Andrey Dymov

Sergei Kuksin

Alberto Maiocchi

Sergei Vladuţ

Sergei Vl Ȃdut

THE LARGE-PERIOD LIMIT FOR EQUATIONS OF DISCRETE TURBULENCE

whether they are published or not. The documents may come

1. Introduction 1.1. The setting. In this paper we continue the study of the Zakharov-L'vov stochastic model for wave turbulence (WT), initiated in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF]; see also a survey [START_REF] Dymov | On the Zakharov-L'vov stochastic model for wave turbulence[END_REF]. We start by recalling the classical and the Zakharov-L'vov stochastic settings of WT. See the introduction to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for more detailed discussions of the two models.

Classical setting. Let T

u(x) = L -d/2 s∈Z d L v s e 2πis⋅x , Z d L = L -1 Z d .
Here the vector of Fourier coefficients v = (v s ) s∈Z d L is given by the Fourier transform of u(x),

v = F(u), v s = L -d/2 T d L u(x)e -2πis⋅x dx for s ∈ Z d L ,
so the Parseval identity takes form ∥u∥

2 = L -d ∑ s∈Z d L |v s | 2 .
We will study solutions u(t, x) whose norms satisfy ∥u(t, ⋅)∥ ∼ 1 as L → ∞. This makes the chosen in (1.1) scaling of Fourier series convenient for our purposes.

consider the cubic NLS equation with modified nonlinearity

(1.2) ∂ ∂t u + i∆u -iλ |u| 2 -2∥u∥ 2 u = 0, x ∈ T d L ,
where u = u(t, x), ∆ = (2π)

-2 ∑ d j=1 (∂ 2 /∂x 2
j
) and λ ∈ (0, 1] is a small parameter. The modification of the nonlinearity by the term 2iλ∥u∥ 2 u keeps the main features of the standard cubic NLS equation, reducing some non-crucial technicalities; see the introduction to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF].

The objective of WT is to study solutions of (1.2) under the limit L → ∞ and λ → 0 on long time intervals. There are plenty of physical works containing some different (but consistent) approaches to the limit; many references may be found in [START_REF] Zakharov | Kolmogorov Spectra of Turbulence[END_REF][START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Newell | Wave Turbulence[END_REF]. Despite the strong interest in physical and mathematical communities to the addressed questions, significant progress in the rigorous justification of the physical predictions was achieved only recently [START_REF] Lukkarinen | Weakly nonlinear Schrödinger equation with random initial data[END_REF][START_REF] Faou | The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation[END_REF][START_REF] Buckmaster | Effective dynamics of the nonlinear Schrödinger equation on large domains[END_REF][START_REF] Faou | Linearized wave turbulence convergence results for three-wave systems[END_REF][START_REF] Deng | On the derivation of the wave kinetic equation for NLS[END_REF][START_REF] Collot | On the derivation of the homogeneous kinetic wave equation[END_REF]4,[START_REF] Buckmaster | Onset of the wave turbulence description of the longtime behaviour of the nonlinear Schrödinger equation[END_REF][START_REF] Deng | Full derivation of the wave kinetic equation[END_REF]. See e.g. the introductions to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Collot | On the derivation of the homogeneous kinetic wave equation[END_REF][START_REF] Deng | Full derivation of the wave kinetic equation[END_REF] for discussions of the obtained results.

Zakharov-L'vov setting. When studying eq. (1.2), members of the WT community talk about "pumping energy to low modes and dissipating it in high modes". To make this rigorous, following Zakharov-L'vov [START_REF] Zakharov | Statistical description of nonlinear wave fields[END_REF], in the present paper as well as in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] we consider the NLS equation (1.2) dumped by a (hyper) viscosity and driven by a random force:

(1.3) ∂ ∂t u + i∆u -iλ |u| 2 -2∥u∥ 2 u = -νA(u) + ν ∂ ∂t η ω (t, x).
Here ν ∈ (0, 1/2] is another small parameter, which should be properly agreed with λ and L. The dissipative linear operator A is defined as

(1.4) A(u(x)) = L -d/2 s∈Z d L γ s v s e 2πis⋅x , v = F(u), γ s = γ 0 (|s| 2 ),
where |s| stands for the Euclidean norm of a vector s and γ 0 (y) is a smooth real increasing function of y > 0, satisfying 1 (1.5) γ 0 ≥ 1 and c(1

+ y) r * ≤ γ 0 (y) ≤ C(1 + y) r * ∀ y > 0 .
The exponent r * > 0 and c, C are positive constants. We also assume that all derivatives of γ 0 have at most polynomial growths at infinity.

The random noise η ω is given by a Fourier series

η ω (t, x) = L -d/2 s∈Z d L b(s)β ω s (t)e 2πis⋅x ,
1 For example, if

γ s = (1 + |s| 2 ) r * , then A = (1 -∆) r * .
where {β s (t), s ∈ Z d L } are standard independent complex Wiener processes 2 and b(x) is a Schwartz function on R d ⊃ Z d L . 3 Solutions u(τ ) of (1.3) are random processes in the space H = L 2 (T d L , C), equipped with the norm ∥ ⋅ ∥. If r * is sufficiently big, equation (1.3) is known to be well posed. Moreover, Ito's formula shows that E∥u(τ )∥

2 is bounded uniformly in τ and L, ν, λ, once E∥u(0)∥ 2 is bounded uniformly in these parameters, see in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF].

We will study the equation on time intervals of order ν -1 . So it is convenient to pass from t to the slow time τ = νt and write eq. (1.3)

as u + iν -1 ∆u -iρ |u| 2 -2∥u∥ 2 u = -A(u) + ηω (τ, x), η ω (τ, x) = L -d/2 s∈Z d L b(s)β ω s (τ )e 2πis⋅x . (1.6)
Here ρ = λν -1 , the upper-dot stands for d/dτ and {β s (τ ), s ∈ Z d L } is another set of standard independent complex Wiener processes. Below we use ρ, ν and L as parameters of the equation.

In the context of equation (1.6), the objective of WT is to study its solutions u(τ ) when (1.7) L → ∞ and ν → 0, while ρ = ρ(ν, L) is scaled appropriately, mostly paying attention to their energy spectra

(1.8) N s (τ ) ∶= E|v s (τ )| 2 , where v(τ ) = F(u(τ )).

Exact meaning of the limit (1.7) is unclear since no relation between the parameters ν and L is postulated by the theory. Motivated by physical works, in the present paper, as in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF], we study formal decompositions in ρ of solutions to eq. (1.6) and of their energy spectra N s under the limit (1.7). See the introduction to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for a discussion of our motivation, and see below Section 4. In [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] we understand the limit (1.7) as (1.9) first L → ∞ and then ν → 0, or L ≫ ν -2 while ν → 0.

There we have shown that principal terms of the decomposition of N s in ρ have a non-trivial limiting behaviour, provided that ρ is scaled as ρ ∼ ν -1/2 , governed by a nonlinear wave kinetic equation (WKE) with added dissipation and a constant forcing. The WKE coincides with that, arising in physical works, so this result agrees well with the predictions of the WT. 3 Often it is assumed that the intensity b(x) of the noise η ω is non-negative, but we do not impose this condition. Note that if b(x) ≡ 0, then our results become trivial since below we will provide (1.3) with the zero initial conditions.

the present paper we are interested in the opposite order of limits, which rarely appears in physical works: (1.10) firstly ν → 0 and then L → ∞.

Roughly speaking, our main result is that under the double limit (1.10) the behaviour of principal terms of the decomposition in ρ for the energy spectrum N s is governed by a modified WKE. The latter is similar to the WKE arising in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] and physical papers, but is different from them. The scaling of ρ now is ρ ∼ L χ d (L), where (1.11)

χ d (L) ≡ 1 if d ≥ 3 and χ d (L) = (ln L) -1/2 if d = 2.
To the best of our knowledge this WKE did not appear in the literature before.

For the proof we start with the result obtained in [START_REF] Kuksin | Resonant averaging for small solutions of stochastic NLS equations[END_REF][START_REF] Huang | Time-averaging for weakly nonlinear CGL equations with arbitrary potentials[END_REF], where the limiting as ν → 0 behaviour of equation (1.6) is examined (while L and ρ are kept fixed). Then we pass to the limit as L → ∞, following the approach of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] and using the developed there tools, such as a specific Feynman diagram presentation. Another key ingredient of the proof is an obtained in [START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF] refinement of the Heath-Brown circle method for quadratic forms [START_REF] Heath-Brown | A new form of the circle method, and its application to quadratic forms[END_REF], and certain upper bounds for the number of integer points on intersections of quadrics. In the next subsections we describe our results and methods in more detail.

In [START_REF] Kuksin | Derivation of a wave kinetic equation from the resonantaveraged stochastic NLS equation[END_REF] a similar result concerning the iterated limit (1.10) was found heuristically, however there ρ was scaled as ρ ∼ L. The present paper shows that the correct scaling is different: ρ ∼ L χ d (L).

Similar regimes, when L → ∞ slowly while ν > 0 fast decays to zero, were studied in [START_REF] Faou | The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation[END_REF][START_REF] Buckmaster | Effective dynamics of the nonlinear Schrödinger equation on large domains[END_REF]. However the elegant description of the limit, obtained there, is far from the prediction of WT. The works [START_REF] Faou | The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation[END_REF][START_REF] Buckmaster | Effective dynamics of the nonlinear Schrödinger equation on large domains[END_REF] should rather be regarded as a kind of averaging (similar to that of Krylov-Bogolyubov) since the considered there time scale is much shorter than the characteristic time scale of WT. Note that in [START_REF] Buckmaster | Effective dynamics of the nonlinear Schrödinger equation on large domains[END_REF] a similar to ours [START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF] refinement of the Heath-Brown method also is crucially used. 1.2. The limit of discrete turbulence. We first consider the limit (1.12) ν → 0 while L and ρ stay fixed.

It is known as the limit of discrete turbulence (see [START_REF] Nazarenko | Wave Turbulence[END_REF]Section 10]) and has been successfully studied in [START_REF] Kuksin | Resonant averaging for small solutions of stochastic NLS equations[END_REF][START_REF] Huang | Time-averaging for weakly nonlinear CGL equations with arbitrary potentials[END_REF]. To explain the result let us take the Fourier transform of equation (1.6):

vs -iν -1 |s| 2 v s + γ s v s = iρL -d 1,2,3 δ ′12 3s v 1 v 2 v3 -|v s | 2 v s + b(s) βs (1.13) for s ∈ Z d L .
Here, as it is common in WT, v j abbreviates v s j , ∑ 1,2,3 stands for ∑ s 1 ,s 2 ,s 3 ∈Z d L , and

(1.14) δ ′12 3s = δ ′ s 1 s 2 s 3 s ∶= 1, if s 1 + s 2 = s 3 + s and {s 1 , s 2 } ≠ {s 3 , s} , 0, otherwise. Note that (1.15) if δ ′12 3s = 1, then {s 1 , s 2 } ∩ {s 3 , s} = ∅. We pass to the interaction representation, (1. [START_REF] Faou | The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation[END_REF] a s (τ ) = v s (τ )e where {β s } is yet another set of standard independent complex Wiener processes (and, again a j stands for a s j ). Note that the energy spectra of solutions to equations (1.13) and (1.18) coincide:

(1. [START_REF] Kuksin | Séminaire Laurent Schwartz EDP et applications[END_REF])

N s (τ ) = E|v s (τ )| 2 = E|a s (τ )| 2 .
Sometimes we will write N s and a s as N s (τ ; ν, L) and a s (τ ; ν, L). The limiting dynamics in equation (1.18) under the limit (1.12) is governed by the effective equation of discrete turbulence. The latter has the form (1.18) with the modified nonlinearity Y res , in which the sum is taken only over resonant vectors s 1 , s 2 , s 3 : ȧs + γ s a s = iρY If b(x) > 0 for all x, then it is known that under the assumptions of Theorem 1.1, equations (1.18) and (1.20) are mixing, so they have unique stationary measures. Then in [START_REF] Kuksin | Resonant averaging for small solutions of stochastic NLS equations[END_REF][START_REF] Huang | Time-averaging for weakly nonlinear CGL equations with arbitrary potentials[END_REF] it is also proven that the assertions of the theorem remain true if a ε (τ ) = N s ( and a 0 (τ ) denote stationary solutions of equations (1.18) and (1.20).

The main result. In view of Theorem 1.1, to understand behaviour of the energy spectrum (1.19) of equation (1.18) under the limit (1.10), it remains to study that of the energy spectrum E|a s (τ ; L)| 2 of the effective equation (1.20) under the limit L → ∞. Instead, following the logic of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], we study the energy spectrum corresponding to a principal part of a decomposition in ρ for the solutions a s (τ ; L) of equation (1.20).

Quasisolutions and their energy spectra. To simplify presentation we assume that initially the system was at rest, i.e. supplement equation (1.20) with the zero initial condition

(1.21) a s (0) = 0 ∀s ∈ Z d L .
We formally decompose the corresponding solution of (1.20) in ρ,

(1.22) a(τ ) = a (0) (τ ) + ρa (1) (τ ) + ρ 2 a (2) (τ ) + . . . , a (k) (0) = 0, a (k) (τ ) = a (k) (τ ; L).
The process a (0) (τ ) satisfies the linear equation

ȧ(0) s + γ s a (0) s = b(s) βs , s ∈ Z d L . So it is Gaussian, (1.23) a (0) s (τ ) = b(s) τ 0 e -γ s (τ -l) dβ s (l),
and its components {a (0)

s } are independent. The process a (1) satisfies

ȧ(1) s + γ s a (1)
s = iY res s (a (0) ) , so that (1.24) a (1)
s (τ ) = iL -d τ 0 e -γ s (τ -l) 1,2,3 δ ′12 3s δ(ω 12 3s )(a (0) 1 a (0) 2 ā(0) 3 ) -|a (0) s | 2 a (0) s (l)dl
is a Wiener chaos of third order (see [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]). Similar for n ≥ 1,

a (n) s (τ ) = iL -d n 1 +n 2 +n 3 =n-1 τ 0 e -γ s (τ -l) × 1,2,3 δ ′12 3s δ(ω 12 3s ) a (n 1 ) 1 a (n 2 ) 2 ā(n 3 ) 3 -a (n 1 ) s a (n 2 ) s ā(n 3 ) s (l) dl, (1.25)
is a Wiener chaos of order 2n + 1.

Next we consider the quadratic truncation of the series (1.22),

(1.26) A s (τ ; L) = A s (τ ) = a (0) s (τ ) + ρa (1)
s (τ ) + ρ 2 a (2)
s (τ ) , which we call the quasisolution 4 of the effective equation (1.20), (1.21). It is traditional in WT to analyse the quasisolution instead of the solution itself, postulating that the former well approximates the latter; see introduction to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for a discussion. The goal of the present paper is to study the behaviour of the energy spectrum of A(τ ),

(1.27) n s,L (τ ) = E|A s (τ ; L)| 2 , s ∈ Z d L , as L → ∞.
Our results, formulated below, show that under this limit the energy spectrum n s,L (τ ) has a non-trivial behaviour (i.e. stays finite and behaves differently from

E|a (0) s | 2 ) only if ρ ∼ L χ d (L)
, where χ d is defined in (1.11). Accordingly, from now on we assume that

(1.28) ρ = εLχ d (L),
where 0 < ε ≤ 1/2 is a small but fixed constant (see a few lines below for its discussion). Then the energy spectrum n s,L expands as

(1.29) n s,L (τ ) = n (0) s,L (τ ) + ε n (1) s,L (τ ) + ε 2 n (2) s,L (τ ) + ε 3 n (3) s,L (τ ) + ε 4 n (4) s,L (τ ), s ∈ Z d L , where (1.30) n (k) s,L (τ ) = Lχ d (L) k k 1 +k 2 =k 0≤k 1 ,k 2 ≤2 Ea (k 1 ) s (τ )ā (k 2 ) s (τ ) .
In particular, by (1.23)

(1.31) n (0) s,L (τ ) = E|a (0) s (τ ; L)| 2 = b(s) 2 γ s 1 -e -2γ s τ , s ∈ Z d L ,
and a simple computation shows that n

s,L (τ ) ≡ 0. For higher order terms we prove that

(1.32) n (2) s,L ∼ 1 and |n (3) s,L |, |n (4)
s,L | ≲ 1 as L → ∞ uniformly in τ ≥ 0; see a discussion in the next subsection. Thus, the parameter ε measures the properly scaled amplitude of the solutions, and indeed it should be small for the methodology of WT to apply. Then, the term ε

2 n (2)
s,L is the crucial nontrivial component of the energy spectrum n s,L while the terms ε 3 n

(3) s,L , ε 4 n (4) s,L
are perturbative. This well agrees with the prediction of physical works concerning various models of WT.

