# Averaging and mixing for stochastic perturbations of linear conservative
systems

Guan Huang  School of Mathematics and Statistics, Beijing Institute of
Technology, Beijing, China,& Peoples’ Friendship University of Russia (RUDN
University), Moscow, Russia<EMAIL_ADDRESS>and  Sergei Kuksin
Université Paris-Diderot (Paris 7), UFR de Mathématiques - Batiment Sophie
Germain, 5 rue Thomas Mann, 75205 Paris, & Peoples’ Friendship University of
Russia (RUDN University), Moscow, Russia<EMAIL_ADDRESS>
###### Abstract.

We study stochastic perturbations of linear systems of the form

$None$
$dv(t)+Av(t)\,dt={\varepsilon}P(v(t))dt+\sqrt{{\varepsilon}}\,\mathcal{B}(v(t))dW(t),\quad
v\in\mathbb{R}^{D},$

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

Dedicated to the memory of M.I.Vishik

on the occasion of his 100-th birthday.

###### Contents

  1. 1 Introduction
  2. 2 Linear systems and their perturbations
  3. 3 Averaging of vector fields with respect to the frequency vector
  4. 4 Effective equation and the averaging theorem
  5. 5 Stationary solutions and mixing
  6. 6 The non-resonant case
  7. 7 Uniform in time convergence.
  8. 8 Averaging for systems with general noises
  9. 9 A sufficient condition for Assumptions 2.1, 5.1 and 7.1
  10. A Proof of Proposition 9.3
  11. B Representation of martingales
  12. C Itô’s formula for complex processes
  13. D Projections to convex sets

##  1\. Introduction

###  1.1. The setting and problems.

The goal of this paper is to present an averaging theory for perturbations of
conservative linear differential equations by locally Lipschitz nonlinearities
and stochastic terms. Namely, we will examine stochastic equations

(1.1)
$dv(t)+Av(t)\,dt={\varepsilon}P(v(t))dt+\sqrt{{\varepsilon}}\,\mathcal{B}(v(t))dW(t),\qquad
v\in\mathbb{R}^{D},$

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

###  1.2. Our results and their deterministic analogies.

We tried to make our work “reader-friendly” and accessible to people with just
a limited knowledge of stochastic calculus. To achieve that in the main part
of the paper we restrict ourselves to the case of equations with additive
noise $\sqrt{\varepsilon}\,\mathcal{B}\,dW(t)$ and exploit there a technical
convenience: we introduce in $\mathbb{R}^{D}$ a complex structure, re-writing
the phase-space $\mathbb{R}^{D}$ as $\mathbb{C}^{D/2}$ (recall that $D$ is
even), in such a way that in the corresponding complex coordinates the
operator $A$ is diagonal, $A=$diag$\\{i\lambda_{j}\\}.$ General equations
(1.1) are discussed in Section 8, where they are treated in parallel with
earlier considered equations with additive noise.

As it is custom in the classical deterministic Krylov–Bogolyubov (K-B)
averaging (e.g. see [4], [1] and [13]), to study solutions
$v(t)\in\mathbb{C}^{D/2}$ we write them in the interaction representation
which preserves the norms of the complex components $v_{j}(\tau)$, but amends
their angles. See below substitution (2.8). The first principal result of the
work is given by Theorem 4.7, where we assume uniform in $\varepsilon$ and in
$t\leq C\varepsilon^{-1}$ bounds on a first few moments of norms of solutions.
The theorem states that as $\varepsilon\to 0$, for $t\leq C\varepsilon^{-1}$
solutions $v(t)$, written in terms of the interaction representation, weakly
converge in distribution to solutions of an additional effective equation. The
latter is obtained from eq. (1.1) by means of certain averaging of vector
field $P$ in terms of the spectrum $\\{i\lambda_{j}\\}$ and in many cases may
be written down explicitly. Proof of Theorem 4.7, given in Section 4, is
obtained by a synthesis of the K-B method (as it is presented e.g. in [13])
and the Khasminski approach to the stochastic averaging [16]; it may serve as
an introduction to the latter. The number of works on the stochastic averaging
is immense, see Section 1.3 for some references. We were not able to find
there the result of Theorem 4.7, but do not insist on its novelty (and
certainly related statements may be found in the literature).

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

In short Section 6 we discuss non-resonant systems (1.1) (where the
frequencies $\\{\lambda_{j}\\}$ are rationally independent). In particular, we
show that then the actions $I_{j}(v(t))$ of solutions $v(t)$ (see below
(1.2)), as $\varepsilon\to 0$, converge in distribution to solutions of a
system of stochastic equations, depending only on actions. The convergence
hold on time-intervals $0\leq t\leq C\varepsilon^{-1}$.

In Section 7 we keep the assumption on the norms of solutions from Section 5.
Supposing that the effective equation is mixing (and without assuming that for
the original equation (1.1)) we prove there Theorem 7.4. It states that the
convergence as in the principal Theorem 4.7 is uniform for $t\geq 0$ (and not
only for $t\leq C\varepsilon^{-1}$).

In Proposition 9.4 we present a simple sufficient condition on eq. (1.1),
based on results from [17], which insures that Theorems 4.7, 5.5 and 7.4 apply
to it.

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

A deterministic analogy of our results, which deals with eq. (1.1) with $W=0$
and describes the behaviour of its solutions on time-intervals of order
$\varepsilon^{-1}$ in the interaction representation in comparison with
solutions for a corresponding effective equation, is given by the K-B
averaging (see [4, 1, 13]).111Theorem 4.7 also applies to such equations, but
then its assertion becomes unnatural. Theorem 5.5 has no analogy for
deterministic systems, but Theorem 7.4 has it. Namely, for the K-B averaging
it is known that if the effective equation has a globally asymptotically
stable equilibrium, then the convergence of solutions for eq. (1.1)W=0,
written in the interaction representation, to solutions of the effective
equation, is uniform in time. This result is known in folklore as the second
K-B theorem and may be found in [6].

The K-B method and Khasminski approach to averaging which we exploit, are
flexible tools. They are applicable to various stochastic systems in finite
and infinite dimensions, including stochastic PDEs, and the specific
realization of the two methods which we use now is inspired by our previous
work on averaging for stochastic PDEs. See [12, 19] for an analogy for SPDEs
of Theorem 4.7, [12] – for an analogy of Theorem 5.5 and [11] – for an analogy
of Theorem 7.4 (also see [7] for more results and references on averaging for
SPDEs).

###  1.3. Relation with classical stochastic averaging.

The averaging in stochastic systems is a well developed topic, usually dealing
with fast-slow stochastic systems, e.g. see publications [16], [10, Section
7], [23, Section II.3], [21], [25], [18] and references there. To explain
relation of that theory with our work let us write eq. (1.1) in the complex
form $v(t)\in\mathbb{C}^{n}$, $n=D/2$ (when the operator $A$ is diagonal) and
then pass to the slow time $\tau=\varepsilon t$ and action-angle coordinates
$(I,\varphi)=(I_{1},\dots,I_{n};\varphi_{1},\dots,\varphi_{n})\in\mathbb{R}^{n}_{+}\times\mathbb{T}^{n}$,
where $\mathbb{R}_{+}=\\{x\in\mathbb{R}:x\geq 0\\}$,
$\mathbb{T}^{n}=\mathbb{R}^{n}/(2\pi\mathbb{Z}^{n})$ and

(1.2)
$\begin{split}I_{k}(v)=\tfrac{1}{2}|v_{k}|^{2}=\tfrac{1}{2}v_{k}\bar{v}_{k},\quad\varphi_{k}(v)=\text{Arg}\;v_{k}\in\mathbb{S}^{1}=\mathbb{R}/2\pi\mathbb{Z},\quad
k=1,\dots,n\end{split}$

(if $v_{k}=0$, we set $\varphi_{k}(v)=0\in\mathbb{S}^{1}$). In these
coordinates equation (1.1) takes the form

(1.3)
$\begin{cases}dI(\tau)=P^{I}(I,\varphi)d\tau+\Psi^{I}(I,\varphi)d\beta(\tau),\\\
d\varphi(\tau)+{\varepsilon}^{-1}\Lambda
d\tau=P^{\varphi}(I,\varphi)d\tau+\Psi^{\varphi}(I,\varphi)d\beta(\tau).\end{cases}$

Here $\beta=(\beta_{1},\dots,\beta_{N})$, where $\\{\beta_{l}\\}$ are
independent standard real Wiener processes, and the coefficients of the system
are given by Itô’s formula. This is a fast-slow system with the slow variable
$I$ and fast variable $\varphi$. The stochastic averaging treats systems like
(1.3), usually adding to the fast part of the $\varphi$-equation a non-
degenerate stochastic term of order $\varepsilon^{-1/2}$. The (first) goal of
a system’s analysis usually is to prove that on time-intervals $0\leq\tau\leq
T$ distributions of the $I$-components of solutions converge as
$\varepsilon\to 0$ to distributions of solutions for a suitably averaged
$I$-equation. After that other goals may be pursued.222E.g. one may study
deviation of the $I$-components of solutions from the averaged dynamics (see
[10, 18]) or, under stronger restrictions on the system, may examine behaviour
of solutions on longer time intervals (see paper [15] and works, descending
from it).

Unfortunately the stochastic averaging does not apply directly to systems
(1.3), coming from equations (1.1), since then the coefficients of the
$\varphi$-equation have singularities when some $I_{k}$ vanishes, and since
the fast $\varphi$-equation is rather degenerate if the vector $\Lambda$ is
resonant. Instead we borrow from the theory the Khasminski method [16] of
stochastic averaging and apply it to eq. (1.1), written in the interaction
representation, thus arriving at the assertion of Theorem 4.7. Averaging
theorem for stationary solutions of eq. (1.3) and for the corresponding
stationary measures are known in the stochastic averaging, but (naturally)
they control the limiting behaviour only of the $I$-components of the
stationary solutions and measures, while our Theorem 5.5 describes a limit of
the whole stationary measure. It seems that no analogy of Theorem 7.4 is known
in the stochastic averaging.

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

Notation. For a Banach space $E$ and $R>0$, $B_{R}(E)$ stands for the open
$R$-ball $\\{e\in E:|e|_{E}<R\\}$, and $\bar{B}_{R}(E)$ – for its closure
$\\{|e|_{E}\leq R\\}$; $C_{b}(E)$ stands for the space of bounded continuous
function on $E$, and $C([0,T],E)$ – for the space of continuous curves
$[0,T]\to E$, given the sup-norm. For any $0<\alpha\leqslant 1$ and $u\in
C([0,T],E)$,

(1.4) $\|u\|_{\alpha}=\sup_{0\leqslant\tau<\tau^{\prime}\leqslant
T}\frac{|u(\tau^{\prime})-u(\tau)|_{E}}{|\tau^{\prime}-\tau|^{\alpha}}+\sup_{\tau\in[0,T]}|u(\tau)|_{E}\leq\infty.$

This is a norm in the Hölder space $C^{\alpha}([0,T],E)$. We denote the
standard $C^{m}$-norm for $C^{m}$-smooth functions on $E$ as
$|\cdot|_{C^{m}(E)}$. By $\mathcal{D}(\xi)$ we denote the law of a random
variable $\xi$, by $\rightharpoonup$ – the weak convergence of measures, and
by $\mathcal{P}(M)$ – the space of Borel measures on a metric space $M$. For a
measurable mapping $F:M_{1}\to M_{2}$ and $\mu\in\mathcal{P}(M_{1})$ we denote
by $F\circ\mu\in\mathcal{P}(M_{2})$ the image of $\mu$ under $F$; i.e.
$F\circ\mu(Q)=\mu(F^{-1}(Q))$.

If $m\geq 0$ and $L$ is $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$, then
$\text{Lip}_{m}(L,E)$ is the collection of maps $F:L\to E$ such that for any
$R\geqslant 1$ we have

(1.5)
$(1+|R|)^{-m}\big{(}\text{Lip}(F|_{\bar{B}_{R}(L)})+\sup_{v\in\bar{B}_{R}(L)}|F(v)|_{E}\big{)}=:\mathcal{C}^{m}(F)<\infty,$

where $\text{Lip}(f)$ is the Lipschitz constant of a mapping $f$ (note that,
in particular, $|F(v)|_{E}\leq\mathcal{C}^{m}(F)(1+|v|_{L})^{m}\,$ for any
$v\in L$). For a complex matrix $A=(A_{ij})$, $A^{*}=(A^{*}_{ji})$ stands for
its Hermitian conjugate: $A^{*}_{ij}=\bar{A}_{ji}$ (so for a real matrix $B$,
$B^{*}$ is the transposed matrix). For a set $Q$ we denote by $\mathbf{1}_{Q}$
its indicator function, and by $Q^{c}$ – its complement. Finally,
$\mathbb{R}_{+}$ ($\mathbb{Z}_{+}$) stands for the set of nonnegative real
numbers (nonnegative integers), and for real numbers $a$ and $b$, $a\vee b$
and $a\wedge b$ indicate their maximum and minimum.

##  2\. Linear systems and their perturbations

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

###  2.1. Assumptions on $A$ and ${W}(t)$

We assume that the unperturbed linear system

(2.1) $(d/dt)v+Av=0,\qquad v\in\mathbb{R}^{D},$

is such that all its trajectories are bounded as $t\to\pm\infty$. Then the
eigenvalues of $A$ are pure imaginary, go in pairs $\pm i\lambda_{j}$ and $A$
has no Jordan cells. We also assume that $A$ is invertible. So

  1. (1)

eigenvalues of $A$ are of the form $\pm i\lambda_{j}$, where
$\mathbb{R}\ni\lambda_{j}\neq 0$;

  2. (2)

in the Jordan normal form of $A$ there are no Jordan cells.

