q| τ =const. ). Let v 0 (τ ), τ 0, be a solution of the effective equation (3.26) such that Dv 0 (0) = µ 0 (which exists by corollary 5.3 and the estimates on µ 0 ). Then, for the same reason as in § 5.1,

and D(I, V )(v 0 (τ )) ≡ q. We do not know if the solution v 0 is stationary, but from the Bogolyubov-Krylov argument we know that for a suitable sequence T j → ∞ we have the convergence

where m 0 is a stationary measure for (3.26). We still have that (I, V ) • m 0 = q, and the measure m 0 satisfies the same a priori estimates as before. Let v(τ ) be a solution for (3.26) such that Dv(0) = m 0 . It is stationary and D(I, V )(v(τ )) ≡ q. Slightly modifying the argument above, we also get that

Writing the convergence (5.3) as

, we note that, as in § 5.1, we also have that

as ν l → 0, for any m and any vectors s1 , . . . , sm perpendicular to Λ. Since for stationary solutions v ν (τ ) we have D(v ν (τ )) τ2 τ1 = D(v ν (τ )), arguing as when proving corollary 5.4 we also get that

then in view of proposition 5.10 and the stationarity of the solutions, we have

If (3.26) has a unique stationary measure m 0 , then the convergences above hold as ν → 0. But in this case a stronger assertion holds.

Theorem 5.8. Let v ν be a stationary solution of (3.1), let D(v ν (τ )) ≡ µ ν and assume that the effective equation (3.26) has a unique stationary measure m 0 . Then

Proof. (i) Consider again the convergence (5.4). We are going to show that the limiting measure µ 0 equals m 0 . Then the limit in (5.4) does not depend on the sequence {ν l → 0}, so it holds as ν → 0, and (5.7) follows.

(ii) Due to lemma 3.2, µ ν ( ) = 0 = µ 0 ( so we may regard µ ν and µ 0 as measures on h r I × T ∞ . Let us fix any n ∈ N and consider measures µ νn , µ 0n and m 0n , which are images of the measures µ ν , µ 0 and m 0 under the projection

(see notation and agreement in § 1.5). We will regard them as measures on R n + ×T n = {(I n , ϕ n )}. To prove that µ 0 = m 0 , it suffices to verify that µ 0n = m 0n for each n.

Let us define A(Λ n ) =: A n , and let the vectors ζ 1 , . . . , ζ n ∈ Z n and the unimodular matrix R be as in lemma 2.1 with A = A n . Let L = L A n : T n → T n-1 be the operator in (2.9), i.e.

(

Writing R T (ϕ n ) = (y 1 , . . . , y n ) T = (y T , y n ) T , where y = (y 1 , . . . , y n-1 ) T , we have L(ϕ n ) = y. We will denote by π 1 the natural projection y → y.

For later purposes we make the following observation. Let µ be a Borel measure on h. Consider its images under rotations Ψ tΛ and projections Π n . In the (I, ϕ)variables the mapping Ψ tΛ becomes id ×(• + tΛ), so

(where Π n • µ is written in the (I n , ϕ n )-variables). By (2.8) the transformation R T of T n conjugates the translation by the vector tΛ n with the translation by te n . Therefore,

where R T = id ×R T .

(iii) Let us apply to the measures µ νn , µ 0n , m 0n the transformation R T :

Recall that by (5.4), N ν l n N 0n as ν l → 0. Our first goal is to calculate the limiting measure N 0n . To do this let us disintegrate N νn and N 0n with respect to the mapping id ×π 1 : R n

Proposition 5.10. Let s ∈ Z ∞ 0 be such that s•Λ = 0 and let G : R M + ×T J(M ) ×S 1 → R be a bounded Lipschitz-continuous function for some M 1. Then

In particular, taking for G Lipschitz functions on S 1 , we get that D(s • ϕ ν (l)) t 0 dθ as ν → 0 for any t > 0.

Appendix A. Proof of proposition 5.10

For this proof, as in § 5.4, we denote by κ(t) various functions of t such that κ → 0 as t → ∞, and denote by κ ∞ (t) functions satisfying κ(t) = o(t -N ) for each N . For events Q and O and a random variable f we write

Without loss of generality we assume that |G| 1 and Lip G 1.

Let us denote by R a suitable function of ν such that R(ν) → ∞ as ν → 0, but

Taking into account the boundedness of G, we get that

As in the proof of lemma 5.6, consider a partition of [0, T ] by the points

where τ K is the last point τ n in [0, T ). The diameter L of the partition is L = √ ν, and the non-random phase τ 0 ∈ [0, L) will be chosen later. Defining

so it remains to estimate

where Q k is a polynomial. These relations and (3.6) imply that

(see the estimate of P(Q n )). Therefore,

and

From here and the definition of the events Q n we find that

For the same reason also

So it remains to estimate the expectation of

Let us expand Ĝ(ψ; τ n ) as a Fourier series Ĝ(ψ) = g k e ikψ , where each g k is a random variable and g 0 = S 1 Ĝ(θ; τ n ) dθ (we discard the dependence on τ n , which is fixed throughout the argument). Then

for a suitable non-random T ε . Indeed, for each non-zero k, one has

12 By the Bernstein theorem, + C(R)ν 1/6 + κ(ν -1/2 ; R, N, ε, s • Λ).

Choosing first R large, and next ε and ν small in such a way that (A 4) holds, we make the quantity above arbitrarily small. This proves the required convergence.

The second assertion of the proposition follows from the first one, since to check the weak convergence of measures on a complete metric space it suffices to take for test functions the Lipschitz functions.