Wave kinetic equation. In view of (1.32), to study the limiting as L → ∞ behaviour of the energy spectrum n s,L (τ ) up to an error of size ε 3 it remains to investigate the behaviour of its principal component n

(0) s,L (τ ) + ε 2 n (2)
s,L (τ ). We show that the latter is governed by a WKE. To state the result let us consider the resonant quadric Let us consider the following non-autonomous cubic wave kinetic integral operator K(τ ), for any τ ≥ 0 sending a function y s , s ∈ R d , to the function

(1.33) Σ s = {(s 1 , s 2 ) ∈ R 2d ∶ (s 1 -s) ⋅ (s 2 -s) = 0}, cf. (1.
K s (τ )y, defined as (1.35) K s (τ )y = 4C d Σ s µ Σ s (ds 1 ds 2 ) Z 4 y 1 y 2 y 3 +Z 3 y 1 y 2 y 4 -Z 2 y 1 y 3 y 4 -Z 1 y 2 y 3 y 4 .
Here y j ∶= y s j with s 4 ∶= s and s 3 ∶= s 1 + s 2 -s, C d is the constant from Theorem B below, the kernels Z j = Z j (τ ; s 1 , s 2 , s 3 , s 4 ) are given by formulas (4.14), (4.15) and satisfy 0

≤ Z j (τ ) ≤ 1. When τ → ∞ the operator K(τ )
exponentially fast converges to a limiting kinetic integral operator K(∞), given by (1.35) with Z j replaced by (γ

s 1 + γ s 2 + γ s 3 + γ s 4 ) -1 for all j: (1.36) K s (∞)y = 4C d Σ s µ Σ s (ds 1 ds 2 ) γ s 1 + γ s 2 + γ s 3 + γ s 4 y 1 y 2 y 3 + y 1 y 2 y 4 -y 1 y 3 y 4 -y 2 y 3 y 4 .
It is similar to the standard four-waves kinetic operator of WT (e.g. see in [START_REF] Nazarenko | Wave Turbulence[END_REF]), which has the form (1.35) with Z j ≡ const, but still is different from the latter since K(∞) depends on the spectrum {γ s } of the dissipation operator A. 5For r ∈ R we denote by C r (R d ) a space of continuous complex functions on R d with finite norm In Section 5, following [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], we show that if r > d, then for any τ the operator K(τ ) defines a continuous 3-homogeneous mapping K(τ

) ∶ C r (R d ) ↦ C r+1 (R d ), and for any y ∈ C r (R d ) the curve τ ↦ K(τ )(y) is Hölder contin- uous in C r (R d ).
Now consider the following damped/driven non-autonomous WKE

(1.38) żs (τ ) = -2γ s z s + ε 2 K s (τ )(z) + 2b(s) 2 , z(0) = 0,
where τ ≥ 0 and s ∈ R d . In Section 5 we prove that for small ε it has a unique solution z s (τ ), which can be written as s,L of the energy spectrum n s,L , given by (1.31), and we prove that z

z s (τ ) = z 0 s (τ ) + ε 2 z 1 s (τ, ε), where z 0 s , z
1 s is ε 4 -close to n (2)
s,L uniformly in τ . Then, in view of (1.32), the energy spectrum n s,L is ε 3 -close to the solution z s (τ ). Namely, under the scaling ρ = εL χ d (L), for any r there exists ε r ∈ (0, 1/2] such that for 0 < ε ≤ ε r we have

(1.39) |n ⋅,L (τ ) -z ⋅ (τ )| r ≤ C r ε 3 ∀ τ ≥ 0, if L ≥ ε -2 for d ≥ 3, and L ≥ e ε -1 for d = 2.
See Theorem 5.9. Since the energy spectrum n s is defined for

s ∈ Z d L with finite L, then the norm in (1.39) is understood as |f | r = sup z∈Z d L |f (z)|⟨z⟩ r . Remark 1.2. If d = 2, the lower bound L ≥ e ε -1
can be relaxed in the following sense. In Appendix D we explain that there is a bounded correction f (τ, L) which can be written explicitly, such that

(1.40) n ⋅,L (τ ) -z ⋅ (τ ) - f (τ, L) ln L r ≤ C r ε 3 ∀ τ ≥ 0, if L ≥ ε -6 .
In Lemma 5.6 we show that for ε ≪ 1 equation (

(1.41) |n ⋅,L (τ ) -z ε ⋅ | r ≤ C r (e -τ + ε 3 ), ∀ τ ≥ 0,
if L is as in Theorem A, see (5.19). The cases d ≥ 3 and d = 2 are similar, but should be treated separately. To shorten the presentation we give a detailed proof of Theorem A only for d ≥ 3, when χ d = 1 and ρ = εL. The proof for d = 2 can be obtained by a simple modification of the argument for d ≥ 3. We sketch it in Appendix D. So from now on, except Section 2 which gives a brief account of the method of Feynman diagrams from [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF], we assume that d ≥ 3.

In paper [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] we examine the behaviour of the energy spectrum n s,L,ν (τ ) of a quasisolution to equation (1.18) under the limit (1.9), assuming that ρ = εν -1/2 . 6 We got there a similar result which states that n ⋅,L,ν (τ ) is ε 4 -close to a solution of the damped/driven four-wave kinetic equation as in Section 6.9.1] (in contrast with eq. (1.38), the kinetic nonlinearity there does not depend on the dissipation A in equation (1.3)).

What next? In this work and in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] we obtained wave kinetic limits for the energy spectra of quasisolutions for the NLS equation (1.6) under limit (1.10) with the scaling ρ = εL χ d (L) and limit (1.9) with the scaling ρ = εν -1/2 .

Our next goal is to show that an exact solution a ⋅ (τ ) of eq. (1.18) is ε 3close to its quasisolution A ⋅ (τ ) (uniformly in L ≥ 2 and τ ∈ [0, T ], for any T > 0). And that a solution of eq. (1.6) is ε 3 -close to the quasisolution of the equation (uniformly in ν, L and τ

∈ [0, T ], if L ≥ ν -2-γ , γ > 0)
. This would imply that the energy spectra of solutions of eq. (1.6) under limit (1.10) and limit (1.9) are ε 3 -close to solutions of the two WKE (namely, eq. (1.38) and the WKE from [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]). To prove this, say, for a solution a ⋅ (τ ) of eq. (1.18) we consider the equation on any fixed time-interval [0, T ] and regard it as a nonlinear equation F T (a ⋅ (⋅)) = 0. Then the quasisolution A satisfies the equation with a disparity ≲ ε 3 . By analogy with some stochastic problems for nonlinear PDEs, recently successfully resolved by the KAM-techniques (e.g. see [START_REF] Kuksin | Séminaire Laurent Schwartz EDP et applications[END_REF]), we believe that KAM also applies to the equation F T = 0. Its application would imply that a is ε 3 -close to A, as stated. We also believe that analysis of the KAM-iterations which build a from A will show that the energy spectrum of the solution a ⋅ (τ ) of eq. (1.18) under the limit L → ∞ converges to a solution of the WKE (1.38). A similar logic should apply to the energy spectra of solutions for eq. (1.6) under the limit (1.9).

1.4. Outline of the proof: Feynman diagrams and number theory.

It is well understood that to write down formulas for the terms n

(k)
s,L of decompositions as (1.29) it is instrumental to use the language of Feynman diagrams. In application to similar problems this goes back at least to the works [START_REF] Erdös | Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation[END_REF][START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF], and then was successfully used for the purposes of WT in [START_REF] Lukkarinen | Weakly nonlinear Schrödinger equation with random initial data[END_REF][START_REF] Deng | On the derivation of the wave kinetic equation for NLS[END_REF][START_REF] Collot | On the derivation of the homogeneous kinetic wave equation[END_REF]4,[START_REF] Buckmaster | Onset of the wave turbulence description of the longtime behaviour of the nonlinear Schrödinger equation[END_REF][START_REF] Deng | Full derivation of the wave kinetic equation[END_REF] and other papers. We use this techniques in the form developed in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] which gives a convenient presentation of the terms n k in a form, convenient to pass to a limit as L → ∞. The term n

s,L is a sum over a single quadric and may be analised without the diagram's machinery. This and some other similar terms play a leading role in our analysis and dictate the form of the limiting WKE. The terms may be written as sums

(1.42) G s (τ, L) = L 2(1-d) z 1 ,z 2 ∈Z d L ∶ z 1 ⋅z 2 =0, z 1 ,z 2 ≠0 Φ s (τ ; z 1 , z 2 ),
well known in works on WT. To study them under the limit L → ∞ we make use of the celebrated circle method of Heath-Brown [START_REF] Heath-Brown | A new form of the circle method, and its application to quadratic forms[END_REF]. Since the result of [START_REF] Heath-Brown | A new form of the circle method, and its application to quadratic forms[END_REF] does not completely fit our purposes, we specified it in the accompanying paper [START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF] (also see [START_REF] Buckmaster | Effective dynamics of the nonlinear Schrödinger equation on large domains[END_REF]Section 5] for another specification of the Heath-Brown method, used for the purposes of WT). This implies Theorem B. For any L ≥ 2,

G s (τ, L) -C d Σ 0 Φ s (τ ; z 1 , z 2 ) µ Σ 0 (dz 1 dz 2 ) ≤ K d ∥Φ s (τ ; ⋅)∥ N 1 ,N 2 L d-5/2 ,
where

Σ 0 = {z 1 , z 2 ∈ R d ∶ z 1 ⋅ z 2 = 0}, µ
Σ 0 is the measure on it, defined by

(1.34) with s = 0, C d is a number-theoretical constant, satisfying C d ∈ (1, 1+ 2 2-d ), the norm ∥ ⋅ ∥ N 1 ,N 2 is defined in (3.
3) and the constants N 1 , N 2 ∈ N depend only on d.

In particular, the term n

s,L (τ ) admits a limit when L → ∞. The terms n Theorems B and C imply (1.32). So to establish Theorem A it remains to show that the term n

≤2 s,L (τ ) ∶= n (0) s,L (τ )+ε 2 n (2)
s,L (τ ) (or equivalently its limit as L → ∞, provided by Theorem B) can be well approximated by a solution of the WKE (1.38). To this end, following the lines of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] (and the logic of the Krylov-Bogolyubov averaging) we consider increments ∆n Although the computation of the increments ∆n ≤2 s,L is similar to that in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], now a mechanism leading to a WKE is rather different. Namely, in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] components of the terms n (k) s,L are approximated by formulas analogous to (1.42), where the summation over the lattice (Z d ) k = {z} is replaced by an integration over R dk . The integrals in those formulae involve fast oscillating Gaussian kernels. The zero sets of these kernels define quadrics, related to the quadrics {(z 1 , z 2 ) ∶ z 1 ⋅ z 2 = 0} in (1.42). Due to the fast oscillations a crucial component of the increments ∆n ≤2 s,L is given by the 7 In fact, in Section 3 we prove an abstract result, more general than the theorem below; see there Theorem 3.2.

associated with short-range correlations in τ of the processes a (0) m (τ ). On the contrary, in the present situation a crucial contribution is given by other terms, associated with long-time correlations of the processes a (0) m (τ ), while the short-range correlations only give a small correction. As the result, the kinetic integral in the WKE (1.38) depends on the viscosity operator A, while that in the WKE in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] does not.

Finally we note that, as we explain in Section 3.2, it is plausible that Theorem C holds for all k ≥ 3. If so, then for ρ = εL the energy spectrum of a solution a(τ ), written as (1.22), defines a formal series in ε, uniformly in L ≥ 2. Then the partial sums of this series, made by the terms of order ε m , m ≤ M , with any fixed M ≥ 2, also satisfy Theorem A with the constants C r , depending on M . Cf. Conjecture 3.8.

Series expansion: approximating equation and diagrammatic representation for solutions

In this section, assuming that d ≥ 2, we approximate processes (1.25) by more convenient processes a (n) , and then obtain a compact and instrumental representation for their correlations in terms of Feynman diagrams (see Lemma 2.2), following [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF]. This representation (as well as its analogy in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF]) is used to estimate various disparity terms, related to quasisolutions A(τ ; L), see (1.26), and to their energy spectra. Our presentation is sketchy, but missing details may be found in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF]. For a general discussion of the language of Feynman diagrams see [START_REF] Janson | Gaussian Hilbert Spaces[END_REF].

ȧs + γ s a s = iρY s (a) + b(s) βs , s ∈ Z d L , Y s (a) = L -d 1,2,3 δ ′12 3s δ(ω 12 3s )a 1 a 2 ā3 . (2.1)
Similar to processes a s we decompose

(2.2) a = a (0) + ρa (1)
+ . . . .

Here a (0) = a (0) and the processes a

s (τ 1 )ā (n) s (τ 2 ) -Ea (m) s (τ 1 )ā (n) s (τ 2 )| ≤ L -N -d+1 χ d (L) N -1 C # (s; n, m) , uniformly in τ 1 , τ 2 ≥ 0.
We prove the proposition in Appendix C for d ≥ 3 and discuss an adaptation of the proof to the case d = 2 in Appendix D. Doing that we use the relation N ∶= m + n ≤ 4 only to apply Theorem 3.2 (or Theorem D.2 if d = 2). So if the assertion (3.9) of the latter theorem holds for larger N 's, then for those N 's estimates (2.3) remains true as well (we believe that (3.9) is fulfilled for all N , see in Section 3.2).

Relations (2.3) imply that moments of processes a (m) s (τ ) well approximate those of processes a (m) s (τ ) as L → ∞. Accordingly, from now on we will mostly study processes a s (τ ) and their decompositions (2.2).

Diagrams for solutions.

For what follows it is convenient to re-write operator Y from (2.1), using a fictitious index s 4 :

Y s (a) = L -d 1,2,3,4 δ ′12 34 δ(ω 12
s (τ ) = m 1 +m 2 +m 3 =m-1 i τ 0 dl e -γ s (τ -l) L -d 1,2,3,4 δ ′12 34 δ(ω 12 34 ) δ s 4 a (m 1 ) 1 a (m 2 ) 2 ā(m 3 ) 3 (l) .
(2.4)

We will call the objects as those in the r.h.s. of (2.4) sums, despite they involve integrating in dl. The r.h.s. of (2.4) contains several sums, corresponding to all admissible choices of numbers m 1 , m 2 , m 3 . We apply Duhamel's formula (2.4) to the terms a

(m i )
s i (l) in the right-hand side of (2.4) with m i > 0, and iterate the procedure till a (m) s (τ ) is expressed through the processes a (0) and ā(0) . Then a (m) s becomes represented as a finite sum of sums; we denote such sums by I s . Below we will associate with each sum I s an appropriately constructed diagram D. Thus we will write a

(m) s (τ ) as (2.5) a (m) s (τ ) = D∈D m I s (D; τ ),
where D m is a set of all diagrams, corresponding to the just explained representation of a (m) via the processes a (0) and ā(0) . Similarly by D n we denote the set of diagrams, parametrizing the terms in the sum, representing ā(n) (τ ) in a form, analogous to (2.5): When iterating the Duhamel formula (2.4) (or its complex conjugation) for a j-th time, we will denote the corresponding time l ∈ [0, τ ] by l j and will write the set of indices {s 1 , s 2 , s 3 , s 4 } as {ξ 2j-1 , ξ 2j , σ 2j-1 , σ 2j }, where we

ā(n) s (τ ) = ∑ D∈D m I s ( D; τ ).
c (2) 0 c (1) 1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 b) c (2) 0 c (0) 1 c (1) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 c) c (2) 0 c (0) 1 c (0) 2 c(1) 1 w2 c (0) 3 w 4 c(0) 3 c(0) 4 Figure 1. The set of diagrams D 2 .
enumerate by ξ i the indices of non-conjugated variables a

(k)
s ′ in (2.4) and by σ i -those of conjugated variables ā(n) s ′′ . We write the corresponding fictitious index s 4 as σ 2j if we apply (2.4), or as ξ 2j if we apply the complex conjugation of (2.4). More precisely, when applying (2.4) we denote s 1 = ξ 2j-1 , s 2 = ξ 2j , s 3 = σ 2j-1 and s 4 = σ 2j , and when applying its complex conjugation, we write

s 1 = σ 2j-1 , s 2 = σ 2j , s 3 = ξ 2j-1 and s 4 = ξ 2j . We will abbreviate (2.6) δ j = δ ′ξ 2j-1 ξ 2j σ 2j-1 σ 2j and ω j = ω ξ 2j-1 ξ 2j σ 2j-1 σ 2j , j ≥ 1,
(these terms correspond to δ ′12 34 and ω 12 34 in (2.4)). We also set (2.7)

ξ 0 = σ 0 ∶= s. Applying (2.4) to a (2)
s and using the notation above with j = 1 we find a

s (τ ) = a (2) ξ 0 (τ ) = m 1 +m 2 +m 3 =1 i τ 0 dl 1 e -γ ξ 0 (τ -l 1 ) L -d ξ 1 ,ξ 2 ,σ 1 ,σ 2 δ 1 δ(ω 1 ) δ ξ 0 σ 2 a (m 1 ) ξ 1 a (m 2 ) ξ 2 ā(m 3 ) σ 1 (2)
(l 1 ) .

(2.8)

Let us consider the summand with m 1 = m 2 = 0 and m 3 = 1. Applying the conjugated formula (2.4) to ā(1) σ 1 and using the introduced notation with j = 2 we get ā(1)

σ 1 (l 1 ) = -i l 1 0 dl 2 e -γ σ 1 (l 1 -l 2 ) L -d ξ 3 ,ξ 4 ,σ 3 ,σ 4 δ 2 δ(ω 2 ) δ σ 1 ξ 4 a (0) ξ 3 ā(0) σ 3 ā(0) σ 4 (l 2 ) .
(2.9) ξ 4 in (2.8) and (2.9). Vertices which are not virtual are called real. Every edge of the diagram couples a nonconjugated (conjugated) vertex c

Inserting (2.
(k) i (c (k) i ) of positive degree k ≥ 1 with a conjugated (non-conjugated) virtual vertex wi ′ (w i ′ ).
It is associated with an application of formula (2.4) (or its complex conjugation) to the variable a (k)

ξ i (or ā(k) σ i ), corresponding to the vertex c (k) i (or c(k) i ). The set of four vertices (2.10) c 2j-1 , c 2j , c2j-1 , w2j or c 2j-1 , w 2j , c2j-1 , c2j
(in dependence whether the virtual vertex is conjugated or not) to which correspond the indices ξ 2j-1 , ξ 2j , σ 2j-1 , σ 2j is called the j-th block ; the diagram D c has two blocks. The index i of a virtual vertex w i ( wi ) is always pair, i = 2j. Each block corresponds to an application of formula (2.4) (or its complex conjugation) to its parent, i.e. to the vertex of positive degree coupled with the virtual vertex of the block. The virtual vertex is conjugated if the parent is non-conjugated and the other way round. The time variable l j is associated with the j-th block.

The leaves are the vertices of zero degree, that is, the vertices c 1 , joint by an edge with w2 in the only block The set D m+1 is constructed inductively in the same way.

B 1 = (c (0) 1 , c (0) 2 , c(0) 1 , w2
i τ 0 dl e -γ s (τ -l) L -d 1,2,3,4 δ ′12 34 δ(ω 12 34 ) δ s 4 a (m 1 ) 1 a (m 2 ) 2 ā(m 3 ) 3 (l), draw the block B 1 = (c (m 1 ) 1 , c (m 2 ) 2 , c(m 3 ) 1 , w2
For further needs we note that due to the factors δ ′12 34 and δ s s 4 in (2.4), the indices ξ i , σ j entering the formula for the sums I s (D) from (2.5) satisfy the relations 1)

δ ′ξ 2j-1 ξ 2j σ 2j-1 σ 2j = 1 ∀j, 2) indices ξ i , σ j corresponding to adjacent in D vertices are equal.
(2.12) 2.3. Feynmann diagrams for expectations. The main objects we are interested in are the correlations Ea (m)

s 1 (τ 1 )ā (n) s 2 (τ 2 ). It can be shown that they vanish if s 1 ≠ s 2 . 8 To represent an expectation Ea (m) s (τ 1 )ā (n) s (τ 2 ) we consider the set of diagrams D m × D n ∶= {D 1 ⊔ D2 ∶ D 1 ∈ D m , D2 ∈ D n }.
Here a diagram D 1 ⊔ D2 is obtained by drawing D 8 As well vanish the correlations Ea (m)

I s (D; τ 1 , τ 2 ) = I s (D 1 ; τ 1 )I s ( D2 ; τ 2 ),
s 1 (τ 1 )a (n) s 2 (τ 2 ) and Eā (m) s 1 (τ 1 )ā (n) s 2 (τ 2 ) for all s 1 , s 2 . a) c (2) 0 c(0) 0 c (1) 1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 b) c (1) 0 w2 c(0) 1 c (0) 2 c (0) 1 c(2) 0 c (1) 3 w 4 c(0) 3 c(0) 4 c (0) 5 c (0) 6 c(0) 5 w6 Figure 2. A diagram from the set a) D 2 × D 0 and b) D 1 × D 2 .
so that a (m)

s (τ 1 )ā (n) s (τ 2 ) = ∑ D∈D m ×D n I s (D; τ 1 , τ 2 ). Our next task is to com- pute EI s (D) for each D ∈ D m × D n .
Randomness enters the term I s (D) via the random variables a (0)

ξ i , ā(0) σ j , corresponding to the leaves of the diagram D = D 1 ⊔ D2 . They are Gaussian with correlations (2.13) Ea (0) s (l 1 )a (0) s ′ (l 2 ) = 0, Ea (0) s (l 1 )ā (0) s ′ (l 2 ) = δ s s ′ Corr(γ s , b(s), l 1 , l 2 ), where (2.14) Corr(γ s , b(s), l 1 , l 2 ) = B s e -γ s |l 1 -l 2 | -e -γ s (l 1 +l 2 ) , B s = b(s) 2 γ s .
So the Wick theorem [START_REF] Janson | Gaussian Hilbert Spaces[END_REF] implies that the expectation EI s (D) is given by a sum over all Wick pairings of variables a (0) ξ i , corresponding to non-conjugated leaves c (0) i , with variables ā(0) σ j corresponding to conjugated leaves c(0) j . Moreover, the leaves c (0) i and c(0) j should belong to different blocks since otherwise the summand corresponding to such Wick pairing vanishes due to (1.15) and item (1) in (2.12). We parametrize the sum over the Wick pairings by the defined below set F(D) of Feynman diagrams. Denoting by J s (F) a term (i.e. a sum), corresponding to a specific Feynman diagram F, we have: do not belong to the same block. To each such partition we associate a diagram F obtained from D by joining with an edge the two leaves in every pair, see fig. 3(a). So in each diagram F ∈ F(D) all vertices of D are joint by edges, every edge couples a conjugated vertex with a non-conjugated in another block (or with a root), and every vertex belongs to exactly one edge.