By these assumptions ${D}=2n$, and in $\mathbb{R}^{2n}$ there exists a base
$\\{\mathbf{e}_{1}^{+},\mathbf{e}_{1}^{-},\dots,\mathbf{e}_{n}^{+},\mathbf{e}_{n}^{-}\\}$
in which the linear operator $A$ takes the block-diagonal form:

$A=\left(\begin{array}[]{c c c}\begin{array}[]{cc}0&-\lambda_{1}\\\
\lambda_{1}&0\end{array}&&0\\\ &\ddots&\\\
0&&\begin{array}[]{cc}0&-\lambda_{n}\\\
\lambda_{n}&0\end{array}\end{array}\right)$

We denote by $(x_{1},y_{1},\dots,x_{n},y_{n})$ the coordinates, corresponding
to this base, and for $j=1,\dots,n$ set $z_{j}=x_{j}+iy_{j}$. Then
$\mathbb{R}^{2n}$ becomes the space of complex vectors $(z_{1},\dots,z_{n})$,
i.e. $\mathbb{R}^{2n}\simeq\mathbb{C}^{n}$. In the complex coordinates the
standard inner product in $\mathbb{R}^{2n}$ reads

(2.2) $\langle
z,z^{\prime}\rangle=\text{Re}\sum_{j=1}^{n}z_{j}\bar{z}_{j}^{\prime},\quad
z,\;z^{\prime}\in\mathbb{C}^{n}.$

Let us denote by

$\Lambda=(\lambda_{1},\dots,\lambda_{n})\in(\mathbb{R}\setminus\\{0\\})^{n}$

the frequency vector of the linear system (2.1). Then in the complex
coordinates $z$ the operator $A$ reads

$Az=\text{diag}\\{i\Lambda\\}z,$

where $\text{diag}\\{i\Lambda\\}$ is the diagonal operator, sending a vector
$(z_{1},\dots,z_{n})$ to $(i\lambda_{1}z_{1},\dots,i\lambda_{n}z_{n}).$
Therefore in $\mathbb{R}^{2n}$, written as the complex space $\mathbb{C}^{n}$,
linear equation (2.1) takes the diagonal form,

$(d/dt)v_{k}+i\lambda_{k}v_{k}=0,\quad\;\;1\leqslant k\leqslant n.$

Below we examine the perturbed eq. (1.1), using these complex coordinates.

We next discuss the random process $W(t)$, written in the complex coordinates.
The standard complex Wiener process  has the form

(2.3) $\beta^{c}(t)=\beta^{+}(t)+i\beta^{-}(t)\in\mathbb{C},$

where $\beta^{+}(t)$ and $\beta^{-}(t)$ are independent standard (real) Wiener
processes, defined on some probability space
$(\Omega,\mathcal{F},\mathbf{P})$. Then
$\bar{\beta}^{c}(t)=\beta^{+}(t)-i\beta^{-}(t)$, and any Wiener process
$W(t)\in\mathbb{C}^{n}$ may be conveniently written in the complex from as

(2.4)
$W_{k}=\sum_{l=0}^{n_{1}}\Psi_{kl}^{1}\beta^{c}_{l}+\sum_{l=1}^{n_{1}}\Psi_{kl}^{2}\bar{\beta}^{c}_{l},\quad
k=1,\dots,n,$

where $\Psi^{1}=(\Psi_{kl}^{1})$ and $\Psi^{2}=(\Psi_{kl}^{2})$ are complex
$n\times n_{1}$ matrices and $\\{\beta^{c}_{l}\\}$ are independent standard
complex Wiener processes. Again, in order to simplify presentation below we
suppose that the noise in eq. (1.1) is of the form

$W(t)=\sum_{l=1}^{n_{1}}\Psi_{kl}\beta^{c}_{l}(t),\quad k=1,\dots,n.$

We do not assume that the matrix $\Psi$ is non-degenerate (in particular, it
may be zero). Then the perturbed equation (1.1) in the complex coordinates
reads as

(2.5)
$dv_{k}+i\lambda_{k}v_{k}dt={\varepsilon}P_{k}(v)dt+\sqrt{{\varepsilon}}\sum\Psi_{kl}d\beta^{c}_{l}(t),\quad
k=1,\dots,n\,,$

where $v=(v_{1},\dots,v_{n})\in\mathbb{C}^{n}$ and $0<\varepsilon\leq 1$.

The results, obtained below for eq. (2.5), remain true for general equations
(1.1) on the price of heavier calculation. The corresponding argument is
sketched in Section 8.

###  2.2. Assumptions on $P$ and on the perturbed equation.

Our first goal is to study equation (2.5) with $0<{\varepsilon}\ll 1$ on a
time interval $0\leq t\leq\varepsilon^{-1}T$, where $T>0$ is a fixed constant.
Introducing the slow time

$\tau={\varepsilon}t$

we write the equation as

(2.6)
$dv_{k}(\tau)+i{\varepsilon}^{-1}\lambda_{k}v_{k}d\tau=P_{k}(v)d\tau+\sum_{l=1}^{n_{1}}\Psi_{kl}d\tilde{\beta^{c}_{l}}(\tau),\qquad
k=1,\dots,n,\;\;0\leq\tau\leq T.$

Here $\\{\tilde{\beta}^{c}_{l}(\tau),l=1,\dots,n_{1}\\}$ is another set of
independent standard complex Wiener processes, which we now re-denote back to
$\\{\beta^{c}_{l}(\tau),l=1,\dots,n_{1}\\}$. We stress that the equation above
is nothing but the original eq. (1.1), where its linear part (2.1) is
conservative and non-degenerate in the sense of conditions (1) and (2),
written in the complex coordinates and slow time. So all results below
concerning eq. (2.6) (and eq. (2.11)) may be reformulated for eq. (1.1) on the
price of heavier notation.

Everywhere below in this paper we assume the following assumption concerning
the well-posedness of eq. (2.6):

######  Assumption 2.1.

  1. (1)

The drift $P(v)=(P_{1}(v),\dots,P_{n}(v))$ is a locally Lipschitz vector
field, belonging to $\text{Lip}_{m_{0}}(\mathbb{C}^{n},\mathbb{C}^{n})$ for
some $m_{0}\geq 0$ (see (1.5)).

  2. (2)

For any $v_{0}\in\mathbb{C}^{n}$, equation (2.6) has a unique strong solution
$v^{\varepsilon}(\tau;v_{0})$, $\tau\in[0,T]$, equal $v_{0}$ at $\tau=0$.
Moreover, there exists $m_{0}^{\prime}>(m_{0}\vee 1)$ such that

(2.7) ${\mathbf{E}}\sup_{0\leqslant\tau\leqslant
T}|v^{{\varepsilon}}(\tau;v_{0})|^{2m^{\prime}_{0}}\leqslant
C_{m^{\prime}_{0}}(|v_{0}|,T)<\infty\qquad\forall\,0<\varepsilon\leq 1,$

where $C_{m^{\prime}_{0}}(\cdot)$ is a non-negative continuous function on
$\mathbb{R}_{+}^{2}$, non-decreasing in both arguments.

Our proofs easily generalize to the case when the vector field $P$ is locally
Lipschitz, satisfying $|P(v)|\leq C(1+|v|)^{m_{0}}$ for all $v$ and some
$C>0,\,m_{0}\geq 0$.333 See [13] for averaging in deterministic perturbation
of eq. (2.1) by locally Lipschitz vector fields. In this case the argument
remains essentially the same (but become a bit longer), and the constants in
estimates depend not only on $m_{0}$, but also on the locally Lipschitz
constant of $P$, which is the function
$R\mapsto$Lip$\,P\\!\mid\\!_{\bar{B}_{R}(\mathbb{C}^{n})}.$

Below $T>0$ is fixed and the dependence of constants on $T$ usually is not
indicated. Solutions of eq. (2.6) are assumed to be strong, unless otherwise
stated. As usual, strong solutions are understood in the sense of an integral
equation. I.e., $v^{\varepsilon}(\tau;v_{0})=v(\tau)$, $0\leq\tau\leq T$, is a
strong solution, equal $v_{0}$ at $\tau=0$, if a.s.

$v_{k}(\tau)+\int_{0}^{\tau}\big{(}i{\varepsilon}^{-1}\lambda_{k}v_{k}(s)-P_{k}(v(s))\big{)}ds=v_{0k}+\sum_{l=1}^{n_{1}}\Psi_{kl}{\beta^{c}_{l}}(\tau),\;\;\;k=1,\dots,n,$

for $0\leq\tau\leq T$.

###  2.3. Interaction representation.

Now in eq. (2.6) we pass to the interaction representation, which means that
we substitute

(2.8) $v_{k}(\tau)=e^{-i\tau{\varepsilon}^{-1}\lambda_{k}}a_{k}(\tau),\quad
k=1,\dots,n.$

Then $v_{k}(0)=a_{k}(0)$ and we obtain the following equations for variables
$a_{k}(\tau)$:

(2.9)
$da_{k}(\tau)=e^{i\tau{\varepsilon}^{-1}\lambda_{k}}P_{k}(v)+e^{i\tau{\varepsilon}^{-1}\lambda_{k}}\sum_{l=1}^{n_{1}}\Psi_{kl}d\beta^{c}_{l}(\tau),\quad
k=1,\dots,n.$

Actions $I_{k}=|a_{k}|^{2}/2$ for solutions of (2.9) are the same as the
actions for solutions of (2.6). Compare to (2.6), in eq. (2.9) we have removed
from the drift the large term ${\varepsilon}^{-1}\text{diag}(i\Lambda)v$ on
the price that now coefficients of the system are fast oscillating functions
of $\tau$.

To rewrite conveniently the equations above we introduce the rotation
operators $\Phi_{w}$: for any real vector
$w=(w_{1},\dots,w_{n})\in\mathbb{R}^{n}$ we denote

(2.10)
$\Phi_{w}:\mathbb{C}^{n}\to\mathbb{C}^{n},\quad\Phi_{w}=\text{diag}\\{e^{iw_{1}},\dots,e^{iw_{n}}\\}.$

Then $(\Phi_{w})^{-1}=\Phi_{-w}$,
$\Phi_{w_{1}}\circ\Phi_{w_{2}}=\Phi_{w_{1}+w_{2}}$, $\Phi_{0}=\mathrm{id},$
each $\Phi_{w}$ is an unitary transformation, so $\Phi_{w}^{*}=\Phi_{w}^{-1}$.
Moreover,

$|(\Phi_{w}z)_{j}|=|z_{j}|,\qquad\forall z,w,j.$

In terms of operators $\Phi$ we write $v(\tau)$ as
$\Phi_{\tau{\varepsilon}^{-1}\Lambda}a(\tau)$, and write system (2.9) as

(2.11)
$da(\tau)=\Phi_{\tau{\varepsilon}^{-1}\Lambda}P(\Phi_{-\tau{\varepsilon}^{-1}\Lambda}a(\tau))d\tau+\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi
d\beta^{c}(\tau),\quad a(\tau)\in\mathbb{C}^{n},$

where $\beta^{c}(\tau)=(\beta^{c}_{1}(\tau),\dots,\beta^{c}_{n_{1}}(\tau))$.
This is the equation which we are going to study for small ${\varepsilon}$ for
$0\leqslant\tau\leqslant T$ under the initial condition

(2.12) $a(0)=v(0)=v_{0}.$

The solution
$a^{\varepsilon}(\tau;v_{0})=\Phi_{-\tau{\varepsilon}^{-1}\Lambda}v^{\varepsilon}(\tau;v_{0})$
of (2.11), (2.12) also satisfies estimate (2.7), for each
$\varepsilon\in(0,1]$.

We recall that a $C^{1}$-diffeomorphism $G$ of $\mathbb{C}^{n}$ transforms a
vector filed $V$ to the field $G_{*}V$, where
$(G_{*}V)(v)=dG(u)\big{(}V(u)\big{)}$ for $u=G^{-1}v.$ In particular,

$\big{(}(\Phi_{\tau{\varepsilon}^{-1}\Lambda})_{*}P\big{)}(v)=\Phi_{\tau{\varepsilon}^{-1}\lambda_{k}}\circ
P(\Phi_{-{\varepsilon}\tau\Lambda}v).$

So equation (2.11) may be written as

$da(\tau)=\big{(}(\Phi_{\tau{\varepsilon}^{-1}\Lambda})_{*}P\big{)}(a(\tau))d\tau+\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi
d\beta^{c}(\tau).$

###  2.4. The compactness

For $0<{\varepsilon}\leqslant 1$ we denote by $a^{{\varepsilon}}(\tau;v_{0})$
a solution of equation (2.11), equals $v_{0}$ at $\tau=0$. Then

$a^{{\varepsilon}}(\tau;v_{0})=\Phi_{\tau{\varepsilon}^{-1}\lambda_{k}}v^{{\varepsilon}}(\tau;v_{0}).$

A unique solution $v^{{\varepsilon}}(\tau;v_{0})$ of (2.6) exists by
Assumption 2.1, so a solution $a^{{\varepsilon}}(\tau;v_{0})$ as well exists
and is unique. Our goal is to examine its law

$Q_{\varepsilon}:=\mathcal{D}(a^{{\varepsilon}}(\cdot;v_{0}))\in\mathcal{P}(C([0,T],\mathbb{C}^{n}))$

as ${\varepsilon}\to 0$. While $v_{0}$ is fixed we will usually write
$a^{{\varepsilon}}(\tau;v_{0})$ as $a^{{\varepsilon}}(\tau)$.

######  Lemma 2.2.

Under Assumption 2.1, the set of probability measures
$Q_{\varepsilon},\,{\varepsilon}\in(0,1]\\}$ is pre-compact in the space
$\mathcal{P}(C([0,T],\mathbb{C}^{n}))$ with respect to the weak topology.