EI s (D) = F∈F(D) J s (F) .
Since

Ea (0) s ā(0) s ′ = 0 if s ≠ s ′
, the indices ξ i , σ j entering the formulas for sums J s (F) satisfy (2.12), where in item (2) the diagram D is replaced by the Feynman diagram F; below we denote these relations as (2.12) F . In particular, due to the item (2), the vector of indices σ = (σ i ) is a function of the vector ξ = (ξ j ). Accordingly below we write σ = σ F (ξ).

c (2) 0 c(0) 0 c (1) 1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 b) c (2) 0 c(0) 0 c (1) 1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0)
Let

F m,n = ⋃ D∈D m ×D n F(D)
be the set of all Feynman diagrams associated with the product a

(m) s ā(n) s . Each diagram F ∈ F m,
n has N ∶= m+n blocks and 4N +2 vertices, including 2N + 2 leaves Half of edges (and of leaves) are conjugated, while another half is not.

By construction a diagram F ∈ F m,n never pairs leaves from the same block. This alone does not exclude that F is such that in (2.12) F the assumptions (1) and ( 2) are incompatible since for some j we may have ξ 2j-1 , ξ 2j = σ 2j-1 or σ 2j once σ = σ F (ξ). Analysis shows that this cannot happen if m + n ≤ 4, but may happen if m + n ≥ 5. Accordingly, we denote by F true m,n ⊂ F m,n the set of Feynman diagrams for which the set of indices ξ i , σ j satisfying the relations (2.12) F is not empty. For any diagram F ∉ F true m,n we have J s (F) = 0 due to the factors δ ′12 34 and δ s s 4 in (2.4). 9 entering the formula for J s (F) are involved, which makes the sum J s (F) difficult for further analysis. In [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] it was found a convenient way to "integrate the sums J s (F)", i.e. to parametrise the indices ξ i , σ j by Nvector z = (z 1 , . . . , z N ) from a domain in (Z d L )

Transformation, resolving linear relations on indices. Let us take a Feynman diagram

F ∈ F m,n , denote N ∶= m + n ≥ 1 and consider the sum J s (F). The relations (2.12) F on indices ξ i , σ j ∈ Z d L , 0 ≤ i, j ≤ 2N ,
N , free from any relations on 9 Recall that ξ 0 = σ 0 = s.

its components. In this section we present this parametrisation, referring the reader to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] for a proof. Since item (2) of (2.12) F is equivalent to the relation σ = σ F (ξ), it suffices to parametrise the set of admissible multi-indices ξ, i.e. of those ξ for which the multi-indices ξ and s = σ F (ξ) satisfy item (1) in (2.12) F . The construction starts with defining for each F ∈ F m,n a skew-symmetric N ×N incidence matrix α F = (α F ij ) whose elements are integers from the set {0, +1, -1}. In terms of this matrix we define the set of polyvectors

(2.15) Z(F) = {z = (z 1 , . . . , z N ) ∈ (Z d L ) N ∶ z j ≠ 0 and (α F z) j ≠ 0 ∀ j} .
Here and below for an M × N -matrix A we denote by Az the polyvector with components (Az

) j ∶= ∑ i A ji z i ∈ Z d .
ξ(z) = s + A F z, z ∈ Z(F).
Here A F is an (2N + 1) × N -matrix, whose elements again are integers from the set {0, +1, -1}. Transformation (2.16) provides a presentation of the terms J s (F), forming the correlation Ea

(m) s (τ 1 )ā (n)
s (τ 2 ), and so for the correlation itself. The corresponding result is proved in Theorem 5.5 of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF]. In our setting its statement, where for the function θ(x, t) is chosen I {0} (x) -the indicator function of the point x = 0 -takes the following form: [START_REF] Faou | The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation[END_REF]) (depending on s and F) is such that the quantity ω j in (2.6), written in the z-coordinates, takes the form

Lemma 2.2. For any integers m, n ≥ 0 satisfying N = m + n ≥ 1, any s ∈ Z d L and τ 1 , τ 2 ≥ 0, 1) for each F ∈ F true m,n parametrisation (2.
(2.17) ω F j (z) = 2z j ⋅ N i=1 α F ji z i = 2z j ⋅ (α F z) j .
2) We have

(2.18) L N Ea (m) s (τ 1 )ā (n) s (τ 2 ) = F∈F true m,n c F J s (τ 1 , τ 2 ; F),
where the constants c F ∈ {±1, ±i} and

(2.19) J s (τ 1 , τ 2 ; F) = R N dl L N (1-d) z∈Z(F), ω F j (z)=0 ∀j F F s (τ 1 , τ 2 , l, z) .
10 That is, abusing notation we denote by A an operator in (Z d ) M with the block-matrix

A ⊗ 1. density F F s (τ 1 , τ 2 , l, z) is a real function, smooth in (s, z) ∈ R d × R dN , satisfying (2.20) |∂ µ s ∂ κ z F F s (τ 1 , τ 2 , l, z)| ≤ C # µ,κ (s)C # µ,κ (z) e -δ ∑ m i=1 |τ 1 -l i |+∑ N i=m+1 |τ 2 -l i | with a suitable δ = δ N > 0, for any vectors µ ∈ (N ∪ {0}) d , κ ∈ (N ∪ {0}) dN and any s ∈ R d , z ∈ R dN , l ∈ R N .
Let us briefly explain the way to construct the parametrization (2.16). We first add to the Feynman diagram F dashed edges that couple non-conjugated vertices with conjugated inside all blocks, as in fig. 3(b). For each block there are two ways of doing that. We prove that there exists a choice (possibly, not unique) of a dashed edge in each block such that the diagram becomes a cycle, as in fig. 3(b). Then, for each j we set x 2j-1 ∶= ξ 2j-1 -σ 2j-1 and x 2j ∶= ξ 2j -σ 2j or x 2j-1 = ξ 2j-1 -σ 2j and x 2j ∶= ξ 2j -σ 2j-1 , according to the choice of the dashed edges in the j-th block, where we substitute σ = σ F (ξ). The fact that the Feynman diagram with added dashed edges forms a cycle implies that the transformation ξ ↦ x is invertible. Item (1) of (2.12) F implies that x 2j = -x 2j-1 . Then we set z j ∶= x 2j-1 and get (2.16). The incidence matrix α F also is constructed in terms of this cycle.

Since the choice of the dashed edges in general is not unique, the parametrization z ↦ ξ is not unique as well. However, if z ′ ↦ ξ is another parametrization, obtained by the procedure above, and α F ′ is the associated incidence matrix, then for each j we have either z

′ j (ξ) = z j (ξ) or z ′ j (ξ) = (α F z(ξ)) j .
In the latter case we also have the symmetric relation [START_REF] Kuksin | Séminaire Laurent Schwartz EDP et applications[END_REF]) the integral over dl and using estimate (2.20), we obtain a form of integrals J s , more convenient for some of the subsequent analysis:

z j (ξ) = (α F ′ z ′ (ξ)) j . Computing in (2.
Corollary 2.3. In terms of Lemma 2.2, the integrals J s from (2.18) can be written as

(2.21) J s (τ 1 , τ 2 ; F) = L N (1-d) z∈Z(F), ω F j (z)=0 ∀j Φ F s (τ 1 , τ 2 , z) ,
where the real-valued functions Φ F s are Schwartz in (s, z) and satisfy

(2.22) |∂ µ s ∂ κ z Φ F s (τ 1 , τ 2 , z)| ≤ C # µ,κ (s)C # µ,κ (z), uniformly in τ 1 , τ 2 ≥ 0, for any vectors µ ∈ (N ∪ {0}) d and κ ∈ (N ∪ {0}) dN .

Main estimates for the sums

In this section we focus on estimates for the sums (2.21) and on their dependence on L and N . We recall that d ≥ 3. It is convenient to study the problem we consider in the following abstract setting. Let α = (α ij ), N ≥ 2, be an N × N skew-symmetric matrix whose elements belong to the set {-1, 0, 1}, without zero lines and rows. 11 Consider a family of quadratic forms on (R

d ) N ω j (z) = z j ⋅ (αz) j , 1 ≤ j ≤ N,
where z is the polyvector (z 1 , . . . , z N ), z j ∈ R d , and (αz

) j ∶= ∑ N i=1 α ji z i . Let us set (3.1) Z = {z ∈ (Z d L ) N ∶ z j ≠ 0 and (αz) j ≠ 0 ∀ j}. Let a function Φ ∶ R N d
→ R be sufficiently smooth and sufficiently fast decaying at infinity (see below for exact assumptions). Our goal is to study asymptotic as L → ∞ behaviour of the sum

(3.2) S L,N (Φ) ∶= L N (1-d) z∈Z∶ ω j (z)=0 ∀j Φ(z). For a function f ∈ C k (R m ), n 1 ∈ N ∪ {0} satisfying n 1 ≤ k and n 2 ∈ R, we set (3.3) ∥f ∥ n 1 ,n 2 = sup z∈R m max |α|≤n 1 |∂ α f (z)|⟨z⟩ n 2 , ⟨x⟩ ∶= max{1, |x|} .
The first crucial result concerns the case

N = 2. Then ω 1 (z) = -ω 2 (z) = α 12 z 1 ⋅ z 2 and α 12 ≠ 0, so (3.4) S L,2 (Φ) = L 2(1-d) z∈Z 2d L ∶ z 1 ⋅z 2 =0 z 1 ≠0, z 2 ≠0 Φ(z).
Then we write the sum above as

∑ z 1 ⋅z 2 =0 -∑ z 1 =0 or z 2 =0 . Since L -d ∑ z∈Z 2d L ∶ z i =0 Φ(z) ≤ C∥Φ∥ 0,d+1 for i = 1, 2, we get (3.5) S L,2 (Φ) -L 2(1-d) z∈Z 2d L ∶ z 1 ⋅z 2 =0 Φ(z) ≤ CL 2-d ∥Φ∥ 0,d+1 .
Now an asymptotic for the sum S L,2 (Φ) immediately follows from Theorem 1.3 in [START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF] where the dimension is 2d, ε = 1/2 and m = 0, by applying it to the sum ∑ z 1 ⋅z 2 =0 in (3.5) (we recall that d ≥ 3):

Theorem 3.1. Let N 1 (d) ∶= 4d(4d 2 + 2d -1) and N 2 (d) ∶= N 1 + 6d + 4. If ∥Φ∥ N 1 ,N 2 < ∞, then there exist constants C d ∈ (1, 1 + 2 2-d ), and
K d > 0 such that (3.6) S L,2 (Φ) -C d Σ 0 Φ(z) µ Σ 0 (dz 1 dz 2 ) ≤ K d ∥Φ∥ N 1 ,N 2 L d-5/2 ,
where Σ 0 is the quadric {z ∈ R 2d ∶ z 1 ⋅ z 2 = 0} and the measure µ Σ 0 is given by (1.34) with s = 0.

11 The theorems below and their proofs remain valid as well for arbitrary skewsymmetric matrices with integer elements without zero lines and rows, but in this case the notation used in the proof becomes heavier.

Appendix C of [START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF] we give the following explicit formula for the number-theoretical constants C d :

(3.7) C d = ζ(d -1)ζ(4d -2) ζ(d)ζ(2d -2) ,
where ζ is the Riemann zeta-function. Due to (3.7)

C d satisfies 1 < C d < 1 + 2 2-d
, as is stated in the theorem. The integral in (3.6) converges if Φ(z) decays at infinity fast enough:

(3.8)

Σ 0 Φ(z) µ Σ 0 (dz 1 dz 2 ) ≤ C r ∥Φ∥ 0,r if r > 2d -1,
see Proposition 3.5 in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]. So, it converges under the theorem's assumptions. From Theorem 3.1 another result can be deduced, whose proof is given in the next Section 3.1: Theorem 3.2. For N = 2, 3, 4 there exist constants C d,N such that

(3.9) |S L,N (Φ)| ≤ C d,N ∥Φ∥ 0, N , for N ∶= ⌊N /2⌋N 2 (d)+(N -2)(d-1)+1, where N 2 is defined in Theorem 3.1.
Since in view of estimate (2.22) the functions Φ F s from Corollary 2.3 satisfy

(3.10) ∥Φ F s (τ 1 , τ 2 , ⋅)∥ n 1 ,n 2 ≤ C # (s), ∀n 1 , n 2 ,
then the two theorems above apply to study correlations (2.18) with N = m + n ≤ 4. In fact, in the case N = 2 the number of Feynmann diagrams is small and the corresponding correlations may be calculated directly without the machinery, developed in Section 2. In Example 3.4 which illustrats this computation, as well as in a number of situations below, we apply Theorem 3.1 in the following setting:

Corollary 3.3. Let S L,2 = L 2(1-d) 1,2,3 δ ′12 3s δ(ω 12 3s )f s (s 1 , s 2 , s 3 ; q),
where ω 12 3s is given by (1.17), q ∈ R n is a parameter (in applications usually this will be the time) and

f s (s 1 , s 2 , s 1 + s 2 -s; q) 12 is a Schwartz function of (s 1 , s 2 , s) satisfying |∂ µ (s 1 ,s 2 ,s) f s | ≤ C # µ (s 1 )C # µ (s 2 )C # µ (s) uniformly in q, for any multi-index µ. Then (3.11) S L,2 -C d Σ s f s (s 1 , s 2 , s 1 + s 2 -s; q) µ Σ s (ds 1 ds 2 ) ≤ C # (s) L d-5/2 ,
uniformly in q, where Σ s and µ Σ s are the quadric (1.33) and the measure

(1.34) on it.

12 The formula for s 3 comes from the relation

δ ′12 3s = 1.
Proof. In the variables z 1 = s 1 -s, z 2 = s 2 -s the quadratic form ω

s (τ )| 2 .
Expanding a

s as in (2.4) and then using (2.13) we get:

E|a (1) s (τ )| 2 = 2L -2d 1,2,3 δ ′12 3s δ(ω 12 3s
)

τ 0 dl 1 τ 0 dl 2 B 123 × 3 j=1 e -γ j |l 1 -l 2 | -e -γ j (l 1 +l 2 ) e γ s (l 1 +l 2 -2τ ) with B 123 = B 1 B 2 B 3
, where B s is defined in (2.14). In the case of τ = ∞ the formula simplifies since by changing the integration variables as r j ∶= τl j and passing to the limit we get

E|a (1) s (∞)| 2 =2L -2d 1,2,3 δ ′12 3s δ(ω 12 3s ) ∞ 0 dr 1 ∞ 0 dr 2 B 123 e -(γ 1 +γ 2 +γ 3 )|r 1 -r 2 | e -γ s (r 1 +r 2 ) = 2L -2d γ s 1,2,3 δ ′12 3s δ(ω 12 3s
)

B 123 γ 1 + γ 2 + γ 3 + γ s .
Then, by Corollary 3.3,

L 2 E|a (1) s (∞)| 2 - 2C d γ s Σ s B 123 µ Σ s (ds 1 ds 2 ) γ 1 + γ 2 + γ 3 + γ s ≤ C # (s) L d-5/2 , s 3 ∶= s 1 + s 2 -s.
3.1. Proof of Theorem 3.2. Let us define the geometric quadrics

Q j ∶= {z ∈ (R d ) N ∶ ω j (z) = 0} and consider their intersection Q = ∩ N j=1 Q j . Note that Q = ∩ N -1 j=1 Q j since the skew symmetry of the matrix α implies ω 1 + . . . + ω N = 0. Denote by B N d R the open cube |z| ∞ < R in R N d , where by | ⋅ | ∞ we denote the l ∞ -norm. Proposition 3.5. If w ∶ R N d ↦ R is such that |w| L ∞ < ∞ and supp(w) ⊂ B N d R , where R ≥ 1, then for N = 2, 3, 4 we have (3.12) z∈Q∩Z w(z) ≤ C(N, d)R ⌊N /2⌋N 2 (d)+(N -2)(d-1) L N (d-1) |w| L ∞ .
Here N 2 is defined in Theorem 3.1 and Z -in (3.1).

Below in this proof for any subset Q ⊂ R md we denote

(3.13) Q L = Q ∩ Z md L .
By suitably rearranging indices i and possibly multiplying ω i by -1, ω 1 may be assumed to be of the form

ω 1 (z) = z 1 ⋅ ∑ i α 1i z i with α 1N = 1. Define v = ∑ i α 1i z i so that (3.14) ω 1 (z) = z 1 ⋅ v and z N = α 1N z N = v - 1<i<N α 1i z i ,
since α 11 = 0 by the skew symmetry of the matrix α.

For N > 2, fix (z 1 , v) ∈ R 2d .
Then the remaining quadratic forms ω j with 1 < j < N as functions of (z 2 , . . . , z N -1 ) ∈ R (N -2)d become polynomials q j of degree at most two, with no constant term. Namely

(3.15) q j (z 2 , . . . , z N -1 ; z 1 , v) = z j ⋅ α j1 z 1 + α jN v + 1<i<N (α ji -α jN α 1i )z i .
For 1 < j < N consider the sets

Qj (z 1 , v) = {(z 2 , . . . , z N -1 ) ∶ q j (z 2 , . . . , z N -1 ; z 1 , v) = 0} ⊂ R (N -2)d ,
and

their intersection Q(z 1 , v) = ∩ 1<j<N Qj (z 1 , v). We denote Q 0 1 = {(z 1 , v) ∈ R 2d ∶ z 1 ⋅ v = 0} (cf. (3. 14
)) and set (3.16)

A 2 = {(z 1 , v) ∈ R 2d ∶ z 1 ≠ 0, v ≠ 0}.
Since |α ij | ≤ 1, then on the support of w we have

|(z 1 , v)| ∞ ≤ (N -1)R. So, recalling (3.13), for N > 2 we get z∈Z∩Q w(z) ≤ C(N, d)|w| L ∞ (z 1 ,v)∈Q 0 1 L ∩B 2d (N -1)R 1 × sup (z 1 ,v)∈Q 0 1 L ∩A 2 ∩B 2d (N -1)R (z 2 ,...,z N -1 )∈ QL (z 1 ,v)∩B (N -2)d R 1 .
(3.17)

For N = 2 the same estimate holds with the second line replaced by 1.

To estimate the sum in the first line, we take any smooth function w 0 (x) ≥ 0, equal one for x ≤ 1 and vanishing for x ≥ 2. Then

(z 1 ,v)∈Q 0 1 L ∩B 2d (N -1)R 1 ≤ (z 1 ,v)∈Q 0 1 L w R (z 1 , v), where w R (z 1 , v) ∶= w 0 |(z 1 , v)|/ (N -
(z 1 ,v)∈Q 0 1 L ∩B 2d (N -1)R 1 ≤ CL 2(d-1) R 2d + R N 2 L -d+5/2 ≤ C ′ L 2(d-1) R N 2 ,
where C, C ′ depend on d, N, N 1 and N 2 . To estimate the second line of (3.17) we use the following lemma, proved in Appendix A. Lemma 3.6. Assume that the matrix α is irreducible. Then for N = 2, 3, 4, any R ≥ 1 and any (z

1 , v) ∈ B 2d (N -1)R satisfying (z 1 , v) ∈ Q 0 1 L ∩ A 2 we have: (3.19) | QL (z 1 , v) ∩ B (N -2)d R | ≤ 2 (N -2)d (N RL) (N -2)(d-1) .
This completes the proof of Proposition 3.5 in the case of irreducible matrix α: indeed, we get

(3.20) z∈Z∩Q w(z) ≤ C(N, d)|w| L ∞ R N 2 +(N -2)(d-1) L N (d-1) .
If the matrix α is reducible, it can be reduced through permutations to a block diagonal matrix with m blocks which are irreducible square matrices of sizes N i satisfying ∑ i N i = N . Since N i ≥ 2 (otherwise there would be a zero row or column in α), m ≤ ⌊N /2⌋. Applying estimate (3.20) to each block we get the assertion of the proposition. Now we derive the theorem from the proposition. Let

ϕ 0 (t) = χ (-∞,1] (t) and for k ≥ 1, ϕ k (t) = χ (2 k-1 ,2 k ] (t). Then 1 = ∑ k ϕ k (t) and Φ = ∞ k=0 f k (z) , f k (z) = ϕ k (|z| ∞ )Φ(z) . Then supp f k ⊂ B k = {|z| ∞ ≤ 2 k } and ∥f k ∥ ∞ ≤ C2 -k
N ∥Φ∥ 0, N , for any N .

Therefore, by Proposition 3.5,

|S L,N (Φ)| ≤ C(N, d)∥Φ∥ 0, N ∞ k=0 2 k(⌊N /2⌋N 2 +(N -2)(d-1)-N ) , which converges if N > ⌊N /2⌋N 2 + (N -2)(d -1
). This completes the proof of Theorem 3.2.

Remark 3.7. For any fixed vector

(z 1 , v), Q(z 1 , v) is a real algebraic set in R (N -2)d of codimension (N -2). If Q(z 1 , v)
were a smooth manifold of that codimension, then estimate (3.19), modified by a multiplicative constant

C Q(z 1 ,v) , would be obvious. But Q(z 1 , v
) is a stratified analytic manifold (with singularities), and to obtain for it a modified version of the estimate (3.19) as above, using analytical tools, seems to be a heavy job since we need a good control for the factor C Q(z 1 ,v) . Instead in Appendix A we prove the lemma, using arithmetical tools.

3.2. On extension of Theorem 3.2 to any N . The restriction on N in the statement of Theorem 3.2 comes from estimate (3.19) in Lemma 3.6, proved only for N = 3, 4. We know that for every N the system of polynomials q j (⋅; z 1 , v), 1 < j < N , defining the set QL (z 1 , v) in Lemma 3.6, linearly independent for any (z 1 , v) and any irreducible incidence matrix α. Also we know that all polynomials q j (⋅; z 1 , v) are irreducible; see Lemmas A.9 and A.10 in Appendix A (there the independence and reducibility are understood over some specific algebraically closed field K, but the argument also works for K replaced by C). These two facts certainly are insufficient to prove Lemma 3.6 for any N , but they naturally lead to Conjecture 3.8. Under assumptions of Lemma 3.6, for any N ≥ 2

| QL (z 1 , v) ∩ B (N -2)d R | ≤ C(N, d)(RL) (N -2)(d-1)
.

One may try to prove this assertion using either arithmetical or analytical tools; cf. Appendix A and Remark 3.7. It is straightforward to see that, if the conjecture is true, then Theorem 3.2 holds for any N , so in view of Lemma 2.2 any expected value

L N Ea (m) s (τ 1 )ā (n) s (τ 2 ) admits a uniform in L upper bound.

Quasisolutions

In this section we start to study a quasisolution A(τ ) = A(τ ; L) of eq. (2.1) with a s (0) = 0, which is the second order truncations of series (2.2):

(4.1) A(τ ) = (A s (τ ), s ∈ Z d L ), A s (τ ) = a (0) s (τ ) + ρa (1)
s (τ ) + ρ 2 a (2)
s (τ ) . We focus on its energy spectrum (4.2)

n s,L = n s,L (τ ) = E|A s (τ )| 2 , s ∈ Z d L
, when L is large and the parameter ρ is chosen to be ρ = εL. Our goal is to show that it approximately satisfies a wave kinetic equation (WKE). Using Proposition 2.1, we will then show that the same applies to the quantities n s,L , considered in the Introduction.

The energy spectrum n s,L is a polynomial in ε of degree four,

(4.3) n s,L = n (0) s,L + ε n (1) s,L + ε 2 n (2) s,L + ε 3 n (3) s,L + ε 4 n (4) s,L , s ∈ Z d L ,
where the terms n given by (2.18), (2.21). Accordingly from now on we always regard the second moments and the terms n

(k) s,L (τ ) are defined by (4.4) n (k) s,L (τ ) = L k k 1 +k 2 =k 0≤k 1 ,k 2 ≤2 Ea (k 1 ) s (τ )ā (k 2 ) s (τ ) .

By Corollary 2.3, the second moments Ea

(k) s,L (τ ) as Schwartz functions of s ∈ R d .
As customary in WT, we aim at considering the limit of n s,L (τ ) as L → ∞, that is, the limits of the terms n

(j) s,L . The term n (0) s,L = n (0)
s,L is given by (1.31) and is L-independent, while by a direct computation we see that (4. [START_REF] Deng | Full derivation of the wave kinetic equation[END_REF] n

s,L = 2REā

(0) s a (1)
s = 0, s ∈ R d ,
(here we use (2.4), the Wick theorem and (1.15)). Writing explicitly

n (i) s,L with 2 ≤ i ≤ 4, we find that n (2) s,L = L 2 E |a (1) s | 2 + 2Rā (0) s a (2) s , n (3)
s,L = 2L 3 REā (1) s a (2)
s , n

s,L = L 4 E|a (2) s | 2 . (4.7) The function R d ∋ s ↦ n (4)
s,L (τ ) is made by two terms. By Corollary 2.3 with N = 2, Theorem 3.1 applies to the both of them. Since d ≥ 3, we get

(4.8) |n (2) s,L (τ ) -n (2)
s (τ )| ≤ C # (s)/L 1/2 ,
where n

s (τ ) ∶= C d F∈F true 1,1 +2R F∈F true 2,0 c F Σ 0 µ Σ 0 (dz 1 dz 2 )Φ F s (τ, τ, z), (2)
and we have used estimate (3.10). Thus, we see that the processes n

(0) s,L , n (1) s,L and n (2)
s,L admit the limits

n (j) s (τ ) ∶= lim L→∞ n (j)
s,L (τ ; L).

The limits satisfy (4.8), and for all τ (4.9) n (0)

s (τ ) = B(s) 1 -e -2γ s (τ 0 +τ ) , n (1)
s (τ ) = 0, |n (2)
s (τ )| ≤ C # (s) ,
where the last inequality follows from Theorem 3.2. We do not know if the terms n

(3) s,L , n (4)
s,L admit limits as L → ∞, but in view of Corollary 2.3 both of them may be estimated through Theorem 3.2:

(4.10) |n (3) s,L (τ )| ≤ C # (s), |n (4)
s,L (τ )| ≤ C # (s) ,
uniformly in L ≥ 2 and τ ≥ 0. We then decompose

n s,L = n ≤2 s,L + n ≥3 s,L , where n ≤2 s,L = n (0) s,L + εn (1) s,L + ε 2 n (2) s,L and n ≥3 s,L = ε 3 n (3) s,L + ε 4 n (4) s,L
(we recall that n

s,L ≡ 0), and similarly define

n ≤2 s ∶= n (0) s + ε 2 n (2)
s . Due to (4.8), (4.11)

|n ≤2 s (τ ) -n ≤2 s,L (τ )| ≤ C # (s)ε 2 L -1/2 ,
by (4.10), (4.12)

|n ≤2 s (τ ) -n s,L (τ )| ≤ C # (s)ε 2 (L -1/2 + ε).
Thus, the cut energy spectrum n ≤2 s governs the limiting as L → ∞ behaviour of the energy spectrum n s,L with precision ε 3 C # (s), where we regard the constant ε ≤ 1/2 (which measures the size of solutions for (1.6) under the proper scaling) as a fixed small parameter. Accordingly, our next goal is to show that n ≤2 s (τ ) approximates the solution of a WKE.

4.1.

Increments of the energy spectra n ≤2 s . In this section we will show that the process n ≤2 s (τ ) approximately satisfies a WKE. We denote s 4 ∶= s, γ j ∶= γ s j and set (4.13)

γ 1234 = γ 1 + γ 2 + γ 3 + γ 4 , ⃗ s = (s 1 , s 2 , s 3 , s 4 ) ∈ (R d ) 4 .
Now, for a fixed τ 0 ≥ 0 and for j = 1, 2, 3, 4 we define the functions

Z j (τ 0 ) = Z j (τ 0 ; ⃗ s) as (4.14) Z j (τ 0 ; ⃗ s) ∶= τ 0 0 dl e -γ j (τ 0 -l) m=1,2,3,4 m≠j sinh(γ m l) sinh(γ m τ 0 ) if τ 0 > 0,
and Z j (0; ⃗ s) = 0. Computing this integral we get

Z j (τ 0 ; ⃗ s) = ⎛ ⎜ ⎝ l≠j 1 1 -e -2γ l τ 0 ⎞ ⎟ ⎠ ⋅ 1 -e -γ 1234 τ 0 γ 1234 - e -2(γ 1234 -γ j )τ 0 -e -γ 1234 τ 0 2γ j -γ 1234 + l≠j e -2(γ 1234 -γ j -γ l )τ 0 -e -γ 1234 τ 0 2(γ j + γ l ) -γ 1234 - e -2γ l τ 0 -e -γ 1234 τ 0 γ 1234 -2γ l , (4.15)
where each fraction from the square brackets should be substituted by τ 0 e -γ 1234 τ 0 if its denominator vanishes.

For any real number r let C r (R d ) be the space of continuous complex functions on R d with the finite norm

(4.16) |f | r = |f (z)⟨z⟩ r | L ∞ .
We naturally extend this norm to f ∈ L ∞ (R d ) and set (4.17)

L ∞,r (R d ) = f ∈ L ∞ (R d ) ∶ |f | r < ∞ .
Consider also the linear operator L, given by

(4.18) (Lv)(s) = 2γ s v(s), s ∈ R d .
Below we often write the value v(s) of a function v at s ∈ R d as v s and the function v itself as

(v s , s ∈ R d ). Now, for v ∈ C r (R d )
, where r > d, and for τ 0 ≥ 0, τ ∈ (0, 1], we define the kinetic integral K

τ (τ 0 )(v) = (K τ s (τ 0 )(v), s ∈ R d ): (4.19) K τ (τ 0 )(v) = τ 0 e -tL K(τ 0 )(v)dt .
Here the operator K(τ

0 ) = K 1 (τ 0 ) + ⋅ ⋅ ⋅ + K 4 (τ 0 ) sends a function v = (v s , s ∈ R d ) to the function K s (τ 0 )(v) = 4C d Σ s µ Σ s (ds 1 ds 2 ) Z 4 (τ 0 ; ⃗ s)v 1 v 2 v 3 + Z 3 (τ 0 ; ⃗ s)v 1 v 2 v 4 -Z 2 (τ 0 ; ⃗ s)v 1 v 3 v 4 -Z 1 (τ 0 ; ⃗ s)v 2 v 3 v 4 =∶ K 4 s (τ 0 )(v) + K 3 s (τ 0 )(v) + K 2 s (τ 0 )(v) + K 1 s (τ 0 )(v) (4.20)
(note the reversed signs for K 2 and K 3 ). Here v j ∶= v(s j ), where s 4 = s and

s 3 ∶= s 1 + s 2 -s 4 (in view of the factor δ ′12 3s ). While µ Σ s is the measure (1.34) on the quadric Σ s = {(s 1 , s 2 ) ∈ R 2d ∶ (s 1 -s) ⋅ (s 2 -s) = 0}. Computing the integral in t in (4.19), we find (4.21) K τ s (τ 0 )(v) = 1 -e -2γ s τ 2γ s K s (τ 0 )(v) = 1 -e -2γ s τ 2γ s 4 j=1 K j s (τ 0 )(v).
We study the kinetic integral K τ in Section 5 while now we formulate a result which is the main step in deriving the wave kinetic limit.

Theorem 4.1. For any 0 < τ ≤ 1 the function

(n ≤2 s , s ∈ R d ) satisfies n ≤2 (τ 0 + τ ) = e -τ L n ≤2 (τ 0 ) + 2 τ 0 e -tL b 2 dt + ε 2 K τ (τ 0 )(n ≤2 (τ 0 )) + ε 2 R, (4.22)
where b 2 = (b 2 (s), s ∈ R d ) and the remainder R s (τ ) satisfies

(4.23) |R(τ )| r ≤ C r τ (τ + ε 2 ) , ∀r.
4.2. Proof of Theorem 4.1. We first fix a value for L and decompose the processes τ ↦ a (i) s (τ 0 + τ ), where τ 0 ≥ 0 and 0 ≤ τ ≤ 1, as (4.24) a (i)

s (τ 0 + τ ) = c (i) s (τ ; τ 0 ) + ∆a (i) s (τ ; τ 0 ), i = 0, 1, 2, s ∈ Z d L . Here c (i) s (τ ; τ 0 ) = e -γ s τ a (i)
s (τ 0 ) and ∆a

i) s (τ ) since τ 0 is fixed. Obviously, c(τ ) ∶= c (0) (τ ) + ρc (1) (τ ) + ρ 2 c (2) (τ )
τ ≥ 0 is a solution of the linear equation (2.1) ρ=0,b(s)≡0 , equal A(τ 0 ) at τ = 0, and ∆a(τ ) = ∑ 2 j=0 ρ j ∆a (j) (τ ) equals A(τ 0 + τ ) -c(τ ). By (4.5), for 0 ≤ i, j ≤ 2 the functions Ec

(i) s c(j) s , Ec (i) s ∆ā (j)
s , E∆a

e -2γ s τ n ≤2 s,L (τ 0 ) = E|c (0) s (τ )| 2 +ρ 2 E |c (1) s (τ )| 2 +2Rc (0) s (τ )c (2) s (τ ) , ∀ s ∈ R d .
Then,

n ≤2 s,L (τ 0 + τ ) -e -2γ s τ n ≤2 s,L (τ 0 ) = E |a (0) s (τ 0 + τ )| 2 -|c (0) s (τ )| 2 + ρ 2 |a (1) s (τ 0 + τ )| 2 -|c (1)
s (τ )| 2 + 2R a (2) s ā(0) s (τ 0 + τ ) -c (2) s c(0) s (τ ) . (4.26) Let us set Y s (u, v, w) ∶= L -d 1,2,3 δ ′12 3s δ(ω 12 3s )u 1 v 2 w3 .
Writing explicitly the processes ∆a

(i) s (τ ), s ∈ Z d L , we find ∆a (0) s (τ ) = b(s) τ 0 +τ τ 0 e -γ s (τ 0 +τ -l) dβ s (l), ∆a (1)
s (τ ) = i τ 0 +τ τ 0 e -γ s (τ 0 +τ -l) Y s (a (0) ) dl, ∆a (2)
s (τ ) = i τ 0 +τ τ 0 e -γ s (τ 0 +τ -l) Y s (a (0) , a (0) , a (1) )
+ Y s (a (0) , a (1) , a (0) ) + Y s (a (1) , a (0) , a (0) ) dl,

(4.27)
where a (i) = a (i) (l). Note that to get explicit formulas for c (i) s (τ ), i = 0, 1, 2, it suffices to replace in the r.h.s.'s of the relations in (4.27) the range of integrating from [τ 0 , τ 0 +τ ] to [0, τ 0 ]. For example, c (0)

s (τ ) = e -γ s τ a (0) s (τ 0 ) = b(s) ∫ τ 0 0 e -γ s (τ 0 +τ -l) dβ s (l).
Using that Ec (i)

s (τ )∆ā (0) s (τ ) = Ec (i) s (τ ) E∆ā (0)
s (τ ) = 0 for any i and s, we obtain

(4.28) E a (2) s ā(0) s (τ 0 + τ ) -c (2) s c(0) s (τ ) = E∆a (2) s (τ )ā (0)
s (τ 0 + τ ), and from (4.24) we get that |a (1)

s (τ 0 + τ )| 2 -|c (1) s (τ )| 2 = |∆a (1) s (τ )| 2 + 2R∆a (1) s c(1) s (τ ), E |a (0) s (τ 0 + τ )| 2 -|c (0) s (τ )| 2 = E|∆a (0) s (τ )| 2 . (4.29)
Then, inserting (4.28) and (4.29) into (4.26) and using that ρ = εL, we find

n ≤2 s,L (τ 0 + τ ) -e -2γ s τ n ≤2 s,L (τ 0 ) = E|∆a (0) s (τ )| 2 + ε 2 Q s,L (τ 0 , τ ), s ∈ R d ,
where

Q s,L (τ 0 , τ ) ∶=L 2 E|∆a (1) s (τ )| 2 + 2RE ∆a (1)
s (τ )c ( 1)

s (τ ) + ∆a (2)
s (τ )ā (0)

s (τ 0 + τ ) , (4.30)
and we recall (4.25). Since

E|∆a (0) s (τ )| 2 = b(s) 2 γ s (1 -e -2γ s τ ) = 2 τ 0 e -tL b 2 (s) dt, then n ≤2 ⋅,L (τ 0 + τ ) -e -τ L n ≤2 ⋅,L (τ 0 ) = 2 τ 0 e -tL b 2 dt + ε 2 Q ⋅,L (τ 0 , τ ), for n ≤2 ⋅,L = (n ≤2 s,L , s ∈ R d ).
In order to pass to the limit L → ∞ we recall the relation (4.11). Then the desired formula (4. 22) is an immediate consequence of the assertion below:

Proposition 4.2. We have

(4.31) lim L→∞ Q s,L (τ 0 , τ ) = K τ s (τ 0 )(n ≤2 (τ 0 )) + R s (τ ), s ∈ R d ,
where the remainder R satisfies (4.23).