######  Proof.

Let us denote the random force in equation (2.11) by
$d\zeta^{{\varepsilon}}(\tau):=\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi
d\beta^{c}(\tau)$, where
$\zeta^{\varepsilon}(\tau)=(\zeta_{l}^{\varepsilon}(\tau),\,l=1,\dots,n_{1})$.
For any $k$ we have

$\zeta^{{\varepsilon}}_{k}(\tau)=\int_{0}^{\tau}d\zeta_{k}^{{\varepsilon}}=\int_{0}^{\tau}e^{is{\varepsilon}^{-1}\lambda_{k}}\sum_{l=1}^{n_{1}}\Psi_{kl}d\beta^{c}_{l}(s).$

So $\zeta^{{\varepsilon}}(\tau)$ is a stochastic integral of a non-random
vector function. Hence, it is a Gaussian random process with zero mean value,
and its increments over non-intersecting time-intervals are independent. For
each $k$,

${\mathbf{E}}|\zeta_{k}^{\varepsilon}(\tau)|^{2}=\int_{0}^{\tau}2\sum_{l=1}^{n_{1}}|\Psi_{kl}|^{2}ds=:2C_{k}^{\zeta}\tau,\qquad
C_{k}^{\zeta}=\sum_{l=1}^{n_{1}}|\Psi_{kl}|^{2}\geq 0,$

and
${\mathbf{E}}\zeta_{k}^{\varepsilon}(\tau)\zeta_{j}^{\varepsilon}(\tau)={\mathbf{E}}\bar{\zeta}_{k}^{\varepsilon}(\tau)\bar{\zeta}_{j}^{\varepsilon}(\tau)=0.$
Therefore $\zeta_{k}^{\varepsilon}(\tau)=C_{k}^{\zeta}\beta^{c}_{k}(\tau)$,
where by Lévy’s theorem (see [14, p. 157]) $\beta^{c}_{k}(\tau)$ is a standard
complex Wiener process. However, the processes $\zeta^{\varepsilon}_{j}$ and
$\zeta^{\varepsilon}_{k}$ with $j\neq k$ are not necessarily independent.

By the basic properties of Wiener process, a.s. the curve
$[0,T]\ni\tau\mapsto\zeta^{\varepsilon}(\omega,\tau)\in\mathbb{C}^{n}$ is
Hölder-continuous with exponent $\tfrac{1}{3}$, and since $C_{k}^{\zeta}$ does
not depend on ${\varepsilon}$, then abbreviating
$(C^{1/3}([0,T],\mathbb{C}^{n})$ to $C^{1/3}$ we have

${\mathbf{P}}\big{(}\zeta^{\varepsilon}(\cdot)\in\bar{B}_{R}(C^{1/3})\big{)}\to
1\;\;\text{as}\;\;R\to\infty,$

uniformly in ${\varepsilon}$. Let us write equation (2.11) as

$da^{{\varepsilon}}(\tau)=V^{{\varepsilon}}(\tau)d\tau+d\zeta^{{\varepsilon}}(\tau).$

By Assumption 2.1 and since
$|a^{\varepsilon}(\tau)|\equiv|v^{\varepsilon}(\tau)|$ we have that

${\mathbf{E}}\sup_{\tau\in[0,T]}|V^{\varepsilon}(\tau)|\leqslant\mathcal{C}^{m_{0}}\\!(P){\mathbf{E}}(1+\sup_{\tau\in[0,T]}|v^{\varepsilon}(\tau)|)^{m_{0}}\leqslant
C(|v_{0}|)<\infty.$

Therefore, by Chebyshev’s inequality,

$\mathbf{P}(\sup_{0\leqslant\tau\leqslant
T}|V^{{\varepsilon}}(\tau)|>R)\leqslant C(|v_{0}|)R^{-1},$

uniformly in ${\varepsilon}\in(0,1]$. Since

$a^{{\varepsilon}}(\tau)=v_{0}+\int_{0}^{\tau}V^{{\varepsilon}}(s)ds+\zeta^{{\varepsilon}}(\tau),$

then we get from the above that

(2.13) $\mathbf{P}\big{(}\|a^{{\varepsilon}}(\cdot)\|_{1/3}>R\big{)}\to
0\quad\text{as}\quad R\to\infty,$

uniformly in ${\varepsilon}\in(0,1]$. By the Ascoli-Arzela theorem the sets
$\bar{B}_{R}(C^{1/3})$ are compact in $C([0,T];\mathbb{C}^{n})$, and in view
of (2.13) for any $\delta>0$ there exists $R_{\delta}$ such that

$Q_{\varepsilon}\big{(}\bar{B}_{R_{\delta}}(C^{1/3})\big{)}\geqslant
1-\delta,\quad\forall{\varepsilon}>0.$

So by Prokhorov’s theorem the set of measures
$\\{Q_{\varepsilon},0<{\varepsilon}\leqslant 1\\}$ is pre-compact in
$\mathcal{P}(C[0,T],\mathbb{C}^{n})$. ∎

By this lemma, for any sequence ${\varepsilon}_{l}\to 0$ there exists a
subsequence ${\varepsilon}_{l}^{\prime}\to 0$ and a measure
$Q_{0}\in\mathcal{P}(C([0,T],\mathbb{C}^{n}))$ such that

(2.14) $Q_{{\varepsilon}_{l}^{\prime}}\rightharpoonup Q_{0}\;\;\text{ as
}\;{\varepsilon}_{l}^{\prime}\to 0.$

##  3\. Averaging of vector fields with respect to the frequency vector

For a vector field
${\tilde{P}}\in\text{Lip}_{m_{0}}(\mathbb{C}^{n},\mathbb{C}^{n})$ we denote

$Y_{\tilde{P}}(a;t)=\big{(}(\Phi_{t\Lambda})_{*}\tilde{P}\big{)}(a)=\Phi_{t\Lambda}\circ{\tilde{P}}(\Phi_{-t\Lambda}a),\quad
a\in\mathbb{C}^{n},\;t\in\mathbb{R},$

and for $T^{\prime}>0$ define partial averaging
$\llangle{\tilde{P}}\rrangle^{T^{\prime}}$ of the vector field ${\tilde{P}}$
with respect to the frequency vector $\Lambda$ as follows:

(3.1)
$\llangle{\tilde{P}}\rrangle^{T^{\prime}}(a)=\frac{1}{T^{\prime}}\int_{0}^{T^{\prime}}Y_{\tilde{P}}(a;t)dt=\frac{1}{T^{\prime}}\int_{0}^{T^{\prime}}\Phi_{t\Lambda}\circ{\tilde{P}}(\Phi_{-t\Lambda}a)dt.$

######  Lemma 3.1.

For any ${T^{\prime}}>0$,
$\llangle{\tilde{P}}\rrangle^{T^{\prime}}(a)\in\text{Lip}_{m_{0}}(\mathbb{C}^{n},\mathbb{C}^{n})$
and
$\mathcal{C}^{m_{0}}(\llangle{\tilde{P}}\rrangle^{T^{\prime}})\leq\mathcal{C}^{m_{0}}({\tilde{P}})$
(see (1.5)).

######  Proof.

If $a\in\bar{B}_{R}(\mathbb{C}^{n})$, then
$\Phi_{-t\Lambda}a\in\bar{B}_{R}(\mathbb{C}^{n})$ for each $t$. So
$|Y_{\tilde{P}}(a;t)|=|(\Phi_{t\Lambda})_{*}{\tilde{P}}(a)|=|{\tilde{P}}(\Phi_{-t\Lambda}a))|,$
and from here

$|\llangle{\tilde{P}}\rrangle^{T^{\prime}}(a)|\leqslant\sup_{0\leq
t\leq{T^{\prime}}}|Y_{\tilde{P}}(a;t)|\leqslant\mathcal{C}^{m_{0}}({\tilde{P}})(1+R)^{m_{0}}.$

Similarly, for any $a_{1},a_{2}\in\bar{B}_{R}(\mathbb{C}^{n})$,

$|Y_{\tilde{P}}(a_{1};t)-Y_{\tilde{P}}(a_{2};t)|=|{\tilde{P}}(\Phi_{-t\Lambda}a_{1})-{\tilde{P}}(\Phi_{-t\Lambda}a_{2})|\leqslant\mathcal{C}^{m_{0}}({\tilde{P}})(1+R)^{m_{0}}|a_{2}-a_{1}|\quad\forall
t\geqslant 0,$

so that

$|\llangle{\tilde{P}}\rrangle^{T^{\prime}}(a_{1})-\llangle{\tilde{P}}\rrangle^{T^{\prime}}(a_{1})|\leqslant\mathcal{C}^{m_{0}}({\tilde{P}})(1+R)^{m_{0}}|a_{1}-a_{1}|.$

This proves the assertion. ∎

Now we define averaging of the vector field ${\tilde{P}}$ with respect to the
frequency vector $\Lambda$ as

(3.2)
$\llangle{\tilde{P}}\rrangle(a)=\lim_{{T^{\prime}}\to\infty}\llangle{\tilde{P}}\rrangle^{T^{\prime}}(a)=\lim_{{T^{\prime}}\to\infty}\frac{1}{{T^{\prime}}}\int_{0}^{T^{\prime}}(\Phi_{t\Lambda})_{*}{\tilde{P}}(a)dt.$

######  Lemma 3.2.

  1. (1)

The limit (3.2) exists for any $a$. Moreover, $\llangle{\tilde{P}}\rrangle$
belongs to $\text{Lip}_{m_{0}}(\mathbb{C}^{n},\mathbb{C}^{n})$ and
$\mathcal{C}^{m_{0}}(\llangle{\tilde{P}}\rrangle)\leq\mathcal{C}^{m_{0}}({\tilde{P}})$.

  2. (2)

If $a\in\bar{B}_{R}(\mathbb{C}^{n})$, then the rate of convergence in (3.2)
depends not on $a$, but only on $R$.

This is the main lemma of deterministic averaging for vector fields. For its
proof see [13, Lemma 3.1].444 In fact, if $\tilde{P}$ is any locally Lipschitz
vector field, then $\llangle{\tilde{P}}\rrangle$ is well defined and locally
Lipschitz with the same locally Lipschitz constant (see discussion after
Assumption 2.1) as $\tilde{P}$. See in [13].

The averaged vector field $\llangle{\tilde{P}}\rrangle$ is invariant with
respect to transformations $\Phi_{\theta\Lambda}$:

######  Lemma 3.3.

For all $a\in\mathbb{C}^{n}$ and $\theta\in\mathbb{R}$,

$\big{(}\Phi_{\theta\Lambda}\big{)}_{*}\llangle{\tilde{P}}\rrangle(a)\equiv\Phi_{\theta\Lambda}\circ\llangle{\tilde{P}}\rrangle\circ\Phi_{-\theta\Lambda}(a)=\llangle{\tilde{P}}\rrangle(a).$

######  Proof.

For definiteness let $\theta>0$. For any ${T^{\prime}}>0$ we have:

$\llangle{\tilde{P}}\rrangle^{T^{\prime}}(\Phi_{-\theta\Lambda}(a))=\frac{1}{{T^{\prime}}}\int_{0}^{T^{\prime}}\Phi_{t\Lambda}\circ{\tilde{P}}(\Phi_{-t\Lambda}\circ\Phi_{-\theta\Lambda}(a))dt=\frac{1}{{T^{\prime}}}\int_{0}^{{T^{\prime}}}\Phi_{t\Lambda}\circ{\tilde{P}}(\Phi_{-(t+\theta)\Lambda}a)dt.$

Since $\Phi_{t\Lambda}=\Phi_{-\theta\Lambda}\circ\Phi_{(t+\theta)\Lambda}$,
then this equals

$\frac{1}{{T^{\prime}}}\Phi_{-\theta\Lambda}\Big{(}\int_{0}^{T^{\prime}}\Phi_{(t+\theta)\Lambda}\circ{\tilde{P}}(\Phi_{-(t+\theta)\Lambda}a)dt\Big{)}=\Phi_{-\theta\Lambda}\circ\llangle{\tilde{P}}\rrangle^{{T^{\prime}}}(a)+O\Big{(}\frac{1}{{T^{\prime}}}\Big{)}.$

Passing to the limit as ${T^{\prime}}\to\infty$ we obtain the assertion. ∎

The statement below asserts that the averaged vector filed
$\llangle{P}\rrangle$ is as least as smooth as ${P}$.

######  Proposition 3.4.

If ${P}\in C^{m}(\mathbb{C}^{n})$ for some $m\in\mathbb{N}$, then
$\llangle{P}\rrangle\in C^{m}(\mathbb{C}^{n})$ and
$|\llangle{P}\rrangle|_{C^{m}(\bar{B}_{R})}\leqslant|{P}|_{C^{m}(\bar{B}_{R})}$,
$\forall R>0$.

######  Proof.

We first fix any $R>0$. Then there exists a sequence of polynomial vector
fields $\\{P_{R,j},j\in\mathbb{N}\\}$ (cf. below item 3) in Section 3.1) such
that $|P_{R,j}-{P}|_{C^{m}(\bar{B}_{R})}\to 0$ as $j\to\infty$. An easy
calculation shows that

(3.3) $|\llangle P_{R,j}\rrangle^{T}-\llangle
P_{R,j}\rrangle|_{C^{m}(\bar{B}_{R})}\to 0\quad\text{as}\;\;T\to\infty,$

for each $j$. Since the transformations $\Phi_{t\Lambda}$ are unitary, then
differentiating the integral in (3.1) in $a$ we get that

(3.4)
$|\llangle{\tilde{P}}\rrangle^{T}|_{C^{m}(\bar{B}_{R})}\leqslant|{\tilde{P}}|_{C^{m}(\bar{B}_{R})}\quad\forall\,T>0,$

for any $C^{m}$-vector field $\tilde{P}$. Therefore

(3.5) $|\llangle P_{R,j}\rrangle^{T}-\llangle
P\rrangle^{T}|_{C^{m}(\bar{B}_{R})}\leq|P_{R,j}-P|_{C^{m}(\bar{B}_{R})}=:\kappa_{j}\to
0\quad\text{as}\;\;j\to\infty,\;\;\;\forall\,T>0.$

So

$|\llangle P_{R,j}\rrangle^{T}-\llangle
P_{R,k}\rrangle^{T}|_{C^{m}(\bar{B}_{R})}\leq 2\kappa_{j\wedge
k}\;\;\;\forall\,T>0.$

From this estimate and (3.3) we find that $|\llangle P_{R,j}\rrangle-\llangle
P_{R,k}\rrangle|_{C^{m}(\bar{B}_{R})}\leq 2\kappa_{j\wedge k}.$ Thus
$\\{\llangle P_{R,j}\rrangle\\}$ is a Cauchy sequence in $C^{m}(\bar{B}_{R})$.
So it $C^{m}$-converges to a limiting field $\llangle P_{R,\infty}\rrangle$.
As $P_{R,j}$ converges to $P$ in $C^{m}(\bar{B}_{R})$, then using again (3.4)
we find that $|\llangle
P_{R,\infty}\rrangle|_{C^{m}(\bar{B}_{R})}\leq|P|_{C^{m}(\bar{B}_{R})}.$ But
by Lemma 3.2 $\llangle P_{R,\infty}\rrangle$ must equal $\llangle P\rrangle$.
Since $R>0$ is arbitrary, the assertion of the proposition follows. ∎

Finally we note that if a vector field $P$ is Hamiltonian, then its averaging
$\llangle P\rrangle$ also is. Looking ahead we state the corresponding result
here, despite the averaging of functions $\langle\cdot\rangle$ is defined
below in Section 3.2.