Proof. The first step in the proof of (4.31) is the following result, established in Appendix B:

Proposition 4.3. One has

(4.32) |Q s,L (τ 0 , τ ) -X s,L (τ 0 , τ )| ≤ C # (s)τ 2 , s ∈ R d ,
where

X s,L (τ 0 , τ ) ∶= 4L 2(1-d) τ 1,2,3 δ ′12 3s δ(ω 12 3s ) Z 4 n (0) 1 n (0) 2 n (0) 3 + Z 3 n (0) 1 n (0) 2 n (0) s -Z 1 n (0) 2 n (0) 3 n (0) s -Z 2 n (0) 1 n (0) 3 n (0) s . (4.33)
The terms Z j = Z j (τ 0 ; s 1 , s 2 , s 3 , s) are defined by (4.14) and n

(0) i ∶= n (0) s i ,L (τ 0 ), n (0) s ∶= n (0) s,L (τ 0 ). By (4.9) n (0) i = n (0)
s i are Schwartz functions in s i . Besides, the functions Z j (τ 0 , ⃗ s) have at most polynomial growth in ⃗ s together with their derivatives, uniformly in τ 0 ≥ 0: Lemma 4.4. For any vector µ ∈ (N ∪ {0}) 4d , uniformly in τ 0 ≥ 0, we have

∂ µ ⃗ s Z j (τ 0 , ⃗ s) ≤ P (⃗ s; µ)
, where P (⃗ s; µ) has at most a polynomial growth in ⃗ s.

the lemma, which is proven in Section B.7, X s,L satisfies the hypotheses of Corollary 3.3. So (4.34)

|X s,L (τ 0 , τ ) -τ K s (τ 0 )(n (0) )| ≤ C # (s)L -1/2 τ .
Next, note that |n (0)

s (τ 0 ) -n ≤2 s (τ 0 )| ≤ C # (
s)ε 2 due to (4.9). Then the estimate on the Lipschitz constants of the operators K j (t), given in (5.4), implies that

|K(τ 0 )(n (0) (τ 0 )) -K(τ 0 )(n ≤2 (τ 0 ))| r ≤ C r ε 2 ∀ r. So that (4.35) |τ K s (τ 0 )(n (0) (τ 0 )) -τ K s (τ 0 )(n ≤2 (τ 0 ))| ≤ C # (s)τ ε 2 .
On the other hand, on account of the definition (4.21), for 0 ≤ τ ≤ 1 we have the bound

(4.36) τ K s (τ 0 )(n ≤2 ) -K τ s (τ 0 )(n ≤2 ) ≤ Cγ s τ 2 |K s (τ 0 )(n ≤2 )| ≤ C # (s)τ 2 ,
where the last inequality follows from the estimate of the norm of the operator K j (t), given in (5.3), and from (4.9).

Putting together (4.32), (4.34), (4.35), (4.36) and letting L grow to infinity, we conclude the proof.

Kinetic equation

At this section we examine the wave kinetic equation (5.1) żs

(τ ) = -(Lz) s + ε 2 K s (τ )(z) + 2b(s) 2 , τ ≥ 0, z(0) = 0
(L is defined in (4.18) and the operator

K = K 1 + ⋅ ⋅ ⋅ + K 4 is defined in (4. 20
)), and next derive from this analysis and (4.22) the proximity of n ≤2 s (τ ) to a solution of (5.1). We will need the following result, which is Lemma 4.2 from [7]: Lemma 5.1. For j, l = 1, . . . , 4 and u j ∈ C r (R d ) consider the operators

J l (u 1 , . . . , u 4 )(s) = Σ s µ Σ s (ds 1 ds 2 ) i≠l u i (s i ) (see (1.34))
, where s 4 = s and s 3 = s 1 + s 2 -s. Then for each l,

(5.2) |J l (u 1 , . . . , u 4 )| r+1 ≤ C r i≠l |u i | r if r > d.
5.1. Kinetic integrals. We recall notation (4.13), (4.14).

Lemma 5.2. For j = 1, . . . 4, any τ ≥ 0 and any ⃗ s = (s 1 , . . . , s 4 )

∈ (R d ) 4 , i) 0 ≤ Z j (τ ; ⃗ s) ≤ min(τ, 1/γ s j ) ≤ 1, ii) |Z j (τ ; ⃗ s) -Z(∞; ⃗ s)| ≤ Ce -2τ , where Z(∞; ⃗ s) = 1/γ 1234 .
Proof. The first assertion follows from (4.14) since sinh(x) is an increasing non-negative function of x ≥ 0, so in the integrand in (4.14) we have 0 ≤ sinh(γ m l)/ sinh(γ m τ ′ ) ≤ 1. For 0 ≤ τ ≤ 1 the second estimate follows from the first one as

|Z j (τ ; ⃗ s) -Z(∞; ⃗ s)| ≤ |Z j (τ ; ⃗ s)| + |Z(∞; ⃗ s)|,
while for τ ≥ 1 it follows from (4.15) since γ 123j -γ j ≥ 1 and γ 1234 -γ j -γ l ≥ 1 for j, l ∈ {1, 2, 3, 4}, j ≠ l.

Since the kernels Z j are non-negative by the first assertion of the lemma above, then denoting κ 1 = κ 2 = 1, κ 3 = κ 4 = -1 we achieve that the operators κ j K j , 1 ≤ j ≤ 4, are positive (in the sense that they send positive functions to positive). Due to the first assertion of the lemma and (5.2), for any τ ≥ 0 they define positive 3-homogeneous mappings C r (R

d ) → C r+1 (R d ) if r > d, and
(5.3) |κ j K j (τ )(v)| r+1 = |K j (τ )(v)| r+1 ≤ C r min(τ, 1)|v| 3
r , j = 1, . . . , 4, for τ ≥ 0. So the mappings K j (τ ) are locally Lipschitz:

(5.4)

|K j (τ )(v 1 )-K j (τ )(v 2 )| r+1 ≤ 3C r min(τ, 1)R 2 |v 1 -v 2 | r , if |v 1 | r , |v 2 | r ≤ R.
Since for j = 1, . . . , 4 and any s ∈ R d ,

• for non-negative functions n, m ∈ L ∞,r (see (4.17)) such that m ≤ n we have

κ j K j s (τ )(m) ≤ κ j K j s (τ )(n) ≤ ∞, • |K j s (τ )(v)| ≤ κ j K j s (τ )(|v|) ≤ ∞ for any complex function v ∈ L ∞ , • |v s | ≤ |v| r ⟨s⟩
-r for all v ∈ L ∞,r , then the relations (5.3), (5.4) remain true for functions from L ∞,r .

Lemma 5.3. If |s l | ≤ R for l = 1, . . . , 4, then (5.5) ∂ ∂τ Z j (τ ; ⃗ s) ≤ Cγ 0 (R 2 ) (see (1.5)).
Proof. For any m ∈ {s 1 , . . . , s 4 } and 0 ≤ l ≤ τ we have ∂ ∂τ

sinh γ m l sinh γ m τ ≤ γ m cosh γ m τ sinh γ m τ ≤ γ m C max(1, 1/(γ m τ )).
Considering separately the cases τ ≥ 1 and 0 ≤ τ < 1, using (4.14) and the estimate above we get the result.

This lemma implies that for any v ∈ C r (R d ) and any j the curve τ ↦

K j (τ )(v) ∈ C r (R d ) is Hölder-continuous:
Lemma 5.4. For any τ 0 ≥ 0, 0 ≤ τ ≤ 1, j = 1, . . . , 4 and any r > d + 1,

(5.6)

|K j (τ 0 + τ )(v) -K j (τ 0 )(v)| r ≤ C r |v| 3 r τ κ * ∀ v ∈ C r (R d ),
where κ * = 1/(1 + 2r * ).

By the homogeneity we may assume that |v|

r = 1. For R ≥ 1 let us set v R (s) = v(s)χ |s|≤R ∈ L ∞ . Then (5.7) |v R | r ≤ 1, |v -v R | r-1 ≤ R -1 .

Now let us write the increment K

j (τ 0 + τ )(v) -K j (τ 0 )(v) as K j (τ 0 + τ )(v) -K j (τ 0 + τ )(v R ) + K j (τ 0 + τ )(v R ) -K j (τ 0 )(v R ) + K j (τ 0 )(v R ) -K j (τ 0 )(v) =∶ ∆ 1 + ∆ 2 + ∆ 3 .
Recalling that (5.3) and ( 5.4) hold for functions from L ∞,r ′ with r ′ > d,

we get from (5.7) that |∆ 1 | r + |∆ 3 | r ≤ C r R -1 . To estimate ∆ 2 we set ∆ R 2 = ∆ 2 χ |s|≤R . Since by (5.3), |∆ 2 | r+1 ≤ 2C r , then |∆ 2 -∆ R 2 | r ≤ 2C r R -1 . For |s| > R the function ∆ R 2 vanishes, while for |s| ≤ R in view of Lemma 5.3 we have |∆ R 2 s | = |∆ 2s | = |K j s (τ 0 + τ )(v R ) -K j s (τ 0 )(v R )| ≤ C r Σ s µ s (dv 1 dv 2 )|Z j (τ 0 + τ ; ⃗ s) -Z j (τ 0 ; ⃗ s)| |v 1 | . . . |v 4 | |v j | χ {|s j |≤R ∀j} ≤ C 1r γ 0 (R 2 )τ Σ s µ s (dv 1 dv 2 ) |v 1 | . . . |v 4 | |v j | ≤ C 2r ⟨s⟩ -r-1 τ R 2r * ,
where to get the last inequality we used (5.2). We have seen that the C r -norm of the increment is bounded by C r (R -1 + τ R 2r * ), for any R ≥ 1. Choosing R = τ -1/(1+2r * ) we achieve (5.6).

5.2. Kinetic equation. Now we will apply the obtained results to the kinetic equation (5.1). Since the function b(⋅)

2 ∶= {b(s) 2 } ∈ C r (R d ) for all
r, since L is the operator of multiplying by the function 2γ s as in (1.4), (1.5), and the operator K satisfies (5.3), (5.4), then for small enough ε > 0 eq. ( 5.1) has a unique solution, belonging to C r (R d ) for each r, which in a Lipschitz way depends on the r.h.s. of the equation, when the latter deviates from b(⋅) 2 . Namely, the following result, where X r stands for the space

C(0, ∞; C r (R d )), given the norm |v(⋅)| X r = sup t≥0 |v(t)| r
, may be easily verified (a proof of a similar fact may be found in Section 4 of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]).

Lemma 5.5. For any r > d, 1) There exists ε * , depending on b(⋅), r and r * , such that for 0 ≤ ε ≤ ε * eq. (5.1) has a unique solution z(τ ), belonging to X r . It satisfies

(5.8) |z| X r ≤ C r |b 2 | r .
2) If z 0 (τ ) is a solution of the linear equation

(5.1)| ε=0 , then |z -z 0 | X r ≤ C r ε 2 . If a curve z ′ (τ ) solves (5.1) with 2b(s) 2 replaced by 2b(s) 2 + ξ s (t),
where ξ ∈ X r and |ξ|

X r ≤ 1, then |z -z ′ | X r ≤ C r |ξ| X r .
The lemma's assertion holds as well for non-zero initial conditions z(0) ∈ C r (R d ) in (5.1), but we do not need this.

Let K(∞) be the operator, obtained from K(τ 0 ) by replacing in (4.20) the kernels Z j (τ 0 ; s), s ∈ R d , by Z(∞; ⃗ s) (see Lemma 5.2). Let r > d and z ε ∈ C r (R d ) be a solution of the limiting stationary equation (5.9)

Lz ε -ε 2 K(∞)(z ε ) = 2b(⋅) 2 in the vicinity of L -1 (2b 2
), existing for small ε by the inverse function the-

orem. Since b 2 (⋅) ∈ ⋂ r C r (R d ) and, as in (5.3), the map K(∞) is one- smoothing, then decreasing ε * if needed we achieve that z ε ∈ ⋂ r C r (R d ) for ε ≤ ε * and
(5.10)

|z ε -2L -1 (b 2 )| m ≤ C m ε 2 ∀ m.
Here and below the constants depend on b and r * .

Let us consider the curve w(t) = z(t) -z ε . It satisfies the equation

ẇ + L(w) = ε 2 K(t)(z) -K(∞)(z ε ) = ε 2 K(t)(z) -K(t)(z ε ) -K(t)(z ε ) -K(∞)(z ε )
and

w(0) = -z ε . Denote K(τ )(z ε ) -K(∞)(z ε ) =∶ -η(τ ).
w(τ ) = v(τ ) + y(τ ), v(τ ) = -e -τ L z ε ,
and re-write the equation on w as an equation on y:

ẏ + Ly = ε 2 K(τ ) v(τ ) + y(τ ) + ε 2 η(τ ), y(0) = 0. Or (5.11) y(t) = ε 2 t 0 e -(t-s)L K(s) v(s) + y(s) + η(s) ds. Let Y r be the space of continuous curves y ∶ R + → C r (R d ),
B(y)(t) = t 0 e -(t-s)L K(s)(y(s)) ds.
Then the equation for y may be written as (5.12)

y(t) = ε 2 B(y)(t) + B(v)(t) + t 0 e -(t-s)L η(s)ds . If |ỹ| Y r = 1, then |B(ỹ(t))| r+1 ≤ t 0 |e -(t-s)L K(s)(ỹ(s))| r+1 ds ≤ C ′ r t 0 e -2(t-s) e -s ds < C ′ r e -t . B ∶ Y r → Y
r+1 is a bounded linear operator if r > d, and accordingly the operator (id -ε 2 B) is a linear isomorphism of Y r if r > d and ε is sufficiently small. It easy to see that B(v) and ∫ t 0 e -(t-s)L η(s)ds both belong to all spaces Y r . Then in view of (5.12), |y| Y r+1 ≤ Cε 2 . Since the operator B is 1-smoothing, then by induction we get that y belongs to all spaces Y r . We have proved that Lemma 5.6. The solution z(τ ), constructed in Lemma 5.5, may be written as

z(τ ) = (id -e -τ L )z ε + y(τ ), where |y(t)| r ≤ C r ε 2 e -t ∀ t ≥ 0, ∀ r .
Here z ε is defined in (5.9) and satisfies (5.10).

5.3.

Energy spectra of quasisolutions and kinetic equation. In this section we prove our main result. Namely, we show that the energy spectrum (4.2) of the quasisolution n s,L (τ ) = E|A s (τ )| 2 of eq. (2.1) with large L is ε 3 -close to the solution z(τ ) of the WKE (5.1), constructed in Lemmas 5.5, 5.6. By (4.12), it suffices to prove this for n s,L replaced by

n ≤2 s . Let us denote w s (τ ) = n ≤2 s (τ ) -z s (τ ); then w s (0) = 0. Recall that ε * is defined in Lemma 5.5. Lemma 5.7. If r > d + 1 and ε ≤ C -1
1r ≤ ε * for an appropriate constant C 1r , then for any τ 0 ≥ 0 and 0 < τ ≤ 1/2, (5.13)

|w(τ 0 + τ )| r ≤ (1 -τ /2)|w(τ 0 )| r + C 2r τ ε 2 (τ κ * + ε 2 ),
where κ * = 1/(1 + 2r * ).

Proof. Since by (5.1)

z(τ 0 + τ ) = e -τ L z(τ 0 ) + 2 τ 0 e -tL b 2 dt + ε 2 τ 0 +τ τ 0 e -(τ 0 +τ -t)L K(t)(z(t))dt,
then in view of (4.22) and (4.19)

(5.14) w(τ + τ 0 ) = e -τ L w(τ 0 ) + ε 2 ∆ + R,
where R is as in (4.22) and

∆ = τ 0 +τ τ 0 e -(τ 0 +τ -t)L K(τ 0 )(n ≤2 (τ 0 )) -K(t)(z(t)) dt.
Note that in view of Lemma 5.5 and estimates (4.9), (5.15) |n ≤2 (τ )| r , |z(τ )| r ≤ C r for all τ and all r, with suitable constants C r . Let us re-write ∆ as follows:

∆ = τ 0 +τ τ 0 e -(τ 0 +τ -t)L K(τ 0 )(n ≤2 (τ 0 )) -K(τ 0 )(z(τ 0 )) dt + τ 0 +τ τ 0 e -(τ 0 +τ -t)L K(τ 0 )(z(τ 0 )) -K(t)(z(τ 0 )) dt + τ 0 +τ τ 0 e -(τ 0 +τ -t)L K(t)(z(τ 0 )) -K(t)(z(t)) dt =∶ ∆ 1 + ∆ 2 + ∆ 3 .
By (5.4) and (5.15), |∆ (5.15). Now let us consider ∆ 2 . By Lemma 5.4, |K(τ 0 )(z(τ 0 )) -

1 | r ≤ C r τ |w(τ 0 )| r . Similar, |∆ 3 | r ≤ C r τ sup τ 0 ≤t≤τ 0 +τ |z(t) -z(τ 0 )| r ≤ C r τ 2 since |z(t) -z(τ 0 )| r ≤ ∫ t τ 0 | -Lz(l) + ε 2 K(l)(z(l)) + 2b 2 | r dl and |z(t)| r+r * ≤ C ′ r by
K(t)(z(τ 0 ))| r ≤ C r (t -τ 0 ) κ * . So ∆ 2 ≤ C r ∫ τ 0 t κ * dt = C ′ r τ 1+κ * . Since L ≥ 21 and τ ≤ 1/2, then |e -τ L w(τ 0 )| r ≤ (1 -τ )|w(τ 0 )| r . Now (5.14), (4.23
) and the bounds on ∆ j imply that

|w(τ 0 + τ )| r ≤ (1 -τ )|w(τ 0 )| r + C r ε 2 τ |w(τ 0 )| r + τ + τ κ * + (τ + ε 2 ) , and (5.13) follows if C -1 1r ≪ 1.
For any 0 < τ ≤ 1/2, any N and for k = 0, . . . , N let us set w k = |w(kτ )| r . Let the function k → w k attains its maximum at a point k which we write as k ∶= k 0 + 1. If k 0 + 1 = 0, then w k ≡ 0. Otherwise in view of (5.13) we have

w k 0 ≤ w k 0 +1 ≤ (1 -τ /2)w k 0 + C 2r τ ε 2 (τ κ * + ε 2 ). So w k 0 ≤ 2C 2r ε 2 (τ κ * + ε 2 ) and max 0≤k≤N |w(kτ )| r = w k 0 +1 ≤ 3C 2r ε 2 (τ κ * + ε 2 )
since τ ≤ 1/2. Applying again (5.13) with τ 0 = kτ and τ replaced by any τ ∈ (0, τ ), and using that in the formula above N is any, we get that

|w(t)| r ≤ 4C 2r ε 2 (τ κ * + ε 2 ), for any t ≥ 0. Sending τ → 0 (and estimating norms | ⋅ | r with r < d + 2 via | ⋅ | d+2
) and then using (4.12) we finally get Theorem 5.8. For any r there exist positive constants

C 1r , C 2r , C 3r such that if ε ≤ C -1 1r , then (5.16) sup τ ≥0 |n ≤2 (τ ) -z(τ )| r ≤ C 2r ε 4 and if L ≥ ε -2 , then (5.17) sup τ
≥0 |n ⋅,L (τ ) -z(τ )| r ≤ C 3r ε 3 .
(5.17) together with Lemma 5.6 give a control over the long time behaviour of the spectra of quasisolutions of (2.1) in terms of the stationary solution z ε of the limiting kinetic equation (see (5.9)):

|n ⋅,L (τ ) -z ε | r ≤ C r (e -τ + ε 3 ), ∀ τ ≥ 0 .
By Proposition 2.1 with d ≥ 3 this result and (5.17) extend to the spectra of quasisolutions of (1.20), defined in (1.27), as expressed in Theorem 5.9. For any r there exist positive constants

C 4r , C 5r such that if ε ≤ C -1 4r and L ≥ ε -2 , then sup τ ≥0 |n ⋅,L (τ ) -z(τ )| r ≤ C 4r ε 3 , (5.18) |n ⋅,L (τ ) -z ε | r ≤ C 5r (e -τ + ε 3 ), ∀ τ ≥ 0 . (5.19)
Relation (5.16) extends to the energy spectra of quasisolutions of (1.20) analogously.

Appendix A. Proof of Lemma 3.6 In this appendix we suppose that the dimension d satisfies d ≥ 2.

A.1. Idea of the proof and general setting. In Lemma 3.6 (up to an obvious scaling) we have to estimate the number of integer points on a quadric inside a large box. The idea is to embed the integral points of the box in an affine space over a large finite field and then apply powerful algebraic geometry techniques to estimate the needed number (note that this identification of bounded integers with elements of a finite field is ubiquitous in coding theory and combinatorics). It is possible mainly due to the fact that this techniques permits to count points defined over a finite field using some geometric information (essentially the dimension, the degree and irredundant decomposition) on the corresponding algebraic set over the algebraic closure of our finite field. We begin with recalling some basic definitions and results concerning such algebraic sets (see, for example, the first chapter of the book [START_REF] Shafarevitch | Basic algebraic geometry[END_REF]).

Affine algebraic sets. Let us fix an algebraically closed field K. Let A m = K m be the m-dimensional affine space over K, and let F 1 , . . . , F s ∈ K[T 1 , . . . , T m ] be non-zero polynomials. Then an affine algebraic set (AAS) X is just the set of common zeros of these polynomials:

(A.1) X = {(a 1 , . . . , a m ) ∈ K m ∶ F 1 (a 1 , . . . , a m ) = . . . = F s (a 1 , . . . , a m ) = 0}.
Irreducibility.

An AAS X is reducible if X = X 1 ∪ X 2 with two non-empty AAS X 1 , X 2 s.t. X 1 ≠ X, X 2 ≠ X.
If it is not the case, X is called irreducible, or an affine algebraic variety (see [START_REF] Shafarevitch | Basic algebraic geometry[END_REF], Section I.3.1).

Theorem A.1. (Irredundant decomposition) Any non-empty AAS X can be presented as

(A.2) X = X 1 ∪ . . . ∪ X l
for irreducible X 1 , . . . , X l such that X i / ⊂ X j for i ≠ j. The decomposition is unique up to order. This decomposition is especially simple for a hypersurface X, i.e. when in (A.1) s = 1. Then F = F 1 (T 1 , . . . T m ) = Π l j=1 Q j for irreducible polynomials Q j which are uniquely defined up to multiplicative constants and permutation since the ring K[T 1 , . . . , T m ] is a unique factorisation domain, see, e.g. Chapter IV of [START_REF] Lang | Algebra[END_REF], and then X j = {Q j = 0}. This uniqueness is true under the condition which we can and will suppose to hold, namely, that the polynomial P does not have multiple divisors, i.e., all Q j , j = 1, . . . , l are distinct. For further references we formulate a corollary of the unique factorisation property (see [START_REF] Shafarevitch | Basic algebraic geometry[END_REF], Section I.3.1) Lemma A.2. i). If X and Y are hypersurfaces, then X = Y if and only if the corresponding polynomials P X and P Y are proportional. Moreover if Y is irreducible and X ⊆ Y , then X = Y.

ii). If deg P X = 2 then there are exactly two possibilities: either X irreducible (in this case it cannot contain a hyperplane), or X = X 1 ∪ X 2 for two affine hyperplanes, defined by affine linear polynomials l 1 and l 2 , and

P X = l 1 l 2 .
Dimension. One can define the dimension r = dim X ∈ {0, 1, . . . , m} as follows: dim X = max{dim X i , i = 1, . . . , l} for (A.2), and for an irreducible

AAS X dim X = max{r ∶ X = X 0 ⊃ X 1 ⊃ . . . ⊃ X r ≠ ∅},
where all X i , i = 0, . . . , r are irreducible AAS and all inclusions are strict.

The codimension of

X is codim X = m -dim X. In particular, if dim X = m then X = A m (indeed, if X ⊂ A m , X ≠ A m the definition implies that dim X < dim A m = m) and if dim X = 0 then X
is a finite set. The codimension of X as in (A.1) is at most s.

From the definition we get immediately (see [START_REF] Shafarevitch | Basic algebraic geometry[END_REF], Section I.6.2)

Lemma A.3. If Y ⊂ X, Y ≠ X and X is irreducible, then dim X > dim Y , codim X < codim Y.
Degree. Let X ⊂ A m be a non-empty AAS, dim X = r. Then its degree deg X is defined as follows:

deg X = max{cardinality of X ∩ L ∶ dim X ∩ L = 0}, where L ⊂ A m is an affine plane with dim L = m -r. Lemma A.4. If X is a hypersurface (i.e. in (A.1) s = 1), then codim X = 1 and deg X = deg F 1 .
The famous Bezout theorem in its the most elementary setting over the field C states that deg

X ≤ Π s i=1 deg F i .
Finite fields' Bezout theorem. From now on the field K is the algebraic closure Fp of a finite field F p , where p is a large prime number (see [START_REF] Lang | Algebra[END_REF], Section V.5).

We will use a version of Bezout's theorem over finite fields which can be deduced from its general form, e.g. [START_REF] Fulton | Intersection Theory[END_REF], and is also explicitly stated and proved in [START_REF] Lachaud | On the number of points of algebraic sets over finite fields[END_REF]Corollary 2.2].

Theorem A.5. Let K = Fp and the AAS X in (A.1) is such that F j ∈ F p [T 1 , . . . , T m ], deg F j = d j , j = 1, . . . , s, and dim X = r. Then |X ∩ F m p | ≤ p r Π s i=1 d i . A.3. Preliminary result. Let q 1 , . . . , q s , s ≥ 1, be polynomials of degree at most two in m ≥ s variables, q i ∈ Z[X 1 , . . . , X m ], with q i (0) = 0, i = 1, ..., s. Consider the geometric quadrics Q i = {x ∈ R m ∶ q i (x) = 0}
and their intersection

Q = ∩ s i=1 Q i . The latter is not empty since 0 ∈ Q. Let B m M ⊂ R m be an open cube {|x| ∞ < M } with some M ≥ 1. Consider the set S m (M, Q) = Q ∩ Z m ∩ B m M .
Let p be a prime and q (p) i ∈ F p [X 1 , . . . , X m ] denote the polynomials q i mod p over the finite field F p . Consider the sets

Q (p) i = {x ∈ K m ∶ q (p) i (x) = 0}
and their intersection Q

(p) = ∩ s i=1 Q (p) i
(recall that now K = Fp is the algebraic closure of F p ). We will be interested mainly in the cardinality of

Q (p) (F p ) ∶= Q (p) ∩ F
m p as a tool to estimate |S m (M, Q)|. Proposition A.6. Let M ≥ 1 and suppose that a prime p > 2M satisfies p = 2M (1 + r(M )), where r(M ) > 0. Suppose also that deg q i ≤ 2 for each i and that the AAS Q (p) is of dimension ms (and of codimension s, that is, the s quadrics Q (p) i intersect properly):

(A.3) dim Q (p) = m -s. Then (A.4) |S m (M, Q)| ≤ 2 m 1 + r(M ) m-s M m-s .
By Bertrand's postulate, for any M ≥ 1 there is a p satisfying 2M < p < 4M , and when applying Proposition A.6 we will always chose ′ ≠ x, then for some i ∈ {1, ..., m} we have x

′ i mod p = x i mod p, but x ′ i ≠ x i . Consequently, |x i -x ′ i | ≥ p > 2M which contradicts the condition x i , x ′ i ∈ B m M . Apllying then Theorem A.5 to X = Q (p) we get the conclusion since |S m (M, Q)| ≤ |Q (p) (F p )| ≤ 2 s p m-s = 2 m 1 + r(M ) m-s M m-s .
A.4. Main estimate for N = 3 and 4. Now we pass to the proof of Lemma 3.6 and denote

| QL (z 1 , v) ∩ B (N -2)d R | = s(R, Q, L). Consider the set S ′ R, Q, L = Q ∩ Z (N -2)d ∩ B (N -2)d RL , Q = Q(Lz 1 , Lv),
and denote by s

′ (R, Q, L) its cardinality. Then s ′ (R, Q, L) = s(R, Q, L) since the map (z 2 , . . . , z N -1 ) ↦ (Lz 2 , . . . , Lz N -1 ) is a bijection between the sets QL (z 1 , v) ∩ B (N -2)d R and S ′ (R, Q, L).
(p) j (a, b) ∶= {(z 2 , . . . , z N -2 ) ∈ K (N -2)d ∶ q (p) j (z 2 , . . . , z N -2 ; a, b) = 0},
where q (p) j (z 2 , . . . , z N -2 ; a, b) are the residues modulo p of the polynomials q j (z 2 , . . . , z N -2 ; a, b), defined by (3.15). We set

Q(p) = ∩ 1<j<N Q(p) j
for the intersection of the algebraic sets.

Lemma A.7. Let N ∈ {3, 4}, (z 1 , v) ∈ Q 0 1 L ∩ A 2 (see (3.16)) and let p be a prime satisfying p > max(|Lz 1 | ∞ , |Lv| ∞ ). Then (A.6) dim Q(p) (Lz 1 , Lv) = (N -2)(d -1) .
The assumption p > max(|Lz 1 | ∞ , |Lv| ∞ ) ensures that Lz 1 and Lv are different from zero in K d . In particular, for

(z 1 , v) ∈ B 2d (N -1)R this assump- tions is satisfied if p > 2M with M = N RL/2.
Proof of Lemma A.7. Let N = 3. Then N -2 = 1 and Q(p) is given by the unique equation q (p) 2 (z 2 ; Lz 1 , Lv) = 0, for a fixed (z 1 , v). By Lemma A.9 the equation is non-trivial, so the conclusion follows from Lemma A.4. = 4. The codimension of the intersection of two quadrics is at most two. We have to show that it is two (and not one). The result will follow from the next three lemmas.

Lemma A.8. Let Q 1 = {q 1 = 0}, Q 2 = {q 2 = 0} be two linearly independent quadrics over K. Then the codimension of Q 1 ∩ Q 2 is one if and only if q1 and q2 have a mutual affine linear factor l(x).

Proof. Let the codimension of the intersection be one. In this case if one of

Q 1 , Q 2 is irreducible, then Q 1 = Q 2 by Lemma A.3 with Y = Q 1 ∩ Q 2 .
However this is impossible by Lemma A.2. i) since q1 and q2 are independent. Therefore by Lemma A.2. ii)

Q 1 = H 1 ∪ H 2 and Q 2 = H ′ 1 ∪ H ′ 2 , with hyperplanes H 1 , . . . , H ′ 2 . If all H i ∩ H ′ j are of codimension two then codim Q 1 ∩ Q 2 = codim ∪(H i ∩ H ′ j ) = min(codim H i ∩ H ′ j ) = 2.
Therefore, at least one of H i ∩ H ′ j is of codimension one and then we have ker(l(x))

= H i ∩ H ′ j ⊂ Q 1 ∩ Q 2 for
an affine linear l(x). Hence l(x) divides both q1 and q2 by Lemma A.2. ii).

The inverse statement is obvious.

Lemma A.9. For any N > 2, if the matrix α is irreducible and

(z 1 , v) ∈ Q 0 1 L ∩A 2 is such that Lz 1 , Lv ≠ 0 in K d , then the polynomials q (p)
j (⋅, Lz 1 , Lv), 1 < j < N are linearly independent over K. In particular, each q (p) j is a non-zero polynomial.

Proof. Consider a linear combination ∑ 1<j<N c j q (p) j . By the homogeneity in (z 2 , . . . , z N -1 ) it vanishes identically if and only if

1<j<N c j z j ⋅ (α j1 (Lz 1 ) + α jN (Lv)) ≡ 0 , 1<i,j<N c j (α ji -α jN α 1i )z j ⋅ z i ≡ 0 . (A.7)
Arguing by induction and using that the matrix α is irreducible we construct a partition E 0 , . . . , E M , M ≥ 1, of the set {1, . . . , N } such that E 0 = {1, N } and for n ≥ 1,

E n = {j ∶ α jl = 0 ∀ l ∈ E n ′ , n ′ ≤ n-2, and ∃ l ′ ∈ E n-1 such that α jl ′ ≠ 0}. Since (z 1 , v) ∈ Q 0 1 L and Lz 1 , Lv ≠ 0 in K d ,
then the term in brackets in the first line of (A.7) is not identically zero for each j ∈ E 1 , so c j = 0 for every j ∈ E 1 . Using this in the second line of (A.7) we get:

M n=2 M m=n-1 j∈E n i∈E m c j α ji z j ⋅ z i ≡ 0 .
This relation holds if and only if (c j -c i )α ji = 0 for all j ∈ E n , 2 ≤ n ≤ M , and i ∈ E m , n -1 ≤ m ≤ M . We know that c j = 0 if j ∈ E 1 . Starting from n = 2 and arguing by induction in n we find that if c i = 0 for all i ∈ E n-1 , then c j = 0 for all j ∈ E n . Indeed, for any j ∈ E n there exists at least one i ∈ E n-1 such that α ji ≠ 0 by the definition of E i , so relation (c j -c i )α ji = 0 implies that c j = 0 if j ∈ E n . That is, c j ≡ 0.

Lemma A.10. For any N > 2, if the matrix α is irreducible and

(z 1 , v) ∈ Q 0 1 L ∩A 2 is such that Lz 1 , Lv ≠ 0 in K d , then the polynomials q (p) j (⋅, Lz 1 , Lv), 1 < j < N , are irreducible.
Proof. Each polynomial q (p) j has degree one or two. If its degree is one the assertion is obvious. Now let the degree be two. Note that in view of (3.15) q (p) j can be written as the scalar products q (p) j = z j ⋅ l j (z 2 , . . . , z N -1 ; z 1 , v) mod p, where l j are surjective affine functions

l j ∶ K d(N -2) ⟶ K d . But such scalar product cannot vanish for d ≥ 2 > 1 on a hyperplane H ⊂ K d(N -2)
which by Lemma A.2. ii) would be the case for a reducible quadric. Indeed, only two cases can occur: a) the coefficient α of z j in l j is non-zero, or b) it is zero but then the coefficient β of some other z i is non-zero.

In case a) take the 2-dimensional plane P (x 1 , x 2 ) in the whole space, generated by two orthogonal vectors from the z j -space, where the first basis vector is parallel to α 1j z 1 + α N j v ≠ 0 (this vector is non-zero since

(z 1 , v) ∈ Q 0 1 L
and Lz 1 , Lv ≠ 0 in K d , and for the case a) we have α 1j , α N j ≠ 0). Then the restriction of q (p) j = 0 on P is α(x

2 1 + x 2
2 ) + c 1 x 1 = 0 with c 1 ≠ 0, which is isomorphic to x 2 1 + x 2 2 = C ≠ 0. This plane quadric in P (x 1 , x 2 ) cannot contain P (x 1 , x 2 ) ∩ H (a line or the whole P (x 1 , x 2 )). Indeed, otherwise, supposing by symmetry that the quadric contains a line x 1 = ax 2 + b, we would have that the polynomial

a 2 x 2 2 + 2abx 2 + b 2 + x 2 2 -C = (a 2 + 1)x 2 2 + 2abx 2 + b 2 -C
vanishes identically. This implies ab = 0, and if a = 0 then the term

(a 2 + 1)x 2 2 = x 2 2 ≠ 0, while for b = 0 the term b 2 -C = -C ≠ 0.
Similarly, in case b) we take the 4-dimensional vector subspace P ′ generated by the two first basis vectors in the z j space and the two first basis vectors in the z i space. The restriction of q (p) j = 0 on P ′ is then

β(x 1 y 1 + x 2 y 2 ) + c 1 x 1 + c 2 x 2 = 0, isomorphic
(a 1 x 2 + b 1 y 1 + b 2 y 2 -c)y 1 + x 2 y 2 -C = a 1 x 2 y 1 + b 1 y 2 1 + b 2 y 1 y 2 -cy 1 + x 2 y 2 -C
vanishes identically, which is clearly wrong.

of the proof of Lemma A.7. Since each q (p) j is a non-zero polynomial of degree one or two, then to prove Lemma A.7 we have to consider three cases. In the first case both polynomials q (p) 2 and q (p) 3 are linear. Then the codimension of the intersection Q(p) is two since they are linearly independent. In the second case both q (p)

2 and q (p)

3 are quadratic. Then, according to Lemma A.8, the codimension still is two since the polynomials are irreducible by Lemma A.10. Finally in the last case, when one polynomial is linear and another one is quadratic, the assertion is clear since then the AAS in question is an intersection of a quadratic irreducible surface with a hyperplane. Thus its codimension is two by Lemma A.2. ii).