######  Proposition 3.5.

If a locally Lipschitz vector field $P$ is Hamiltonian, i.e.
$P(z)=i\frac{\partial}{\partial\bar{z}}H(z)$ for some $C^{1}$-function $H$,
then $\llangle P\rrangle$ also is Hamiltonian and $\llangle
P\rrangle=i\frac{\partial}{\partial\bar{z}}\langle H\rangle.$

For a proof see [13, Theorem 5.2].

###  3.1. Calculating the averagings

1) The frequency vector $\Lambda=(\lambda_{1},\dots,\lambda_{n})$ is called
completely resonant if its components $\lambda_{j}$ are proportional to some
$\lambda>0$. I.e. if $\lambda_{j}/\lambda\in\mathbb{Z}$ for all $j$. In this
case all trajectories of the original linear system (2.1) are periodic, the
operator $\Phi_{t\Lambda}$ is $2\pi/\lambda$-periodic in $t$ and so

(3.6)
$\llangle{\tilde{P}}\rrangle(a)=\llangle{\tilde{P}}\rrangle^{2\pi/\lambda}(a)=\frac{\lambda}{{2\pi}}\
\int_{0}^{2\pi/\lambda}(\Phi_{t\Lambda})_{*}{\tilde{P}}(a)dt.$

Completely resonant linear systems (2.1) and their perturbations (1.1) often
occur in applications. In particular – in non-equilibrium statistical physics.
There the dimension $D=2n$ is large, all $\lambda_{j}$’s are equal, and the
Wiener process $W(t)$ in (1.1) may be very degenerate (it may have only two
non-zero components). E.g. see [9], where more references may be found.

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

(3.7) $\sum_{i=1}^{n}m_{j}\lambda_{j}\neq
0\qquad\forall(m_{1},\dots,m_{n})\in\mathbb{Z}^{n}\setminus\\{0\\}$

(that is, the real numbers $\lambda_{j}$ are rationally independent). Then

(3.8)
$\llangle{\tilde{P}}\rrangle(a)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{T}^{n}}(\Phi_{w})_{*}\tilde{P}(a)\,dw,\qquad\mathbb{T}^{n}=\mathbb{R}^{n}/(2\pi\mathbb{Z}^{n}).$

Indeed, if ${\tilde{P}}$ is a polynomial vector field, then (3.8) easily
follows from (3.2) by a direct component-wise calculation. The general case is
a consequence of this result since any vector field may be approximated by
polynomial fields. Details are left to the reader (cf. Lemma 3.5 in [13],
where ${\tilde{P}}^{res}$ equals to the r.h.s. of (3.8) if the vector
$\Lambda$ is non-resonant).

The r.h.s. of (3.8) obviously is invariant with respect to all rotations
$\Phi_{w^{\prime}}$, so it depends not on the vector $a$, but only on the
corresponding torus

(3.9) $\\{z\in\mathbb{C}^{n}:I_{j}(z)=I_{j}(a)\quad\forall\,j\\}$

(see (1.2)) to which $a$ belongs,

(3.10)
$\big{(}\Phi_{w}\big{)}_{*}\llangle{\tilde{P}}\rrangle(a)\equiv\llangle{\tilde{P}}\rrangle(a)\qquad\forall
w\in\mathbb{C}^{n}\;\;\;\text{if $\Lambda$ is non-resonant}.$

See below Section 6 for discussing equations (1.1) with non-resonant vectors
$\Lambda$.

3) If the field $\tilde{P}$ in (3.2) is polynomial, i.e.

(3.11) $\tilde{P}_{j}(a)=\sum_{|\alpha|,|\beta|\leq
N}C_{j}^{\alpha,\beta}a^{\alpha}\bar{a}^{\beta},\qquad j=1,\dots,n,$

for some $N\in\mathbb{N}$, where $\alpha,\beta\in\mathbb{Z}_{+}^{n}$,
$a^{\alpha}=\prod a_{j}^{\alpha_{j}}$ and $|\alpha|=\sum|\alpha_{j}|$, then
$\llangle{\tilde{P}}\rrangle=\tilde{P}^{res}$. Here $\tilde{P}^{res}$ is a
polynomial vector field such that for each $j$, $\tilde{P}^{res}_{j}(a)$ is
given by the r.h.s. of (3.11), where the summation is taken over all
$|\alpha|,|\beta|\leq N$, satisfying $\Lambda\cdot(\alpha-\beta)=\lambda_{j}.$
This easily follows from an explicit calculation of the integral in (3.1) (see
[13, Lemma 3.5]).

###  3.2. Averaging of functions

Similarly to definition (3.2), for a locally Lipschitz function
$f\in\text{Lip}_{m}(\mathbb{C}^{n},\mathbb{C})$, $m\geq 0$, we define its
averaging with respect to a frequency vector $\Lambda$ as

(3.12) $\langle
f\rangle(a)=\lim_{{T^{\prime}}\to\infty}\frac{1}{{T^{\prime}}}\int_{0}^{T^{\prime}}f(\Phi_{-t\Lambda}a)dt,\quad
a\in\mathbb{C}^{n}.$

Then by the same argument as above we obtain:

######  Lemma 3.6.

If $f\in\text{Lip}_{m}(\mathbb{C}^{n},\mathbb{C})$, then

  1. (1)

The limit (3.12) exists for every $a$, and for
$a\in\bar{B}_{R}(\mathbb{C}^{n})$ the rate of convergence in (3.12) depends
not on $a$, but only on $R$.

  2. (2)

$\langle f\rangle\in\text{Lip}_{m}(\mathbb{C}^{n};\mathbb{C})$ and
$\mathcal{C}^{m}(\langle f\rangle)\leq\mathcal{C}^{m}(f)$.

  3. (3)

If $f$ is $C^{m}$-smooth for some $m\in\mathbb{N}$, then $\langle f\rangle$
also is, and the $C^{m}$-norm of the latter is bounded by the $C^{m}$-norm of
the former.

  4. (4)

The function $\langle f\rangle$ commutes with the operators
$\Phi_{\theta\Lambda}$, $\theta\in\mathbb{R}$, in the sense that $\langle
f\circ\Phi_{\theta\Lambda}\rangle=\langle
f\rangle\circ\Phi_{\theta\Lambda}=\langle f\rangle$.

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

(3.13)
$\langle{f}\rangle(a)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{T}^{n}}{f}(\Phi_{-w}a)\,dw.$

The r.h.s. of (3.13) is the averaging of function $f$ in angles. It is
constant on the tori (3.9).

##  4\. Effective equation and the averaging theorem

In this section we show that the limiting measure $Q_{0}$ in (2.14) is
independent of the choice of the sequence ${\varepsilon}^{\prime}_{l}\to 0$,
so $\mathcal{D}(a^{\varepsilon})\rightharpoonup Q_{0}\;\text{as}\
{\varepsilon}\to 0,$ and represent $Q_{0}$ as the law of a solution of an
auxiliary effective equation. The drift in this equation is the averaged drift
in eq. (2.6). Now we construct its dispersion.

The diffusion matrix for equation (2.11) is the $n\times n$ complex matrix

$\mathcal{A}^{\varepsilon}(\tau)=(\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi)\cdot(\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi)^{*}.$

Denoting

(4.1)
$\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi=\big{(}e^{i\tau{\varepsilon}^{-1}\lambda_{l}}\Psi_{lj}\big{)}=:\big{(}\psi^{\varepsilon}_{lj}(\tau)\big{)}=\psi^{\varepsilon}(\tau),$

we have

$\mathcal{A}^{\varepsilon}_{kj}(\tau)=\sum_{l=1}^{n_{1}}\psi^{\varepsilon}_{kl}(\tau)\bar{\psi^{\varepsilon}_{jl}}(\tau)=e^{i\tau{\varepsilon}^{-1}(\lambda_{k}-\lambda_{j})}\sum_{l=1}^{n_{1}}\Psi_{kl}\bar{\Psi}_{jl}.$

So for any $\tau>0$,

$\frac{1}{\tau}\int_{0}^{\tau}\mathcal{A}^{\varepsilon}_{kj}(s)ds=\big{(}\sum_{l=1}^{n_{1}}\Psi_{kl}\bar{\Psi}_{jl}\big{)}\frac{1}{\tau}\int_{0}^{\tau}e^{is{\varepsilon}^{-1}(\lambda_{k}-\lambda_{j})}ds,$

and we immediately see that

(4.2)
$\frac{1}{{\tau}}\int_{0}^{\tau}\mathcal{A}^{\varepsilon}_{kj}(\tau)d\tau\to
A_{kj}\quad\text{as}\quad{\varepsilon}\to 0,$

where

(4.3)
$A_{kj}=\begin{cases}\sum_{l=1}^{n_{1}}\Psi_{kl}\bar{\Psi}_{jl},&\text{if
}\lambda_{k}=\lambda_{j},\\\ 0,&\text{otherwise}.\end{cases}$

Clearly, $A_{kj}=\bar{A}_{jk}$, so $A$ is a Hermitian matrix. If
$\lambda_{k}\neq\lambda_{j}$ for $k\neq j$, then

(4.4) $A=\text{diag}\\{b_{1},\dots,b_{n}\\},\quad
b_{k}=\sum_{l=1}^{n_{1}}|\Psi_{kl}|^{2}.$

For any vector $\xi\in\mathbb{C}^{n}$ we get from (4.2) that $\langle
A\xi,\xi\rangle\geqslant 0$ since obviously
$\langle\mathcal{A}^{\varepsilon}(\tau)\xi,\xi\rangle=|\psi^{\varepsilon}(\tau)\xi|^{2}\geqslant
0$ for each ${\varepsilon}$. Therefore $A$ is a nonnegative Hermitian matrix,
and there exists another nonnegative Hermitian matrix $B$ (called the
principal square root of $A$) such that $BB^{*}=B^{2}=A$. The matrix $B$ is
non-degenerate if $\Psi$ is.

######  Example 4.1.

If $\Psi$ is a diagonal matrix diag$\,\\{\psi_{1},\dots,\psi_{n}\\}$,
$\psi_{j}\in\mathbb{R}$, then $\mathcal{A}^{\varepsilon}(\tau)=|\Psi|^{2}.$ In
this case $A=|\Psi|^{2}$ and
$B=|\Psi|=$diag$\,\\{|\psi_{1}|,\dots,|\psi_{n}|\\}$.

In fact, it is not needed that the matrix $B$ is square and Hermitian, and the
argument below remains true if we take for $B$ any complex $n\times N$-matrix
(with any $N\in\mathbb{N}$), satisfying the equation

$BB^{*}=A.$

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

(4.5) $da_{k}-\llangle
P\rrangle_{k}(a)d\tau=\sum_{l=1}^{n}B_{kl}d\beta^{c}_{l},\quad k=1,\dots,n.$

Here the matrix $B$ is as above and $\llangle P\rrangle$ is the resonant
averaging of the vector field $P$. We will usually consider this equation with
the same initial condition as equations (2.6) and (2.11):

(4.6) $a(0)=v_{0}.$

Since the vector field $\llangle P\rrangle$ is locally Lipschitz and the
dispersion matrix $B$ is constant, then a strong solution of eq. (4.5), (4.6),
if exists, is unique.

Note that the effective dispersion $B$ in (4.5) is a square root of an
explicit matrix, and due to item 3) of Section 3.1 if the vector field $P(v)$
is polynomial, then the effective drift $\llangle P\rrangle(a)$ also is given
by an explicit formula.

######  Proposition 4.2.

The limiting probability measure $Q_{0}$ in (2.14) is a weak solution of
effective equation (4.5), (4.6).

We recall that a measure $Q\in\mathcal{P}(C([0,T],\mathbb{C}^{n}))$ is a weak
solution of equation (4.5), (4.6) if $Q=\mathcal{D}(\tilde{a})$, where the
random process $\tilde{a}(\tau)$, $0\leq\tau\leq T$, is a weak solution of
(4.5), (4.6). 555Concerning weak solutions of SDEs e.g. see e.g. [14, Section
5.3].

A proof of this result is preceded by a number of lemmas. Till the end of this
section we assume Assumption 2.1. As in Section 3 we denote

(4.7)
$Y(a;\tau{\varepsilon}^{-1}):=(\Phi_{\tau{\varepsilon}^{-1}\Lambda})_{*}P(a).$

Then equation (2.11) for $a^{\varepsilon}$ reads as

(4.8)
$da^{{\varepsilon}}(\tau)-Y(a^{{\varepsilon}},\tau{\varepsilon}^{-1})d\tau=\Phi_{\tau{\varepsilon}^{-1}\Lambda}\Psi
d\beta^{c}(\tau).$

Denote

(4.9)
$\tilde{y}(a,\tau{\varepsilon}^{-1})=Y(a,\tau{\varepsilon}^{-1})-\llangle
P\rrangle(a)=(\Phi_{\tau{\varepsilon}^{-1}\Lambda})_{*}P(a)-\langle\\!\langle
P\rangle\\!\rangle(a).$

The following key lemma shows that integrals of
$\tilde{y}(a^{\varepsilon},\tau{\varepsilon}^{-1})$ with respect to $\tau$
become small with ${\varepsilon}$, uniformly in the segment of integrating.