Remark A.11. The proof of Lemma A.7 follows from three lemmas. Two of them are valid for any N > 2, but Lemma A.8 holds only for N = 4 (and tautologically holds for smaller N ). Still the bi-linear (or linear) nature of the polynomials q (p) j and direct analysis of the AAS Q(p) , jointly with the two lemmas, valid for any N > 2, allow to prove by hand Lemma A.7 for "not too high" values of N , and thus, to prove for those N 's Theorem 3.2. Unfortunately, for the moment we cannot prove the theorem for all N > 2; cf. Conjecture 3.8. The proof of the proposition is somewhat cumbersome since we have to consider a number of different terms and different cases. During the proof we will often skip the upper index (0), so by writing a and a s we will mean a (0) and a

s . We will also skip the dependence on τ 0 by writing c (i) s (τ ; τ 0 ) and ∆a

(i) s (τ ; τ 0 ) as c (i) s (τ ) and ∆a (i) s (τ ). Besides, for a complex function (w s 1 ,...,s k , s j ∈ Z d L ) we denote ⨊ s 1 ,...,s k ∈Z d L w s 1 ,...,s k = L -kd s 1 ,...,s k ∈Z d L w s 1 ,...,s k ,
and we introduce the symmetrisation

Y sym s (u, v, w; t) = L -d 3 1,2,3 δ ′12 3s δ(ω 12
3s ) (u 1 v 2 w3 + v 1 w 2 ū3 + w 1 u 2 v3 ) .
We recall that Q s,L is given by formula (4.30) and first consider the term E∆a

s (τ )ā s (τ 0 + τ ). Inserting the identity a (1) (τ 0 + l) = c (1) (l) + ∆a (1) (l)

into formula (4.27) for ∆a

N s ∶= i E ās (τ 0 + τ ) τ 0 e -γ s (τ -l) 3Y sym s (a(τ 0 + l), a(τ 0 + l), ∆a (1)
(l)) dl and

Ñs ∶= i E ās (τ 0 + τ ) τ 0 e -γ s (τ -l) 3Y sym s (a(τ 0 + l), a(τ 0 + l), c (1)
(l)) dl .

Thus, (B.2)

Q s,L = L 2 E|∆a (1) s (τ )| 2 + 2RN s + 2RE∆a (1)
s (τ )c ( 1)

s (τ ) + 2R Ñs , s ∈ R d .
We will analyse the four terms above term by term.

s (τ )| 2 = E τ 0 +τ τ 0 dl τ 0 +τ τ 0 dl ′ e -γ s (2τ 0 +2τ -l-l ′ ) Y s (a(l)) Y s (a(l ′ )).
Writing the functions Y s explicitly and applying the Wick theorem, in view of (2.13) we find E|∆a

s (τ )| 2 = 2L -2d 1,2 δ ′12 3s δ(ω (1)
)

τ 0 +τ τ 0 dl τ 0 +τ τ 0 dl ′ e -γ s (2τ 0 +2τ -l-l ′ ) Ea 1 (l)ā 1 (l ′ ) Ea 2 (l)ā 2 (l ′ ) Eā 3 (l)a 3 (l ′ ),
and note that

τ 0 +τ τ 0 dl τ 0 +τ τ 0 dl ′ e -γ s (2τ 0 +2τ -l-l ′ ) ≤ τ 2 .
On account of (2.13), we can bound

E|∆a (1) s (τ )| 2 ≤ 2τ 2 ⨊ 1,2 δ ′12 3s δ(ω 12
= N 1 s + 2N 2 s , s ∈ R d , where N 1 s = i E ās (τ 0 + τ ) τ 0 e -γ s (τ -l) Y s (a(τ 0 + l), a(τ 0 + l), ∆a (1) (l)) dl , N 2 s = i E ās (τ 0 + τ ) τ 0 e
-γ s (τ -l) Y s (∆a (1) (l), a(τ 0 + l), a(τ 0 + l)) dl . × E a 1 (τ 0 + l)a 2 (τ 0 + l)∆ā

(1)

3 (l)ā s (τ 0 + τ ) = L -2d 1,2 1 ′ ,2 ′ δ ′12 3s δ ′1 ′ 2 ′ 3 ′ 3 δ(ω 12
3s )δ(ω

1 ′ 2 ′ 3 ′ 3 ) τ 0 dl l 0 dl ′ e -γ s (τ -l) e -γ 3 (l-l ′ ) (B.5) × E a 1 (τ 0 + l)a 2 (τ 0 + l)ā 1 ′ (τ 0 + l ′ )ā 2 ′ (τ 0 + l ′ )a 3 ′ (τ 0 + l ′ )ā s (τ 0 + τ ) .
By the Wick theorem, we need to take the summation only over

s 1 ′ , s 2 ′ , s 3 ′ satisfying s 1 ′ = s 1 , s 2 ′ = s 2 , s 3 ′ = s or s 1 ′ = s 2 , s 2 ′ = s 1 , s 3 ′ = s. Since in both cases we get δ ′1 ′ 2 ′ 3 ′ 3 = δ ′12 3s and ω 1 ′ 2 ′ 3 ′ 3 = ω 12 3s
, we find

N 1 s = 2⨊ 1,2 δ ′12 3s δ(ω 12 3s ) τ 0 dl l 0 dl ′ e -γ s (τ -l)-γ 3 (l-l ′ ) × Ea 1 (τ 0 + l)ā 1 (τ 0 + l ′ ) Ea 2 (τ 0 + l)ā 2 (τ 0 + l ′ ) Ea s (τ 0 + l ′ )ā s (τ 0 + τ ).
Arguing as in Section B.2 we find

(B.6) |N 1 s | ≤ C # (s)L -2 τ 2 .
Term N 2 s . By literally repeating the argument we have applied to N 1 s we find that

N 2 s = i L -d 1,2 δ ′12 3s δ(ω 12 3s ) τ 0 dl e -γ s (τ -l) × E ∆a (1) 1 (l)a 2 (τ 0 + l)ā 3 (τ 0 + l)ā s (τ 0 + τ ) = -L -2d 1,2 1 ′ ,2 ′ δ ′12 3s δ ′1 ′ 2 ′ 3 ′ 1 δ(ω 12 3s )δ(ω 1 ′ 2 ′ 3 ′ 1 ) τ 0 dl l 0 dl ′ e -γ s (τ -l) e -γ 1 (l-l ′ ) × E a 1 ′ (τ 0 + l ′ )a 2 ′ (τ 0 + l ′ )ā 3 ′ (τ 0 + l ′ )a 2 (τ 0 + l)ā 3 (τ 0 + l)ā s (τ 0 + τ ) .
By the Wick theorem we should take summation either under the condition

s 1 ′ = s 3 , s 2 ′ = s, s 3 ′ = s 2 or s 1 ′ = s, s 2 ′ = s 3 , s 3 ′ = s 2 . Since in both cases δ ′1 ′ 2 ′ 3 ′ 1 = δ ′12 3s and ω 1 ′ 2 ′ 3 ′ 1 = -ω 12 3s , then N 2 s = -2⨊ 1,2 δ ′12 3s δ(ω 12 3s ) τ 0 dl l 0 dl ′ e -γ s (τ -l) e -γ 1 (l-l ′ ) (B.7) Ea 2 (τ 0 + l)ā 2 (τ 0 + l ′ ) Ea 3 (τ 0 + l ′ )ā 3 (τ 0 + l) Ea s (τ 0 + l ′ )ā s (τ 0 + τ ) .
Again we get

(B.8) |N 2 s | ≤ C # (s)L -2 τ 2 .
B.4. The third term of Q s,L in (B.2). We have

E∆a (1) s c(1) s (τ ) = E τ 0 +τ τ 0 e -γ s (τ 0 +τ -l) Y s (a(l)) dl τ 0 0 e -γ s (τ 0 +τ -l ′ ) Y s (a(l ′ )) dl ′ .
This expression coincides with (B.3) in which the integral ∫

τ 0 +τ τ 0 dl ′ is re- placed by ∫ τ 0 0 dl ′ . Then, E∆a (1)
s (τ )c ( 1)

s (τ ) = 2⨊ 1,2 δ ′12 3s δ(ω 12 3s
)

τ 0 +τ τ 0 dl τ 0 0 dl ′ e -γ s (2τ 0 +2τ -l-l ′ ) Ea 1 (l)ā 1 (l ′ ) Ea 2 (l)ā 2 (l ′ ) Eā 3 (l)a 3 (l ′ ),
Expressing the correlations Ea j (l)ā j (l ′ ) through (2.13), we get

E∆a (1) s (τ )c (1)
s (τ ) =2⨊ 1,2 δ ′12 3s δ(ω 12
s (τ ) = 2⨊ 1,2 δ ′12 3s δ(ω (1) s (τ )c (1)
= Ñ 1 s + 2 Ñ 2 s , s ∈ R d , where Ñ 1 s = i E ās (τ 0 + τ ) τ 0 e -γ s (τ -l) Y s (a(τ 0 + l), a(τ 0 + l), c (1)
(l)) dl ,

Ñ 2 s = i E ās (τ 0 + τ ) τ 0 e -γ s (τ -l) Y s (c (1) (l), a(τ 0 + l), a(τ 0 + l)) dl . Term Ñ 1 s .
Writing explicitly the function Y s and then c(1) we get

Ñ 1 s = L -2d 1,2 1 ′ ,2 ′ δ ′12 3s δ ′1 ′ 2 ′ 3 ′ 3 δ(ω 12
3s )δ(ω

1 ′ 2 ′ 3 ′ 3 ) τ 0 dl 0 -τ 0 dl ′ e -γ s (τ -l) e -γ 3 (l-l ′ ) × E a 1 (τ 0 + l)a 2 (τ 0 + l)ā 1 ′ (τ 0 + l ′ )ā 2 ′ (τ 0 + l ′ )a 3 ′ (τ 0 + l ′ )ā s (τ 0 + τ ) .
this is the same expression as (B.5), with the integration over dl ′ ranging from -τ 0 to 0 instead of from 0 to l. Thus, by the Wick theorem, we obtain

Ñ 1 s = 2⨊ 1,2 δ ′12 3s δ(ω 12 3s ) τ 0 dl 0 -τ 0 dl ′ e -γ s (τ -l)-γ 3 (l-l ′ ) × Ea 1 (τ 0 + l)ā 1 (τ 0 + l ′ ) Ea 2 (τ 0 + l)ā 2 (τ 0 + l ′ ) Ea s (τ 0 + l ′ )ā s (τ 0 + τ ).
Following the line of Section B.4 we express the correlations through (2.13) and get

Ñ 1 s =2⨊ 1,2 δ ′12 3s δ(ω 12 3s )B 12s τ 0 dl e -2γ s (τ -l)-γ 123s l τ 0 0 dl ′ e -γ 123s τ 0 e γ 3 l ′ e γ s l ′ -e -γ s l ′ j=1,2 e γ j l ′ -e -γ j l ′ =2⨊ 1,2 δ ′12 3s δ(ω 12
l 0 replaced by ∫ 0 -τ 0 : Ñ 2 s = -2⨊ 1,2 δ ′12 3s δ(ω 12 3s ) τ 0 dl 0 -τ 0 dl ′ e -γ s (τ -l) e -γ 1 (l-l ′ ) Ea 2 (τ 0 + l)ā 2 (τ 0 + l ′ ) Ea 3 (τ 0 + l ′ )ā 3 (τ 0 + l) Ea s (τ 0 + l ′ )ā s (τ 0 + τ ).
Again we get

(B.14) Ñ 2 s = -2⨊ 1,2 δ ′12 3s δ(ω 12
Q s,L -4L 2 T s ⨊ 1,2 δ ′12 3s δ(ω 12 3s ) B 123 T s + B 12s T 3 -2B 23s T 1 ≤ C # (s)τ 2 .
(B.15)

Note that the terms Z j defined in (4.14) can be written as

(B.16) Z j = T j ∏ k≠j (1 -e -2γ k τ 0 ) .
The relations (2.13)-(2.14) imply that for any permutation

(k 1 , k 2 , k 3 , k 4 ) of (1, 2, 3, s) we have B k 1 k 2 k 3 = n (0) k 1 n (0) k 2 n (0) k 3 / ∏ m=k 1 ,k 2 ,k 3 (1 -e -2γ m τ 0 ), where n (0) k i = n (0)
k i ,L (τ 0 ). Together with (B.16) this implies (B.17)

B k 1 k 2 k 3 T k 4 = Z k 4 n (0) k 1 n (0) k 2 n (0) k 3 .
By symmetry, the term 2B 23s T ′ (γ j )∂ s j γ j , where ∂ s j γ j (as well as higher order derivatives of γ j ) have at most polynomial growth at infinity. Then, using the definition (4.14) of Z j we find 1 -e -2γl

∂ µ ⃗ s Z j (τ 0 , ⃗ s) =
(1 -e -2γτ 0 ) m+1 ≤ C k,m,p , 50ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUT uniformly in τ 0 ≤ γ -1 . We have seen that the product in (B.18) is bounded uniformly in ⃗ s, l and τ 0 , so the integral over l is also bounded uniformly in ⃗ s and τ 0 .

Appendix C. Proof of Proposition 2.1

The proof uses the theory of Feynman diagrams, presented in Section 2. For N = 0 the assertion is trivial. For N ≥ 1 in Proposition 8.7 of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] it is proven that E a Here Ñ = Ñ (F) < N , quadratic forms ωF k (z) are defined by relations (2.17), where N is replaced by Ñ and the matrix (α We first show that the term J 2 s (F) is bounded by the r.h.s. of (2.3). To this end we write Z + (F) = ∪ K Z K , where the union is taken over all subsets 13 These diagrams are defined similarly to the Feynman diagrams from Section 2.3.1, but now we allow to couple leaves not only from different blocks but also from the same block. approximation, jointly with a modification of the Heath-Brown result from [START_REF] Heath-Brown | A new form of the circle method, and its application to quadratic forms[END_REF], given in Theorem 1.4 of [START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF], implies the following version of Theorem s,L ). We see that the correlations take the form (D.1), so Theorems D.1 and D.2 apply to study them.

The rest of the proof of Theorem 5.8 literally repeats that for the case d ≥ 3, except the appearance of the (ln L)

-k/2 factors, coming from the new definition of ρ. Now the estimates, using Theorem 3.1, should be relaxed since the estimate provided by Theorem D.1 is slightly weaker than that of Theorem 3.1. In particular, in the r.h.s. of (4.8), (4.11) and (4.12) the factor L -1/2 should be replaced by (ln L) (see Theorem A).

Theorem 5.9, as Theorem 5.8, remains unchanged, except the lower bound for L which is modified as above. Indeed, the theorem follows from Theorem 5.

  d L = R d /(LZ d ) be the d-dimensional torus, d ≥ 2,of period L ≥ 2. We denote by ∥u∥ the normalized L 2 -norm of a complex function u on T

2 i.e. β s = β 1 s + iβ 2 s

 12 , where {β j s , s ∈ Z d L , j = 1, 2} are standard independent real Wiener processes.

2 s 3 s 2 + |s 2 | 2 -|s 3 | 2 -|s| 2 =3s a 1 a 2 ā3 e itω 12 3s

 23222212 ∶= |s 1 | -2(s 1 -s) ⋅ (s 2 -s),where the last equality holds ifδ ′12 3s = 1 since then s 3 = s 1 + s 2 -s. Then equation (1.13) takes the form ȧs + γ s a s = iρY s (a, ν -1 τ ) + b(s) βs , s ∈ Z d L , Y s (a, t) = L -|a s | 2 a s ,(1.18)

12 3s )a 1 a 2 Theorem 1 . 1 .

 12211 res s (a) + b(s) βs , s ∈ Z ā3 -|a s | 2 a s . (1.20) Here δ(ω 12 3s ) = 1 if ω 12 3s = 0 and δ(ω 12 3s ) = 0 otherwise. The following result is proven in [21, 17]. If d ≥ 1 and r * ≥ d/2 + 1, then equations (1.18) and (1.20) are well posed. Under the limit (1.12), on time intervals of order 1, i) a solution a ν (τ ) of (1.18) converges in distribution to a solution a 0 (τ ) of (1.20) with the same initial data at τ = 0; ii) the energy spectrum E|a ν s (τ )| 2 = N s (τ ; ν, L) converges to the energy spec-

( 1 .

 1 37) |f | r = sup z∈R d |f (z)|⟨z⟩ r , where ⟨z⟩ = max(|z|, 1).

1 s ∼ 1 and z 0 s

 10 solves the linear equation (1.38)| ε=0 . It is easy to see that z 0 s equals the component n (0)

  30)). Namely, by iterating the Duhamel formula (1.25) we express a (n) (τ ) in terms of the Gaussian processes a (0) s , and next evoking the Wick formula for moments of a (0) s write the terms n (k) s,L as multiple sums. Then the just mentioned diagram techniques allows to 'integrate' these sums. That is, to write any n (k) s,L as a sum over an intersections of k -1 quadrics in (Z d L )

  in (1.29) correspond to multiple intersections of quadrics, and the Heath-Brown method does not apply to them. Still the diagram technique allows to write these terms in a convenient compact form. Then next in Section 3 and Appendix A we use Theorem B jointly with another powerful result from the number theory -Bezout's theorem for finite fields -to prove 7 Theorem C. For k = 3, 4, |n (k) s,L (τ )| ≤ C # (s).

  ≤2 s,L ∶= n ≤2 s,L (τ + θ) -n ≤2 s,L (τ ) and express them through the processes a (0) m via the Duhamel formula (1.25) and the Wick theorem. Then the increments approximately take the form (1.42), and we use Theorem B to show that they are close to the r.h.s. of the WKE, multiplied by θ.

2. 1 .

 1 Approximate a-equation. We start by considering an approximation of the original equation (1.20) by an equation, where the term L -d |a s | 2 a s is removed:

s 1 s a n 2 s ān 3 s

 13 (τ ) with n ≥ 1 are built by the recursive formula (1.25) with the term a n being dropped. I.e. with the nonlinearity Y res s replaced by the Y s above. Results of [8] together with Theorem 3.2 below (which is an abstract version of Theorem C from the introduction) imply Proposition 2.1. For all m, n ≥ 0, satisfying N ∶= m + n ≤ 4, (2.3) |Ea (m)

  34 )δs 4 a 1 a 2 ā3 , where δ s 4 is the Kronecker symbol. Then analogous of the expression (1.25) for a (m) , m ≥ 1, takes the form a (m)

2. 2 . 1 .

 21 Construction of the sets of diagrams D m and D n . We start with discussing the set D 2 and the sums I s (D) with D ∈ D 2 .

9 ) 0 σ 2 and δ σ 1

 901 into the summand in (2.8) with m 1 = m 2 = 0 and m 3 = 1, we get a sum I s (D; τ ) which we associate with the diagram D from fig. 1(c); further on we will denote this diagram by D c . The non-conjugated vertices c (k) i of the diagram are associated with the variables a (k) ξ i in (2.8), (2.9); the corresponding to them indices are ξ i . The conjugated vertices c(n) j are associated with the variables ā(n) σ j and the corresponding indices are σ j . In particular, the root c (2) 0 is associated with a (2) ξ 0 = a (2) s and the corresponding index is ξ 0 . In the notation c the upper indices k and n which we call the degrees of the vertices c vertices w2 and w 4 are called conjugated (non-conjugated) virtual vertices and the corresponding indices are σ 2 and ξ 4 ; these vertices are associated with the Kronecker symbols δ ξ

  diagrams from fig. 1(a,b) correspond to the summands in (2.8) with m 1 = 1, m 2 = m 3 = 0 and m 1 = m 3 = 0, m 2 = 1; they are constructed by the same rules as the diagram D c . The three diagrams from fig. 1 form the set D 2 . The set of diagrams D 2 , corresponding to ā(2) s (τ ), is obtained by conjugating the vertices in the three diagrams above and re-ordering the elements of each block in such a way that the pair of non-conjugated vertices is followed by the pair of conjugated vertices, i.e. the blocks have the form (2.10). The sets D m and D n with arbitrary m, n ≥ 0 and the diagrams which are their elements, are constructed similarly. Namely, the sets D 0 and D 0 are trivial -they contain one diagram each, made by the root c sets D 1 and D 1 also contain only one diagram each; e.g. the diagram in D 1 consists of the root c (0)

), and join w2 by an edge with the root c (m+1) 0 . 1 ∈ D m 1 ,(m 1 ) 1 and identify its root with c (m 1 ) 1 .

 01111 Next consider the sets D m 1 , D m 2 , D m 3 and do the following: a) Firstly take D m 1 . If m 1 = 0, do nothing. Otherwise choose any diagram D place it below c Do this for each diagram in D m 1 , thus obtaining |D m 1 | diagrams with roots in c (m+1) 0 . b) Then consider the set D m 2 and do the same with the just obtained |D m 1 | diagrams, identifying their roots with the vertex c (m 2 ) 2 , and nextthe set D m 3 , identifying the roots with c(m 3 ) 3 . c) It remains to convert thus obtained |D m 1 | × |D m 2 | × |D m 3 | diagrams to elements of the set D m+1 by re-numerating properly their blocks and accordingly re-numerating the vertices in the blocks as in (2.10). Do this by numerating the blocks from top to the bottom and from left to right, as in the examples above with m = 2. d) Doing the same for all blocks, corresponding to all possible (m+2)(m+ 1)/2 terms (2.11), get the diagrams, forming the set D m+1 .

1

 1 and D2 side by side, where the blocks of D 1 are enumerated from 1 to m, while those of D2 together with the corresponding time variables l j are enumerated from j = m + 1 to m + n. The vertices together with the corresponding indices ξ 2j-1 , ξ 2j , σ 2j-1 , σ 2j are enumerated accordingly, see fig. 2. The diagram D 1 ⊔ D2 has two roots c

2. 3 . 1 .

 31 Definition of Feynman diagrams. To construct the set of Feynman diagrams F(D), corresponding to some diagram D = D 1 ⊔ D2 , we consider all possible partitions of the set of leaves of D to non-intersecting pairs (c (0) i , c(0) j ), such that the paired leaves c

3 w4Figure 3 .

 33 Figure 3. a) A Feynman diagram F obtained from the diagram D in fig. 2(a). b) A cycle obtained from the Feynman diagram F.

12 3s with s 3 = s 1 + s 2 -s reads ω 12 3s = -2z 1 ⋅

 12312121 z 2 (see(1.17)). Then, taking into account that the relation δ ′12 3s = 1 is equivalent to the relations z 1 , z 2 ≠ 0 and s 3 = s 1 + s 2 -s, we find that the sum S L,2 takes the form (3.4). Applying next Theorem 3.1 and changing in (3.6) back to the variables s 1 , s 2 we get(3.11).Example 3.4. Let us calculate the asymptotic as L → ∞ of E|a

  1)R 2d . Since for R ≥ 1 and any a ∈ N ∪ {0}, b ≥ 0 we have ∥w R ∥ a,b ≤ C(a, b, N, d)R b , then in view of Theorem 3.1 and (3.8), (3.18)

s

  is defined via relation (4.24). Below we write c

s

  (τ ; τ 0 ) and ∆a

s

  (τ ; τ 0 ) as c

(

  In view of Lemmas 5.2 and 5.1, |η(τ )| r ≤ C r e -2τ for τ ≥ 0. Next, regarding the difference K(τ )(z(τ )) -K(τ )(z ε ) as an operator, linear in w(τ ) = z(τ ) -z ε and quadratic in (z(τ ), z ε ), we write it as K(τ )(w(τ )). Then by (5.4) and (5.8), |K(τ )w| r+1 ≤ C r |w| r , ∀r > d. Finally, we substitute

(A. 5 )′ 1

 51 r(M ) < 1. Moreover, by the Prime Number Theorem, for large M one can chose r(M ) = o(1). Proof of Proposition A.6. Let Π ∶ S m (M, Q) ⟶ F m p be defined by Π(x 1 , . . . , x m ) = (x 1 mod p, . . . , x m mod p).Then Π is injective and its image is contained in Q(p) ∩ F m p ⊂ F m p .Indeed, the last assertion is clear and the injectivity is established as follows: if (x mod p, . . . , x ′ m mod p) = (x 1 mod p, . . . , x m mod p) but x

1 L

 1 Let us estimate s ′ (R, Q, L) through Proposition A.6 with m = (N -2)d and s = N -2, where N = 3 or 4. To this end it suffices to find M ≥ RL and p > 2M such that assumption (A.3) is fulfilled for any (z 1 , v) ∈ B 2d (N -1)R satisfying (z 1 , v) ∈ Q 0 ∩ A 2 . Lemma A.7 below establishes this for M = N RL/2 and any p > 2M . Then, applying (A.4) with r(M ) < 1 (see (A.5)), we conclude the proof of Lemma 3.6. For a prime p and a, b ∈ F d p let us consider algebraic sets Q(p) j over K = Fp : Q

Appendix B . B. 1 .

 .1 Proof of Proposition 4.3 and Lemma 4.4 We prove Proposition 4.3 in Sections B.1-B.6 and Lemma 4.4 in Section B.7. Beginning of the proof of Proposition 4.3.

s

  (τ )ā s (τ 0 + τ ) = N s + Ñs , where (B.1)

N 1 s

 1 . Writing explicitly the function Y s and then ∆ā

  s (τ -l)-γ 123s l = τ e -2γ s τ if 2γ s = γ 123s e -2γ s τ -e -γ 123s τ γ 123s -2γ s elsewhere . For the integral in the second line, let us denote T j, k ∈ {1, 2, 3, s}. Then, (B.11) 0 ≤ T j ≤ 1/γ 123s . Due to (B.9) and (B.10) we get (B.12) E∆a

|µ| n 1 ( 1 -τ m+p 0 ( 1 - 2 k+1 τ n 0 ( 1 -

 1101201 +n 2 +n 3 +n s =1 P (⃗ s; n 1 , . . . , n s ) m l) sinh(γ m τ 0 ) , (B.18) where P (⃗ s; a 1 , . . . , a n ) denotes a function of ⃗ s, dependent on parameters (a 1 , . . . , a n ), having at most a polynomial growth at infinity. Using the relation sinh(γl) sinh(γτ 0 ) = e -γ(τ 0 -l) -e -γ(l+τ 0 ) 1-e -2γτ 0 we find by induction that d n dγ n sinh(γl) sinh(γτ 0 ) = k+m+p=n c k,m,p I k,m,p (l, τ 0 , γ), where c k,m,p are constants, (B.19) I k,m,p = (τ 0 -l) e -2γτ 0 ) m+1 , and p ≠ 0 only if m ≠ 0. For τ 0 ≥ γ -1 the terms I k,m,p are bounded in absolute values by absolute constants C k,m,p , where we recall that 0 ≤ l ≤ τ 0 and γ ≥ 1. Let nowτ 0 ≤ γ -1 . In this case, since k + m + p = n, |I k,m,p | ≤ 2 (l + τ 0 ) k e -2γτ 0 ) m+1 ≤ e -2γτ 0 ) m+1 .So, in the case m ≤ n -1 we have |I k,m,p | ≤ C k,m,p uniformly in τ 0 ≤ γ -1 . If m = n (so k = p = 0) we use another estimate, following from (B.19): |I k,m,p | ≤ C e -γ(τ 0 -l) -e -γ(l+τ 0 ) (1 -e -2γτ 0 ) m+1 τ m 0 = Cτ m 0 e -γ(τ 0 -l)

F

  is a certain (finite) set of extended Feynman diagrams, 13 c F is a complex number of unit norm and J s (F), J 2 s (F) are sums, similar to(2.19). In Section 8.6.3 of[START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition[END_REF] are established the following bounds for these sums:(C.2) |J 2 s (F)| ≤ C # (s)L ki z i ≠ 0 ∀1 ≤ k ≤ Nwhile the quadratic forms ω F k and the skew-symmertic matrix α F are defined in Section 2.4. Note that possibly the diagram F does not belong to the set F true m,n , so that the matrix α F may have zero columns and lines. On the other hand, (C.3) |J s (F)| ≤ C # (s)L

F

  ij ) -by a certain Ñ × Ñ -matrix ( αF ji ), also satisfying αF ji = -αF ij ∈ {0, ±1} for all i, j. Accordingly the set Z+ (F) ⊂ (Z d L ) Ñ is defined as Z + (F)above, but with N and α F ij replaced by Ñ and αF ij .

KFF( 2 s

 2 ⊂ {1, . . . , N } andZ K (F) = {z ∶ z k = N i=1 α ki z i = 0 ∀k ∈ K and z k ≠ 0, ki z i ≠ 0 ∀k ∉ K}.Then the r.h.s. of (C.2) takes the form Note that on the setZ K (F) we have ω k (z) = 0 for all k ∈ K and ω k (z) = 2z k ⋅ ∑ i∉K α F ki z i for k ∉ K.Thus, the sum over z in (C.4) takes the form of the sum in(3.2), where z = (z j ) j∉K and N is replaced byNκ, κ = #K.We recall that N ≤ 4 and K / = ∅, so that Nκ takes values 0, 1, 2 or 3. For the sets K satisfying Nκ = 0 we haveZ K (F) = {0}, so the sum (C.3) is bounded by C # (s)L -N d . Since the matrix α F is skew-symmetric, then in the case Nκ = 1 we have Z K (F) = ∅,so the sum (C.3) vanishes. When Nκ = 2 or 3 we apply Theorem 3.2 and see that the sum over z in (C.4) is bounded by CL -(N -κ)(1-d) . So |J Same argument implies that the r.h.s. of (C.3) also is bounded by the quantity C # (s)L -N +1-d (note that decomposing the r.h.s. of (C.3) as in (C.4) we get a new term with K = ∅, but for it Nκ = N ≤ 4 and Theorem 3.2 still applies). Appendix D. Case d = 2 A difference between the cases d ≥ 3 and d = 2 comes from Theorem 3.1 since in the asymptotic, given by the latter, an additional log-factor appears when d = 2. To handle it we redefine the sum in (3.2), defining S L,N (Φ), by multiplying it by (ln L) -N /2 . So when d = 2 S L,N takes the form (D.1) S L,N (Φ) ∶= L N (1-d) (ln L) N /2 z∈Z∶ ω j (z)=0 ∀j Φ(z). Accordingly the (d = 2)-analogy of (3.5) reads (D.2) S L,2 (Φ) -L 2(1-d) ln L z∈Z 2d L ∶ z 1 ⋅z 2 =0 Φ(z) ≤ CL 2-d ln L ∥Φ∥ 0,d+1 = C ln L ∥Φ∥ 0,3 .

2 Σ 0 Φ

 20 3.1 for d = 2: Theorem D.1. Let d = 2. Then there exist constants N 1 , N 2 > 4 such that if ∥Φ∥ N 1 ,N 2 < ∞,(D.3) S L,2 (Φ) -C (z) µ Σ 0 (dz 1 dz 2 ) ≤ K 2 ∥Φ∥ N 1 ,N 2 ln L ,where C 2 > 0 is a number theoretical constant and K 2 > 0.Note that estimate (3.8) stays true when d = 2.Theorem D.2. In the case d = 2 assertion of Theorem 3.2 remains true, if the sum S L,N is defined as in (D.1) and N 2 is the constant from Theorem D.1.Proof. The only difference with the proof of Theorem 3.2 comes from estimate (3.18) since the latter is obtained by applying Theorem 3.1, and in the case d = 2 we should apply Theorem D.1 instead. Namely, now the r.h.s. of (3.18) takes the form CL2(d-1) ln L R 2d +R N 2 (ln L) -1 ≤ C ′ R N 2 L 2(d-1) ln L.Since Lemma 3.6 remains unchanged, then for d = 2 the r.h.s. of estimate (3.20), which holds for irreducible matrices α, should be multiplied by ln L.In the case of reducible matrix α we apply the latter estimate to each irreducible block, which gives the factor (ln L) ⌊N /2⌋ in the r.h.s. of (3.12), since the number of blocks does not exceed ⌊N /2⌋. However, the final estimate of Theorem 3.2 remains unchanged because of the factor (ln L) -N /2 in the definition (D.1) of the sum S L,N .Since in the case d = 2 we choose ρ = εL/ ln L, then the terms n (k) s,L are given by formula (4.4), multiplied by (ln L)-k/2 . The proof of Proposition 2.1 is analogous to that presented in Appendix C for d = 2. The only difference being the use of Theorem D.2 in place of Theorem 3.2. Lemma 2.2 remains unchanged, so the correlations L k (ln L) s (τ 2 ), m + n = k, are given by formula (2.18), multiplied by (ln L)-k/2 (recall that the sum of these correlations makes n (k)

- 1 .

 1 This results in the stronger lower bound for L in Theorem 5.8: now it is L ≥ e ε -1 instead of L ≥ ε -2

D. 1 .Φ 1 2 L 1 1 (

 11211 8 and Proposition 2.1, and the term χ 2 (L) -N +1 = (ln L) (N -1)/2 , appearing in estimate (2.3) for d = 2 does not change the assertion of Theorem 5.8. Discussion of Remark 1.2. In fact, Theorem 1.4 from [10] provides more delicate information about S L,2 than what is stated in Theorem D.1.Namely, if d = 2 then due to[START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF], is a certain function satisfying |σΦ 1 (L)| ≤ C 1 ∥Φ∥ N 1 ,N 2, uniformly in L. See[START_REF] Dymov | A refinement of Heath-Brown's theorem on quadratic forms[END_REF] for an explicit (but complicated) formula for σΦ 1 . Consequently, S L,2 (Φ) -C 2 Σ 0 Φ(z) µ Σ 0 (dz 1 dz 2 ) -σΦ 1 (L) ln L ≤ C ∥Φ∥ N 1 ,N )| ≤ C∥Φ∥ N 1 ,N 2 in view of (D.2). Then estimate (4L) and Φ(τ ) is the function satisfying n ≤2 s,L (τ ) = S L,2 (Φ(τ )) that comes from Corollary 2.3. By(2.22) and the estimate for σΦ 1 above, the function f (τ, L) is bounded uniformly in τ . The rest of the proofs of Theorems 5.8 and 5.9 remain unchanged while the estimate (D.4) leads to the assertion of the remark.

  Σ s , ds 1 ds 2 | Σ s denotes the volume element on Σ s , corresponding to the standard Euclidean structure on R 2d .

[START_REF] Huang | Time-averaging for weakly nonlinear CGL equations with arbitrary potentials[END_REF]

, and a measure µ Σ s on it, given by

(1.34) µ Σ s (ds 1 ds 2 ) = |s 1 -s| 2 + |s 2 -s| 2 -1/2 ds 1 ds 2 |

  Next, it turns out that the vectors z ∈ Z(F) may be used to parametrize the set of admissible indices ξ i by means of an affine mapping(2.16)

10

The matrix α F has no zero rows and zero columns if and only if F ∈ F true m,n , and, accordingly, the set Z(F) is non-empty if and only if F ∈ F true m,n .

  vanishing at zero, with finite norm |y| Y r = sup t≥0 e

t |y(t)| r . Let B be the linear operator

  to x 1 y 1 + x 2 y 2 = C which can not contain P ′ (x 1 , x 2 , y 1 , y 2 ) ∩ H. Indeed, else, supposing by symmetry thatP ′ (x 1 , x 2 , y 1 , y 2 )∩H ⊃ {x 1 = a 1 x 2 +b 1 y 1 +b 2 y 2 -c} we get that the following quadratic function of x 2 , y 1 , y 2 :

  3s )B 123 , where B 123 = B 1 B 2 B 3 . Since B 123 with s 3 = ss 1 -s 2 is a Schwartz function of s, s 1 , s 2 then Theorem 3.2 with N = 2 applies and we find

	(B.4)	E|∆a	(1) s (τ )| 2 ≤ C	# (s)L -2 τ	2 .
	B.3. The second term of Q s,L in (B.2). To study the term 2RN s we use sym the same strategy as above. Namely, expressing in (B.1) the function 3Y s
	via Y

s , we write N s as N s

  The fourth term of Q s,L in (B.2). To study the term 2R Ñs , as in Section B.3, we write Ñs as Ñs

3s )B 123 T s T s . B.5.

  1 in (B.15) can be replaced by B 23s T 1 + is a consequence of Theorem 3.2. This concludes the proof of Proposition 4.3. B.7. Proof of Lemma 4.4. Note that ∂ s j f (γ j ) = f

	B 13s T	2 . Then, inserting (B.17) in (B.15) we get
	|Q s,L -X s | ≤ 4L 2 (T s -τ )⨊ 1,2	δ 3s δ(ω ′12 3s ) T 12	s B 123 + T	3 B 12s -2T	s B 23s
		+ C	# (s)τ	2 ,
	with X s defined in (4.33). Finally, we point out that |(T s -τ )T due to (B.11) and since |T s -τ | ≤ 3τ 2 γ 123s . So, the bound	j | ≤ 3τ	2 ,
	L 2 (T s -τ )⨊ 1,2	δ 3s δ(ω ′12 3s ) T 12	s B 123 + T	3 B 12s -2T	s B 23s ≤ C	# (s)τ	2 ,

ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUT In

ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUT 1.3.

Earlier the kinetic operator K(∞) was heuristically obtained in[START_REF] Kuksin | Derivation of a wave kinetic equation from the resonantaveraged stochastic NLS equation[END_REF].

In[START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] the notation is slightly different: there we set ρ = ε 1/2 ν

12ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUT ţerms,

14ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUT a)

24ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUTProof.

34ANDREY DYMOV, SERGEI KUKSIN, ALBERTO MAIOCCHI, AND SERGEI VL ȂDUTProof.

Acknowledgements. The work of AD was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614). SK was supported by Agence Nationale de la Recherche through the grant 17-CE40-0006.