######  Lemma 4.3.

For a solution $a^{\varepsilon}(\tau)$ of equation (2.11), (2.12) we have

${\mathbf{E}}\max_{0\leqslant\tau\leqslant
T}\Big{|}\int_{0}^{\tau}\tilde{y}(a^{{\varepsilon}}(s),s{\varepsilon}^{-1})ds\Big{|}\to
0\quad\text{as}\quad{\varepsilon}\to 0.$

The lemma is proved in Subsection 4.1.

Now let us introduce a natural filtered measurable space for the problem we
consider,

(4.10)
$(\tilde{\Omega},\mathcal{B},\\{\mathcal{B}_{\tau},0\leqslant\tau\leqslant
T\\}),$

where $\tilde{\Omega}$ is the Banach space
$C([0,T],\mathbb{C}^{n})=\\{a:=a(\cdot)\\}$, $\mathcal{B}$ is its Borel
$\sigma$-algebra and $\mathcal{B}_{\tau}$ is the sigma-algebra, generated by
the r.v.’s $\\{a(s):0\leqslant s\leqslant\tau)\\}$. Consider the process on
$\tilde{\Omega}$, defined by the l.h.s. of (4.5)

(4.11) $N^{\llangle P\rrangle}(\tau;a)=a(\tau)-\int_{0}^{\tau}{\llangle
P\rrangle}(a(s))ds,\qquad a\in\tilde{\Omega},\;\tau\in[0,T].$

Note that for any $0\leq\tau\leq T$, $N^{\llangle P\rrangle}(\tau;\cdot)$ is a
$\mathcal{B}_{\tau}$-measurable continuous functional on
$C([0,T];\mathbb{C}^{n})$.

######  Lemma 4.4.

The random process $N^{\llangle P\rrangle}(\tau;a)$ is a martingale on the
space (4.10) with respect to the limiting measure $Q_{0}$ in (2.14).

######  Proof.

Let us fix a $\tau\in[0,T]$ and take a $\mathcal{B}_{\tau}$-measurable
function $f^{\tau}\in C_{b}(\tilde{\Omega})$. We will show that

(4.12) ${\mathbf{E}}^{Q_{0}}\Big{(}N^{\llangle
P\rrangle}(t;a)f^{\tau}(a)\Big{)}=\mathbf{E}^{Q_{0}}\Big{(}N^{\llangle
P\rrangle}(\tau;a)f^{\tau}(a)\Big{)}\quad\text{for any}\;\;\tau\leqslant
t\leqslant T,$

which would imply the assertion. To establish this let us first consider the
process

$N^{Y,{\varepsilon}}(\tau;a^{\varepsilon}):=a^{{\varepsilon}}(\tau)-\int_{0}^{\tau}Y(a^{{\varepsilon}},s{\varepsilon}^{-1})ds,$

who is a martingale in view of (4.8). As

$N^{Y,{\varepsilon}}(\tau;a^{\varepsilon})-N^{{\llangle
P\rrangle}}(\tau;a^{\varepsilon})=\int_{0}^{\tau}\big{[}{\llangle
P\rrangle}(a^{\varepsilon}(s))-Y(a^{\varepsilon}(s),s\varepsilon^{-1})\big{]}ds,$

then by Lemma 4.3,

(4.13) $\max_{0\leq\tau\leq
T}\mathbf{E}\big{|}N^{Y,{\varepsilon}}(\tau;a^{\varepsilon})-N^{{\llangle
P\rrangle}}(\tau;a^{\varepsilon})\big{|}=o_{\varepsilon}(1).$

Here and below in this proof $o_{\varepsilon}(1)$ stands for a quantity which
goes to zero with $\varepsilon$. Since $N^{Y,{\varepsilon}}$ is a martingale,
then by this relation

$\begin{split}{\mathbf{E}}\big{(}N^{{\llangle
P\rrangle}}(t;a^{{\varepsilon}})f^{\tau}(a^{{\varepsilon}})\big{)}+o_{\varepsilon}(1)&={\mathbf{E}}\big{(}N^{Y,{\varepsilon}}(t;a^{{\varepsilon}})f^{\tau}(a^{{\varepsilon}})\big{)}\\\
&={\mathbf{E}}\big{(}N^{Y,{\varepsilon}}({\tau};a^{{\varepsilon}})f^{\tau}(a^{{\varepsilon}})\big{)}={\mathbf{E}}\big{(}N^{{\llangle
P\rrangle}}({\tau};a^{{\varepsilon}})f^{\tau}(a^{{\varepsilon}})\big{)}+o_{\varepsilon}(1).\end{split}$

So

(4.14) $\mathbf{E}^{Q_{\varepsilon}}\big{(}N^{{\llangle
P\rrangle}}(t;a)f^{\tau}(a)-N^{{\llangle
P\rrangle}}({\tau};a)f^{\tau}(a)\big{)}=o_{\varepsilon}(1).$

To get (4.12) we will pass in this relation to a limit as $\varepsilon\to 0$.
For this end for $M>0$ consider the function

$G_{M}(t)=\begin{cases}t,&\text{if }|t|\leq M,\\\
M\,\text{sgn}\,t,&\text{otherwise}.\end{cases}$

Since by Assumption 2.1 and Lemma 3.2

${\mathbf{E}}^{Q_{{\varepsilon}}}\big{(}\sup_{\tau\in[0,T]}|N^{\llangle
P\rrangle}(\tau;a)|^{2}\Big{)}\leqslant{\mathbf{E}}^{Q_{{\varepsilon}}}\big{(}C_{P}(1+\sup_{\tau\in[0,T]}|a(\tau)|^{2(m_{0}\vee
1)})\Big{)}\leqslant C_{P,m_{0}}(|v_{0}|),$

then for any $\varepsilon$ we have

(4.15) $\mathbf{E}^{Q_{\varepsilon}}\big{|}(1-G_{M})\circ\big{(}N^{\llangle
P\rrangle}(t;a)f^{\tau}(a)-N^{\llangle
P\rrangle}(\tau;a)f^{\tau}(a)\big{)}\big{|}\leq CM^{-1}.$

As $Q_{\varepsilon^{\prime}_{l}}\rightharpoonup Q_{0}$, then by Fatou lemma
this estimate stays true for $\varepsilon=0$.

Relations (4.14) and (4.15) show that

$\mathbf{E}^{Q_{\varepsilon}}\big{(}G_{M}\circ(N^{\llangle
P\rrangle}(t;a)f^{\tau}(a)-N^{\llangle
P\rrangle}(\tau;a)f^{\tau}(a))\big{)}=o_{\varepsilon}(1)+o_{M^{-1}}(1).$

From this and convergence (2.14) we derive the relation

$\mathbf{E}^{Q_{0}}\big{(}G_{M}\circ(N^{\llangle
P\rrangle}(t;a)f^{\tau}(a)-N^{\llangle
P\rrangle}(\tau;a)f^{\tau}(a))\big{)}=o_{M^{-1}}(1),$

which jointly with estimate (4.15)ε=0 imply (4.12) when we send $M$ to
$\infty$. The lemma is proved. ∎

######  Definition 4.5.

A measure $Q$ on the space (4.10) is called a solution of the martingale
problem for effective equation (4.5) with the initial condition (4.6) if
$a(0)=v_{0}$ $Q$–a.s. and

1) the process $\\{N^{\llangle
P\rrangle}(\tau;a)\in\mathbb{C}^{n},\;\tau\in[0,T]\\}$ (see (4.11)) is a
vector–martingale on the filtered space (4.10) with respect to the measure
$Q$;

2) for any $k,j=1,\dots,n$ the process

(4.16) $N^{\llangle P\rrangle}_{k}(\tau;a)\overline{N^{\llangle
P\rrangle}_{j}}(\tau;a)-2\int_{0}^{\tau}(BB^{*})_{kj}ds,\;\;\;\;\tau\in[0,T]$

(where $BB^{*}=A$) is a martingale on the space (4.10) with respect to the
measure $Q$, as well as the process $N^{\llangle
P\rrangle}_{k}(\tau;a){N^{\llangle P\rrangle}_{j}}(\tau;a).$666 That is,
$\langle N^{\llangle P\rrangle}_{k}(\tau;a),\overline{N^{\llangle
P\rrangle}_{j}}(\tau;a)\rangle(\tau)=2\int_{0}^{\tau}(BB^{*})_{kj}$ and
$\langle N^{\llangle P\rrangle}_{k}(\tau;a),{N^{\llangle
P\rrangle}_{j}}(\tau;a)\rangle(\tau)=0.$ See Appendix B.

This is a classical definition, written in complex coordinates. See [24] and
[14, Section 5.4], where we profited from [14, Remark 4.12 ] and the result of
[14, Problem 4.13] since by Lemma 3.2 the vector field ${\llangle P\rrangle}$
in (4.5) is locally Lipschitz.

We have

######  Lemma 4.6.

The limiting measure $Q_{0}$ in (2.14) is a solution of the martingale problem
for effective equation (4.5), (4.6).

######  Proof.

Since condition 1) in Definition 4.5 has been verified in Lemma 4.4, it
remains to check condition 2). For the second term in (4.16) we have, as
${\varepsilon}\to 0$,

(4.17)
$\int_{0}^{\tau}\big{(}\psi^{\varepsilon}(s)(\psi^{\varepsilon}(s))^{*}\big{)}_{kj}ds=\int_{0}^{\tau}e^{i{\varepsilon}^{-1}(\lambda_{k}-\lambda_{j})s}(\psi\psi^{*})_{kj}ds\to\tau
A_{kj},$

where the matrix $(A_{kj})$ is given by (4.3). Let us pass to the first term.
Since by (2.11) and (4.1)

$N^{Y,{\varepsilon}}(\tau)=v_{0}+\int_{0}^{\tau}\psi^{\varepsilon}(s)d\beta^{c}(s),\qquad\psi^{\varepsilon}_{lj}(s)=e^{is{\varepsilon}^{-1}\lambda_{l}}\Psi_{lj},$

then by the complex Itô formula (see Appendix C) and Assumption 2.1, for any
$k,j\in\\{1,\dots,n\\}$ the process

(4.18)
$N_{k}^{Y,{\varepsilon}}(\tau)\overline{N_{j}^{Y,{\varepsilon}}}(\tau)-2\int_{0}^{\tau}(\psi^{\varepsilon}(s)(\psi^{\varepsilon}(s))^{*})_{kj}ds,$

is a martingale. As when verifying 1), we will compare (4.16) with (4.18). To
do this let us consider

$\begin{split}N_{k}^{\llangle
P\rrangle}(\tau;a^{\varepsilon})&\overline{N_{j}^{{\llangle
P\rrangle}}}(\tau;a^{\varepsilon})-N_{k}^{Y,{\varepsilon}}(\tau;a^{\varepsilon})\overline{N_{j}^{Y,{\varepsilon}}}(\tau;a^{\varepsilon})\\\
&=\big{(}a_{k}^{\varepsilon}(\tau)-\int_{0}^{\tau}{\llangle
P\rrangle}_{k}(a^{\varepsilon}(s))\,ds\big{)}\big{(}\bar{a}_{j}^{\varepsilon}(\tau)-\int_{0}^{\tau}\bar{\llangle
P\rrangle}_{j}(a^{\varepsilon}(s))\,ds\big{)}\\\
&-\big{(}a_{k}^{\varepsilon}(\tau)-\int_{0}^{\tau}Y_{k}(a^{\varepsilon}(s),s\varepsilon^{-1})\,ds\big{)}\big{(}\bar{a}_{j}^{\varepsilon}(\tau)-\int_{0}^{\tau}\bar{Y}_{j}(a^{\varepsilon}(s),s\varepsilon^{-1})\,ds\big{)}=:M_{kj}(a^{\varepsilon},\tau).\end{split}$

Closely repeating the proof of (4.13) we get that

$\sup_{0\leq\tau\leq
T}\mathbf{E}\big{|}M_{kj}(a^{\varepsilon};\tau)\big{|}=o_{\varepsilon}(1)\quad\text{as}\;\;\varepsilon\to
0.$

Since (4.18) is a martingale, then this relation and (4.17) imply that (4.16)
is a martingale due to the same reasoning by which relations (4.13) and the
fact that $N^{Y,\varepsilon}(\tau;a^{\varepsilon})$ is a martingale imply that
$N^{\llangle P\rrangle}(\tau;a)$ also is one. To pass to a limit as
$\varepsilon\to 0$ the proof uses that the random variables like
$N_{k}^{Y,{\varepsilon}}(\tau;a^{\varepsilon})\overline{N_{j}^{Y,{\varepsilon}}}(\tau;a^{\varepsilon})$
are integrable uniformly in $\varepsilon>0$ due to Assumption 2.1, where
$m^{\prime}_{0}>m_{0}$.

Similarly for any $k$ and $j$, the process $N_{k}^{\llangle
P\rrangle}(\tau){N_{j}^{\llangle P\rrangle}}(\tau)$ also is a martingale. The
lemma’s assertion is established. ∎

Now we can prove Proposition 4.2:

######  Proof.

[of Proposition 4.2] It is well known that a solution of the martingale
problem for a stochastic differential equation is its weak solution. Instead
of referring to a corresponding theorem (see [24] or [14, Section 5.4]), again
following [16], we give a short direct proof, based on another strong result
from the stochastic calculus. By Lemma 4.6 and the martingale representation
theorem for complex processes (see Appendix B), we know that there exists an
extension $(\hat{\Omega},\hat{\mathcal{B}},\hat{\mathbf{P}})$ of the
probability space $(\tilde{\Omega},\mathcal{B},Q_{0})$ and on it exist
standard independent complex Wiener processes
$\beta^{c}_{1}(\tau),\dots,\beta^{c}_{n}(\tau)$ such that

$da_{j}(\tau)-{\llangle
P\rrangle}_{j}(a)d\tau=\sum_{l=1}^{n}B_{jl}d\beta^{c}_{l}(\tau),\quad
j=1,\dots,n,$

where the dispersion $B$ is a nonnegative Hermitian matrix, satisfying
$BB^{*}=A$. Therefore the measure $Q_{0}$ is a weak solution of effective
equation (4.5). We thus proved the assertion of the proposition. ∎

By Lemma 3.2, in effective equation (4.5) the drift term ${\llangle
P\rrangle}$ is locally Lipschitz. So its strong solution (if exists) is
unique. By Proposition 4.2 the measure $Q_{0}$ is a weak solution of eq.
(4.5). Hence, by the Yamada-Watanabe theorem [14, Section 5.3.D], [24, Chapter
8] a strong solution for the effective equation exists, and its weak solution
is unique. Therefore the limit $Q_{0}=\lim_{{\varepsilon}_{l}\to
0}Q_{{\varepsilon}_{l}}$ does not depend on the sequence ${\varepsilon}_{l}\to
0$. So the convergence holds as $\varepsilon\to 0$ and we thus have
established

######  Theorem 4.7.

For any $v_{0}\in\mathbb{C}^{n}$ the solution $a^{\varepsilon}(\tau;v_{0})$ of
problem (2.11), (2.12) satisfies

(4.19) $\mathcal{D}(a^{{\varepsilon}}(\cdot;v_{0}))\rightharpoonup Q_{0}\text{
in }\mathcal{P}(C([0,T],\mathbb{C}^{n}))\quad\text{ as }{\varepsilon}\to 0,$

where $Q_{0}$ is the law of a unique weak solution $a^{0}(\tau;v_{0})$ of
effective equation (4.5), (4.6).

######  Remark 4.8.

1) A straightforward analysis of the theorem’s proof shows that it goes
without changes if $a^{{\varepsilon}}(\tau)$ solves eq. (2.11) with an initial
data $v_{\varepsilon 0}$ which converges to $v_{0}$ when $\varepsilon\to 0$.
So

(4.20) $\mathcal{D}(a^{{\varepsilon}}(\cdot;v_{\varepsilon 0}))\rightharpoonup
Q_{0}\text{ in }\mathcal{P}(C([0,T],\mathbb{C}^{n}))\quad\text{ as
}{\varepsilon}\to 0,\;\;\text{if}\;\;v_{\varepsilon 0}\to
v_{0}\;\;\text{when}\;\;\varepsilon\to 0.$

2) Denoting as before
$\mathcal{D}(a^{\varepsilon}(\cdot;v_{0}))=Q_{\varepsilon}\in\mathcal{P}({C([0,T];\mathbb{C}^{n})})$
we use the Skorokhod representation theorem (see [2, Section 6]) to find a
sequence $\varepsilon_{j}\to 0$ and processes $\xi_{j}(\tau)$, $0\leq\tau\leq
T$, $j=0,1,\dots$, such that $\mathcal{D}(\xi_{0})=Q_{0},$
$\mathcal{D}(\xi_{j})=Q_{\varepsilon_{j}},$ and $\xi_{j}\to\xi_{0}$ in
$C([0,T];\mathbb{C}^{n})$, a.s. Then (2.7) and Fatou’s lemma imply that

(4.21)
$\mathbf{E}\|a^{0}\|^{2m_{0}^{\prime}}_{C([0,T];\mathbb{C}^{n})}=\mathbf{E}^{Q_{0}}\|a\|^{2m_{0}^{\prime}}_{C([0,T];\mathbb{C}^{n})}=\mathbf{E}\|\xi_{0}\|^{2m_{0}^{\prime}}_{C([0,T];\mathbb{C}^{n})}\leq
C_{m^{\prime}_{0}}(|v_{0}|,T).$

The result of Theorem 4.7 admits an immediate generalisation to the case when
the initial data $v_{0}$ in (2.12) is a random variable:

######  Amplification 4.9.

Let $v_{0}$ be a r.v., independent from the Wiener process $\beta^{c}(\tau)$.
Then still the convergence (4.19) holds.

######  Proof.

It suffices to establish (4.19) when $a^{\varepsilon}$ is a weak solution of
the problem. Now, let
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbf{P}^{\prime})$ be another
probability space and $\xi_{0}^{\omega^{\prime}}$ be a r.v. on
$\Omega^{\prime}$, distributed as $v_{0}$. Then
$a^{\varepsilon\omega}(\tau;\xi_{0}^{\omega^{\prime}})$ is a weak solution of
(2.11), (2.12), defined on the probability space
$\Omega^{\prime}\times\Omega$. Take $f$ to be a bounded continuous function on
$C([0,T],\mathbb{C}^{n})$. Then by the theorem above, for each
$\omega^{\prime}\in\Omega^{\prime}$

$\lim_{\varepsilon\to
0}\mathbf{E}^{\Omega}f(a^{\varepsilon\omega}(\cdot;\xi_{0}^{\omega^{\prime}}))=\mathbf{E}^{\Omega}f(a^{0\omega}(\cdot;\xi_{0}^{\omega^{\prime}})).$

Since $f$ is bounded, then by the Lebesgue dominated convergence theorem we
have

$\lim_{\varepsilon\to
0}\mathbf{E}f(a^{\varepsilon}(\cdot;v_{0}))=\lim_{\varepsilon\to
0}\mathbf{E}^{\Omega^{\prime}}\mathbf{E}^{\Omega}f(a^{\varepsilon\omega}(\cdot;\xi_{0}^{\omega^{\prime}}))=\mathbf{E}^{\Omega^{\prime}}\mathbf{E}^{\Omega}f(a^{0\omega}(\cdot;\xi_{0}^{\omega^{\prime}}))=\mathbf{E}f(a^{0}(\cdot;v_{0})).$

This implies the required convergence (4.19). ∎

The convergence, stated in the last amplification, holds uniformly in the
class of random initial data $v_{0}$, a.s. bounded by a fixed constant. To
state the result we have to introduce a distance in the space of measures.

######  Definition 4.10.

Let $M$ be a Polish (i.e. a complete and separable) metric space. For any two
measures $\mu_{1},\mu_{2}\in\mathcal{P}(M)$ we define the dual-Lipschitz
distance between them as

$\|\mu_{1}-\mu_{2}\|_{L,M}^{*}:=\sup_{f\in C_{b}(M),\,|f|_{L}\leqslant
1}\Big{|}\langle f,\mu_{1}\rangle-\langle f,\mu_{2}\rangle\Big{|}\leq 2,$

where $|f|_{L}=|f|_{L,M}=\,$Lip$\,f+\|f\|_{C(M)}$.

In the definition and below we denote

(4.22) $\langle f,\mu\rangle:=\int_{M}f(m)\mu(dm).$

######  Example 4.11.

Consider the Polish spaces $C([0,T],\mathbb{C}^{n})$ and $\mathbb{C}^{n}$ and
the mappings

$\Pi_{t}:C([0,T],\mathbb{C}^{n})\to\mathbb{C}^{n},\quad a(\cdot)\mapsto
a(t),\qquad 0\leq t\leq T.$

Noting that
$|f\circ\Pi_{t}|_{L,\mathbb{C}^{n}}\leq|f|_{L,C([0,T],\mathbb{C}^{n})}$ for
each $t$ we get that

(4.23)
$\|\Pi_{t}\circ\mu_{1}-\Pi_{t}\circ\mu_{2}\|_{L,\mathbb{C}^{n}}^{*}\leq\|\mu_{1}-\mu_{2}\|_{L,C([0,T],\mathbb{C}^{n})}^{*}$

for all $\mu_{1},\mu_{2}\in\mathcal{P}(C([0,T],\mathbb{C}^{n}))$ and all
$0\leq t\leq T$ (where $\Pi_{t}\circ\mu_{j}\in\mathcal{P}(\mathbb{C}^{n})$
stands for the image of $\mu_{j}$ under the mapping $\Pi_{t}$).

The dual-Lipschitz distance converts $\mathcal{P}(M)$ to a complete metric
space and induces on it a topology, equivalent to the weak convergence of
measures, e.g. see [8, Section 11.3] and [5, Section 1.7]).

######  Proposition 4.12.

Under the assumption of Amplification 4.9 let the r.v. $v_{0}$ be such that
$|v_{0}|\leq R$ a.s., for some $R>0$. Then the rate of convergence in (4.19)
with respect to the dual-Lipschitz distance depends only on $R$.

######  Proof.

The proof of Amplification 4.9 shows that it suffices to verify that for a
non-random initial data $v_{0}\in\bar{B}_{R}(\mathbb{C}^{n})$ the rate of
convergence in (4.19) depends only on $R$. Assume the opposite. Then there
exist a $\delta>0$, a sequence $\varepsilon_{j}\to 0$ and vectors
$v_{j}\in\bar{B}_{R}(\mathbb{C}^{n})$ such that

(4.24)
$\|\mathcal{D}(a^{\varepsilon_{j}}(\cdot;v_{j}))-\mathcal{D}(a^{0}(\cdot;v_{j}))\|_{L,C([0,T],\mathbb{C}^{n})}^{*}\geqslant\delta.$

By the same argument as in the proof of Lemma 2.2, we know that the two sets
of probability measures $\\{\mathcal{D}(a^{\varepsilon_{j}}(\cdot;v_{j}))\\}$
and $\\{\mathcal{D}(a^{0}(\cdot;v_{j}))\\}$ are pre-compact in
$C([0,T],\mathbb{C}^{n})$. Therefore, there exists a sequence $k_{j}\to\infty$
such that $\varepsilon_{k_{j}}\to 0$, $v_{k_{j}}\to v_{0}$ and

$\mathcal{D}(a^{\varepsilon_{k_{j}}}(\cdot;v_{k_{j}}))\rightharpoonup\tilde{Q}_{0},\;\;\;\mathcal{D}(a^{0}(\cdot;v_{k_{j}}))\rightharpoonup
Q_{0}\;\text{in}\;\mathcal{P}(C([0,T],\mathbb{C}^{n})).$

Then

(4.25)
$\;\|\tilde{Q}_{0}-Q_{0}\|_{L,C([0,T],\mathbb{C}^{n})}^{*}\geqslant\delta.$

Since in the well-posed eq. (4.5) the drift and dispersion are locally
Lipschitz, then the law $\mathcal{D}(a^{0}(\cdot;v^{\prime}))$ is continuous
with respect to the initial condition $v^{\prime}$.777This is well known, and
can be easily proved using the estimate in Remark 4.8.2).  Therefore $Q_{0}$
is the unique weak solution of the effective equation (4.5) with initial
condition $a^{0}(0)=v_{0}$. By (4.20) the measure $\tilde{Q}_{0}$ also is a
weak solution of the problem (4.5), (4.6). Hence, $Q_{0}=\tilde{Q}_{0}$. This
contradicts (4.25) and proves the assertion. ∎

We continue with an obvious application of Theorem 4.7 to solutions
$v^{\varepsilon}(\tau;v_{0})$ of the original eq. (2.6). Consider the action-
mapping $(z_{1},\dots,z_{n})\mapsto(I_{1},\dots,I_{n})=:I$ (see (1.2)). Since
the interaction representation (2.8) does not change the actions, then we have
from the theorem that

######  Corollary 4.13.

For any $v_{0}$, as ${\varepsilon}\to 0$

(4.26)
$\mathcal{D}(I\big{(}v^{{\varepsilon}}(\cdot;v_{0}))\big{)}\rightharpoonup
I\circ\mathcal{D}(a(\cdot;v_{0}))\quad\text{in}\quad\mathcal{P}(C([0,T],\mathbb{R}_{+}^{n})),$

where $a(\cdot;v_{0})$ is a unique weak solution of effective equation (4.5),
(4.6).

######  Example 4.14.

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

###  4.1. Proof of Lemma 4.3

In this subsection we denote by $\mathcal{H}_{k}(r;c_{1},\dots)$,
$k=1,2,\dots$, non-negative functions of $r>0$ which go to zero with $r$ and
depend on parameters $c_{1},\dots$ (dependence of $\mathcal{H}_{k}$’s on $T$
and $P$ is not indicated). Also, for an event $Q$ we denote
${\mathbf{E}}_{Q}f(\xi)={\mathbf{E}}\mathbf{1}_{Q}f(\xi)$.

For any ${M_{1}}\geqslant 1$ we set

$\mathcal{E}^{1}=\mathcal{E}^{1\varepsilon}_{M_{1}}=\\{\omega\in\Omega:\sup_{0\leqslant\tau\leqslant
T}|a^{{\varepsilon}}(\tau)|\leqslant{M_{1}}\\}.$

By Assumption 2.1 and Chebyshev’s inequality,

$\mathbf{P}(\Omega\setminus\mathcal{E}^{1})\leqslant\mathcal{H}_{1}({M_{1}}^{-1};|v_{0}|).$

Recalling that $\tilde{y}$ was defined in (4.9), by Lemma 3.2 we have that

$|\tilde{y}(a^{{\varepsilon}}(s),s{\varepsilon}^{-1})|\leqslant|Y(a^{{\varepsilon}}(s),s{\varepsilon}^{-1})|+|{\llangle
P\rrangle}(a^{{\varepsilon}}(s))|\leqslant
2{\mathcal{C}}^{m_{0}}(P)|a^{\varepsilon}(s)|^{m_{0}}.$

So, abbreviating $\tilde{y}(a^{{\varepsilon}}(s),s{\varepsilon}^{-1})$ to
$\tilde{y}(s)$, in view of (2.7) we have:

$\begin{split}&{\mathbf{E}}_{\Omega\setminus{\mathcal{E}^{1}}}\max_{\tau\in[0,T]}|\int_{0}^{\tau}\tilde{y}(s)ds|\leqslant\int_{0}^{T}{\mathbf{E}}\big{(}\mathbf{1}_{\Omega\setminus{\mathcal{E}^{1}}}|\tilde{y}(s)|\big{)}ds\\\
&\leqslant
2{\mathcal{C}}^{m_{0}}(P)\big{(}P(\Omega\setminus{\mathcal{E}^{1}})\big{)}^{1/2}(\int_{0}^{T}{\mathbf{E}}|a^{{\varepsilon}}(s)|^{2m_{0}}ds)^{1/2}\\\
&\leqslant
2{\mathcal{C}}^{m_{0}}(P)(\mathcal{H}_{1}({M_{1}}^{-1};|v_{0}|))^{1/2}=:\mathcal{H}_{2}({M_{1}}^{-1};|v_{0}|).\end{split}$

Now we should estimate
${\mathbf{E}}_{{\mathcal{E}^{1}}}\max_{\tau\in[0,T]}|\int_{0}^{\tau}\tilde{y}(s)ds|$.
For an $M_{2}\geqslant 1$ consider the event

$\mathcal{E}^{2}=\mathcal{E}^{2\varepsilon}_{M_{2}}=\\{\omega\in\Omega:\|a^{\varepsilon}\|_{1/3}\leqslant
M_{2}\\}$

(see (1.4)). Then by (2.13)

$\mathbf{P}(\Omega\setminus{\mathcal{E}^{2}})\leqslant\mathcal{H}_{3}(M_{2}^{-1};|v_{0}|).$

Therefore,

${\mathbf{E}}_{\Omega\setminus\mathcal{E}^{2}}\max_{\tau\in[0,T]}|\int_{0}^{\tau}\tilde{y}(s)ds|\leqslant\big{(}\mathbf{P}(\Omega\setminus{\mathcal{E}^{2}})\big{)}^{1/2}(C_{P}\int_{0}^{T}{\mathbf{E}}|a^{{\varepsilon}}(s)|^{2m_{0}}ds)^{1/2}\leqslant\mathcal{H}_{4}(M_{2}^{-1};|v_{0}|)\,.$

It remains to bound $\
{\mathbf{E}}_{{\mathcal{E}^{1}}\cap{\mathcal{E}^{2}}}\max_{\tau\in[0,T]}\big{|}\int_{0}^{\tau}\tilde{y}(s)ds\big{|}.$

We set

$N=[T/\sqrt{\varepsilon}]+1,\qquad L=T/N.$

Then $C^{-1}\sqrt{\varepsilon}\leqslant L\leqslant C\sqrt{{\varepsilon}}$ and
$c^{-1}/\sqrt{\varepsilon}\leqslant N\leqslant c/\sqrt{\varepsilon}$ for some
constants $C$ and $c$. We consider a partition of interval $[0,T]$ by points
$\tau_{l}=lL$, $l=0,\dots,N$, and denote

$\eta_{l}=\int_{\tau_{l}}^{\tau_{l+1}}\tilde{y}(s)ds,\quad l=0,\dots,N-1.$

For any $\tau\in[0,T]$ let us find $l=l(\tau)$ such that
$\tau\in[\tau_{l},\tau_{l+1}]$. Then

$|\int_{0}^{\tau}\tilde{y}(s)ds|\leqslant|\eta_{1}|+\dots+|\eta_{l}|+|\int_{\tau_{l}}^{\tau}\tilde{y}(s)ds|.$

If $\omega\in\mathcal{E}^{1}$, then
$|\int_{\tau_{l}}^{\tau}\tilde{y}(s)ds|\leqslant
2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}L.$ Therefore

(4.27)
${\mathbf{E}}_{{\mathcal{E}^{1}}\cap{\mathcal{E}^{2}}}\max_{0\leqslant\tau\leqslant
T}\big{|}\int_{0}^{\tau}\tilde{y}(s)ds\big{|}\leqslant
2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}L+\sum_{l=0}^{N-1}{\mathbf{E}}_{\mathcal{E}^{1}\cap\mathcal{E}^{2}}|\eta_{l}|,$

and it remains to estimate the integrals
${\mathbf{E}}_{\mathcal{E}^{1}\cap\mathcal{E}^{2}}|\eta_{l}|$ for
$l=0,\dots,N-1$. Observe that

$|\eta_{l}|\leqslant\Big{|}\int_{\tau_{l}}^{\tau_{l+1}}[\tilde{y}(a^{\varepsilon}(s),s{\varepsilon}^{-1})-\tilde{y}(a^{\varepsilon}(\tau_{l}),s{\varepsilon}^{-1})]ds\Big{|}+\Big{|}\int_{\tau_{l}}^{\tau_{l+1}}\tilde{y}(a^{\varepsilon}(\tau_{l}),s{\varepsilon}^{-1})ds\Big{|}=:|U_{l}^{1}|+|U_{l}^{2}|.$

Since
$\tilde{y}(a^{\varepsilon},\tau{\varepsilon}^{-1})=(\Phi_{\tau{\varepsilon}^{-1}\Lambda})_{*}P(a^{\varepsilon})-{\llangle
P\rrangle}(a^{\varepsilon})$ and $P,{\llangle
P\rrangle}\in\text{Lip}_{m_{0}}(\mathbb{C}^{n},\mathbb{C}^{n})$, then for
$\omega\in{\mathcal{E}^{1}}\cap\mathcal{E}^{2}$ the integrand in $U_{l}^{1}$
is bounded by

$2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}\sup_{\tau_{l}\leqslant
s\leqslant\tau_{l+1}}|a^{\varepsilon}(s)-a^{\varepsilon}(\tau_{l})|\leqslant
2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}M_{2}L^{1/3}.$

So

$|U_{l}^{1}|\leqslant 2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}M_{2}L^{4/3}.$

Now consider integral $U_{l}^{2}$. By the definition of
$\tilde{y}(a^{\varepsilon},\tau{\varepsilon}^{-1})$, we have

$U_{l}^{2}=\int_{\tau_{l}}^{\tau_{l+1}}Y(a^{{\varepsilon}}(\tau_{l}),s{\varepsilon}^{-1})ds\,-\,L{\llangle
P\rrangle}(a^{{\varepsilon}}(\tau_{l})\big{)}=:Z^{1}+Z^{2}.$

For integral $Z^{1}$, making the change of variable
$s=\tau_{l}+{\varepsilon}x$ with $x\in[0,\frac{L}{{\varepsilon}}]$, we have

$Z^{1}={\varepsilon}\int_{0}^{L/{\varepsilon}}Y(a^{{\varepsilon}}(\tau_{l}),\tau_{l}{\varepsilon}^{-1}+x)dx.$

Since

$Y(a^{{\varepsilon}}(\tau_{l}),\tau_{l}{\varepsilon}^{-1}+x)=\Phi_{\tau_{l}{\varepsilon}^{-1}\Lambda}\circ\Phi_{x\Lambda}P\big{(}\Phi_{-x\Lambda}(\Phi_{-\tau_{l}{\varepsilon}^{-1}\Lambda}a^{\varepsilon}(\tau_{l}))\big{)},$

then

$Z^{1}=L\Phi_{\tau_{l}{\varepsilon}^{-1}\Lambda}\Big{(}\frac{{\varepsilon}}{L}\\!\int_{0}^{L/{\varepsilon}}\Phi_{x\Lambda}P\big{(}\Phi_{-x\Lambda}\Big{(}\Phi_{-\tau_{l}{\varepsilon}^{-1}\Lambda}a^{\varepsilon}(\tau_{l})\big{)}\big{)}dx\Big{)}=L\Phi_{\tau_{l}{\varepsilon}^{-1}\Lambda}\llangle
P\rrangle^{L/{\varepsilon}}\big{(}\Phi_{-\tau_{l}{\varepsilon}^{-1}\Lambda}a^{\varepsilon}(\tau_{l})\big{)}$

(see (3.1)). As $L/{\varepsilon}\sim{\varepsilon}^{-1/2}\gg 1$ and
$\big{|}\Phi_{-\tau_{l}{\varepsilon}^{-1}\Lambda}a^{\varepsilon}(\tau_{l})\big{|}=|a^{\varepsilon}(\tau_{l})|\leqslant{M_{1}}$
for $\omega\in{\mathcal{E}^{1}}\cap\mathcal{E}^{2}$, then by Lemma 3.2 the
partial averaging $\llangle P\rrangle^{L/{\varepsilon}}$ is close to the
complete averaging $\llangle P\rrangle.$ Thus

$\begin{split}\big{|}Z^{1}-L\Phi_{\tau_{l}{\varepsilon}^{-1}\Lambda}&{\llangle
P\rrangle}\big{(}\Phi_{-\tau_{l}{\varepsilon}^{-1}\Lambda}(a^{\varepsilon}(\tau_{l}))\big{)}\big{|}=\big{|}Z^{1}-L(\Phi_{\tau_{l}{\varepsilon}^{-1}\Lambda})_{*}{\llangle
P\rrangle}(a^{\varepsilon}(\tau_{l}))\big{|}\\\ =&\big{|}Z^{1}-L{\llangle
P\rrangle}(a^{\varepsilon}(\tau_{l}))\big{|}\leq
L\mathcal{H}_{5}(\sqrt{{\varepsilon}};{M_{1}},|v_{0}|),\end{split}$

where we used Lemma 3.3 to get the second equality. Since $Z^{2}=-L{\llangle
P\rrangle}(a^{\varepsilon}(\tau_{l}))$, then

$|U_{l}^{2}|=|Z^{1}+Z^{2}|\leq
L\mathcal{H}_{5}(\sqrt{{\varepsilon}};{M_{1}},|v_{0}|).$

We then have obtained that

${\mathbf{E}}_{\mathcal{E}^{1}\cap{\mathcal{E}^{2}}}|\eta_{l}|\leqslant
2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}M_{2}L^{4/3}+L\mathcal{H}_{5}(\sqrt{{\varepsilon}};{M_{1}},|v_{0}|).$

Together with (4.27) it gives us that

$\begin{split}{\mathbf{E}}_{{\mathcal{E}^{1}}\cap{\mathcal{E}^{2}}}\Big{(}\max_{0\leqslant\tau\leq
T}\big{|}\int_{0}^{\tau}\tilde{y}(s)ds\big{|}\Big{)}&\leqslant
2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}L+2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}M_{2}L^{1/3}+\mathcal{H}_{5}(\sqrt{{\varepsilon}};{M_{1}},|v_{0}|).\end{split}$

Therefore

$\begin{split}{\mathbf{E}}\max_{0\leqslant\tau\leqslant
T}\big{|}\int_{0}^{\tau}\tilde{y}(s)ds\big{|}&\leqslant\mathcal{H}_{2}({M_{1}}^{-1};|v_{0}|)+\mathcal{H}_{4}(M_{2}^{-1};|v_{0}|)\\\
&+2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}{(M_{2}+1)}{\varepsilon}^{1/6}+\mathcal{H}_{5}(\sqrt{{\varepsilon}};{M_{1}},|v_{0}|).\end{split}$

Now for any $\delta>0$, we perform the following procedure:

  1. (1)

choose ${M_{1}}$ so large that
$\mathcal{H}_{2}({M_{1}}^{-1};|v_{0}|)\leqslant\delta$;

  2. (2)

choose $M_{2}$ so large that
$\mathcal{H}_{4}(M_{2}^{-1};|v_{0}|)\leqslant\delta$;

  3. (3)

finally, choose ${\varepsilon}_{\delta}>0$ so small that

$2{\mathcal{C}}^{m_{0}}(P){M_{1}}^{m_{0}}(M_{2}+1){\varepsilon}^{1/3}+\mathcal{H}_{5}(\sqrt{{\varepsilon}};{M_{1}},|v_{0}|)\leqslant\delta\quad\text{if}\;\;\;0<{\varepsilon}\leqslant{\varepsilon}_{\delta}.$

We have seen that for any $\delta>0$,
${\mathbf{E}}\max_{0\leqslant\tau\leqslant
T}\big{|}\int_{0}^{\tau}\tilde{y}(a^{{\varepsilon}}(s),s{\varepsilon}^{-1})ds\big{|}\leq
3\delta$ if ${\varepsilon}\leq{\varepsilon}_{\delta}$. So

${\mathbf{E}}\max_{0\leqslant\tau\leqslant
T}\big{|}\int_{0}^{\tau}[Y(a^{{\varepsilon}}(s),s{\varepsilon}^{-1})-{\llangle
P\rrangle}(a^{\varepsilon}(s))]ds\big{|}\to
0\quad\text{as}\quad{\varepsilon}\to 0,$

which completes the proof of Lemma 4.3.

##  5\. Stationary solutions and mixing

In this section we study relation between stationary solutions of equation
(2.6) and of the effective equation. We recall that a solution $a(\tau)$,
$\tau\geqslant 0$, of equation (2.6) (or of effective equation (4.5)) is
stationary if $\mathcal{D}(a(\tau))\equiv\mu$ for all $\tau\geqslant 0$ and
some measure $\mu\in\mathcal{P}(\mathbb{C}^{n})$, called a stationary measure
for the corresponding equation.

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

######  Assumption 5.1.

  1. (1)

The drift $P(v)$ is a locally Lipschitz vector filed, belonging to
$\text{Lip}_{m_{0}}(\mathbb{C}^{n},\mathbb{C}^{n})$ for some
$m_{0}\in\mathbb{N}$.

  2. (2)

For any $v_{0}\in\mathbb{C}^{n}$ equation (2.6) has a unique strong solution
$v^{\varepsilon}(\tau;v_{0})$, $\tau\geq 0$, equal $v_{0}$ at $\tau=0$. There
exists $m_{0}^{\prime}>(m_{0}\vee 1)$ such that

(5.1)
${\mathbf{E}}\sup_{{T^{\prime}}\leqslant\tau\leqslant{T^{\prime}}+1}|v^{\varepsilon}(\tau;v_{0})|^{2m^{\prime}_{0}}\leqslant
C_{m^{\prime}_{0}}(|v_{0}|)$

for every ${T^{\prime}}\geqslant 0$ and ${\varepsilon}\in(0,1]$, where
$C_{m_{0}^{\prime}}(\cdot)$ is a continuous non-decreasing function.

  3. (3)

Equation (2.6) is mixing. That is, it has a stationary solution
$v^{\varepsilon}_{st}(\tau)$,
$\mathcal{D}(v^{\varepsilon}_{st}(\tau))\equiv\mu^{\varepsilon}\in\mathcal{P}(\mathbb{C}^{n})$,
and

(5.2)
$\mathcal{D}(v^{\varepsilon}(\tau;v_{0}))\rightharpoonup\mu^{\varepsilon}\text{
in }\mathcal{P}(\mathbb{C}^{n})\;\;\text{ as }\tau\to+\infty,$

for every $v_{0}$.

Under Assumption 5.1 equation (2.6) defines in $\mathbb{C}^{n}$ a mixing
Markov process with the transition probability
$\Sigma_{\tau}(v)\in\mathcal{P}(\mathbb{C}^{n})$, $\tau\geq 0$,
$v\in\mathbb{C}^{n},$ where
$\Sigma_{\tau}(v)=\mathcal{D}v^{\varepsilon}(\tau;v);$ e.g. see [14, Section
5.4.C]. Let us denote by $X$ the complete separable metric space
$X=C([0,\infty),\mathbb{C}^{n})$ with the distance

(5.3)
$\text{dist}(a_{1},a_{2})=\sum_{N=1}^{\infty}2^{-N}\frac{\|a_{1}-a_{2}\|_{C([0,N],\mathbb{C}^{n})}}{1+\|a_{1}-a_{2}\|_{C([0,N],\mathbb{C}^{n})}},\qquad
a_{1},a_{2}\in X\,,$

and consider on $X$ continuous function ${g}(a)=\sup_{0\leq t\leq
1}|a(t)|^{2m^{\prime}_{0}}.$ Denoting
$\mu^{\tau}_{\varepsilon}=\mathcal{D}(v^{\varepsilon}(\tau;0))$ we have by the
Markov property that

(5.4) ${\mathbf{E}}\sup_{T^{\prime}\leqslant\tau\leqslant
T^{\prime}+1}|v^{\varepsilon}(\tau;0)|^{m}=\int_{\mathbb{C}^{n}}\mathbf{E}{g}(v^{\varepsilon}(\cdot;v_{0}))\mu^{T^{\prime}}_{\varepsilon}(dv_{0}),$

and

(5.5) ${\mathbf{E}}\sup_{0\leqslant\tau\leqslant
1}|v^{\varepsilon}_{st}(\tau;0)|^{m}=\int_{\mathbb{C}^{n}}\mathbf{E}{g}(v^{\varepsilon}(\cdot;v_{0}))\mu^{\varepsilon}(dv_{0}).$

The l.h.s. of (5.4) is estimated in (5.1). To estimate (5.5) we will pass in
the r.h.s. of (5.4) to the limit as ${T^{\prime}}\to\infty$, using (5.2). To
do that we start with a lemma

######  Lemma 5.2.

Let $n_{1},n_{2}\in\mathbb{N}$, $\mathcal{B}\subset\mathbb{R}^{n_{1}}$ is a
closed convex set which contains more than one point, and
$F:\mathcal{B}\to\mathbb{R}^{n_{2}}$ is a Lipschitz mapping. Then $F$ may be
extended to a map $\tilde{F}:\mathbb{R}^{n_{1}}\to\mathbb{R}^{n_{2}}$ in such
a way that

a) Lip$\,\tilde{F}$ = Lip$\,F$,

b) $\tilde{F}(\mathbb{R}^{n_{1}})=F(\mathcal{B})$.

######  Proof.

Let $\Pi:\mathbb{R}^{n_{1}}\to\mathcal{B}$ be the projection, sending any
point of $\mathbb{R}^{n_{1}}$ to a nearest point in $\mathcal{B}$. Then
Lip$\,\Pi$ =1, see Appendix D, and obviously $\tilde{F}=F\circ\Pi$ is a
required extension of $F$. ∎

Since $\mathcal{C}^{m_{0}}(P)=:C_{*}<\infty$, then for any $M\in\mathbb{N}$
the restriction of $P$ to $\bar{B}_{M}(\mathbb{C}^{n})$ and its Lipschitz
constant both are bounded by $(1+M)^{m_{0}}C_{*}$. In view of the lemma above
we may extend $P\\!\mid_{\bar{B}_{M}(\mathbb{C}^{n})}$ to a Lipschitz mapping
$P^{M}:\mathbb{C}^{n}\to\mathbb{C}^{n}$ such that

$\text{Lip}\,(P^{M})\leq(1+M)^{m_{0}}C_{*},\quad\sup|P^{M}(v)|\leq(1+M)^{m_{0}}C_{*}.$

Considering for a solution $v(\tau)$ of equation (2.6) the stopping time
$\tau_{M}=\inf\\{t\geq 0:|v(t)|\geq M\\}$ and denoting by $v^{\varepsilon M}$
the stopped solution $v^{\varepsilon
M}(\tau;v_{0})=v^{\varepsilon}(\tau\wedge\tau_{M};v_{0}),$ we note that the
process $v^{\varepsilon M}$ will not change if in (2.6) we replace $P(v)$ by
$P^{M}(v)$. So $v^{\varepsilon M}(\tau;v_{0})$ is a stopped solution of a
stochastic equation with Lipschitz coefficients, and thus a.s. the curve
$v^{\varepsilon M\omega}(\cdot;v_{0})\in X$ continuously depends on $v_{0}$,
for each $M\in\mathbb{N}$. As

${g}(v^{\varepsilon M})\leq{g}(v^{\varepsilon})\quad\text{ a.s.,}$

then in view of (5.2) and (5.1), for every $M$ and any $N>0$,

$\int\mathbf{E}(N\wedge{g})(v^{\varepsilon
M}(\cdot;v))\,\mu^{\varepsilon}(dv)=\lim_{{T^{\prime}}\to\infty}\int\mathbf{E}(N\wedge{g})(v^{\varepsilon
M}(\cdot;v))\,\mu^{T^{\prime}}_{\varepsilon}(dv)\leq C_{m}(0)$

(to get the last inequality from (5.1) we used the Markov property). Passing
in the l.h.s. to the limit as $N\to\infty$ using the monotone convergence
theorem we see that

(5.6) $\int\mathbf{E}{g}(v^{\varepsilon
M}(\cdot;v))\,\mu^{\varepsilon}(dv)\leq C_{m}(0).$

Since for every $v$ a.s. ${g}(v^{\varepsilon
M}(\cdot;v))\nearrow{g}(v^{\varepsilon}(\cdot;v))\leq\infty$ as $M\to\infty,$
then using again the latter theorem we derive from (5.6) that

$\int\mathbf{E}{g}(v^{\varepsilon}(\cdot;v))\,\mu^{\varepsilon}(dv)\leq
C_{m}(0).$

Evoking (5.5) we get

######  Lemma 5.3.

The stationary solution $v^{\varepsilon}_{st}(\tau)$ satisfies estimate (5.1)
with $C_{m_{0}^{\prime}}(|v_{0}|)$ replaced by $C_{m_{0}^{\prime}}(0)$.

Let us consider the interaction representation for $v^{\varepsilon}_{st}$,
$v^{\varepsilon}_{st}(\tau)=\Phi_{-\tau{\varepsilon}^{-1}\Lambda}a^{\varepsilon}(\tau)$
(note that $a^{\varepsilon}$ is not a stationary process!). Then
$a^{\varepsilon}(\tau)$ satisfies equation (2.11), so for any $N\in\mathbb{N}$
the collection of measures
$\\{\mathcal{D}(a^{\varepsilon}|_{[0,N]}),0<{\varepsilon}\leqslant 1\\}$ is
tight in view of (5.1) (for the same reason as in Section 2.4). We choose a
sequence ${\varepsilon}_{l}\to 0$ (depending on $N$) such that

$\mathcal{D}(a^{{\varepsilon}_{l}}|_{[0,N]})\rightharpoonup Q_{0}\text{ in
}\mathcal{P}\big{(}C([0,N],\mathbb{C}^{n})\big{)}.$

Applying the diagonal process and replacing $\\{{\varepsilon}_{l}\\}$ by a
subsequence, which we still denote $\\{{\varepsilon}_{l}\\}$, we achieve that
$\mathcal{D}a^{{\varepsilon}_{l}}\rightharpoonup
Q_{0}\quad\text{in}\quad\mathcal{P}(X)$ (see (5.3)).

Since $a^{\varepsilon}(0)=v_{st}^{\varepsilon}(0)$, then

$\mu^{{\varepsilon}_{l}}\rightharpoonup\mu^{0}:=Q_{0}|_{\tau=0}.$

Let $a^{0}(\tau)$ be a process in $\mathbb{C}^{n}$ such that
$\mathcal{D}(a^{0})=Q_{0}.$ Then

(5.7)
$\mathcal{D}(a^{{\varepsilon}_{l}}(\tau))\rightharpoonup\mathcal{D}(a^{0}(\tau))\quad\forall\,0\leq\tau<\infty.$

In particular, $\mathcal{D}(a^{0}(0))=\mu^{0}$.

######  Proposition 5.4.

  1. (1)

The limiting process $a^{0}$is a stationary weak solution of effective
equation (4.5) and $\mathcal{D}(a^{0}(\tau))\equiv\mu^{0}$,
$\tau\in[0,\infty)$. In particular, the limiting points of the collection of
stationary measures $\\{\mu^{\varepsilon},{\varepsilon}\in(0,1]\\}$ as
${\varepsilon}\to 0$ are stationary measures of the effective equation.

  2. (2)

Any limiting measure $\mu^{0}$ is invariant under operators
$\Phi_{\theta\Lambda}$, $\theta\in\mathbb{R}$. So
$\mathcal{D}(\Phi_{\theta\Lambda}a^{0}(\tau))=\mu^{0}$ for all
$\theta\in\mathbb{R}$ and $\tau\in[0,\infty)$.

######  Proof.

(1) Using Lemma 5.3 and repeating the argument in the proof of Proposition 4.2
we obtain that $a^{0}$ is a weak solution of the effective equation. It
remains to prove its stationarity.

Take any bounded Lipschitz function $f$ on $\mathbb{C}^{n}$ and consider

${\mathbf{E}}\int_{0}^{1}f(v_{st}^{{\varepsilon}_{l}}(\tau))d\tau={\mathbf{E}}\int_{0}^{1}f(\Phi_{-\tau{{\varepsilon}_{l}}^{-1}\Lambda}a^{{\varepsilon}_{l}}(\tau))d\tau.$

Using the same argument as in the proof of Lemma 4.3 (but applied to averaging
of functions rather than that of vector-fields), we obtain that

(5.8)
$\begin{split}{\mathbf{E}}\int_{0}^{1}&f\big{(}v_{st}^{{\varepsilon}_{l}}(\tau)\big{)}d\tau-{\mathbf{E}}\int_{0}^{1}\langle
f\rangle\big{(}a^{{\varepsilon}_{l}}(\tau)\big{)}d\tau\\\
&={\mathbf{E}}\int_{0}^{1}\big{(}f\big{(}\Phi_{-\tau{\varepsilon}_{l}^{-1}\Lambda}a^{{\varepsilon}_{l}}(\tau))-\langle
f\rangle(a^{{\varepsilon}_{l}}(\tau)\big{)}\big{)}d\tau\to
0\quad\text{as}\;\;{\varepsilon}_{l}\to 0.\end{split}$

By Lemma 3.6,

$\langle f\rangle\big{(}a^{{\varepsilon}_{l}}(\tau)\big{)}=\langle
f\rangle\big{(}\Phi_{\tau{\varepsilon}_{l}^{-1}\Lambda}v_{st}^{{\varepsilon}_{l}}(\tau)\big{)}=\langle
f\rangle\big{(}v_{st}^{{\varepsilon}_{l}}(\tau)\big{)}$

for every $\tau$. Since the process $v_{st}^{{\varepsilon}_{l}}(\tau)$ is
stationary, then

${\mathbf{E}}f(v_{st}^{{\varepsilon}_{l}}(\tau))=\text{Const}$ and
${\mathbf{E}}\langle
f\rangle\big{(}a^{{\varepsilon}_{l}}(\tau)\big{)}={\mathbf{E}}\langle
f\rangle(v_{st}^{{\varepsilon}_{l}}(\tau))=\text{Const}^{\prime}$.

So we get from (5.8) that

(5.9)
${\mathbf{E}}f\big{(}v_{st}^{{\varepsilon}_{l}}(\tau)\big{)}-{\mathbf{E}}\langle
f\rangle\big{(}a^{{\varepsilon}_{l}}(\tau)\big{)}\to 0\;\text{ as
}{\varepsilon}_{l}\to 0\quad\forall\,\tau.$

For any $\tau$ let us consider
$\tilde{f}_{\tau}=f\circ\Phi_{\tau{\varepsilon}_{l}^{-1}\Lambda}$. Then
$f(a^{{\varepsilon}_{l}}(\tau))=\tilde{f}_{\tau}(v^{{\varepsilon}_{l}}_{st}(\tau))$.
Since $\langle f\rangle=\langle\tilde{f}_{\tau}\rangle$ by Lemma 3.6.(4), then
applying (5.9) to $\tilde{f}_{\tau}$ we get:

$\lim_{{\varepsilon}_{l}\to
0}\mathbf{E}f(a^{{\varepsilon}_{l}}(\tau))=\lim_{{\varepsilon}_{l}\to
0}\mathbf{E}\tilde{f}_{\tau}(v_{st}^{{\varepsilon}_{l}}(\tau))=\lim_{{\varepsilon}_{l}\to
0}\mathbf{E}\langle\tilde{f}_{\tau}\rangle(a^{{\varepsilon}_{l}}(\tau))=\lim_{{\varepsilon}_{l}\to
0}\mathbf{E}\langle f\rangle(a^{{\varepsilon}_{l}}(\tau)).$

From this relation, (5.9) and (5.7) we find that
$\mathbf{E}f(a^{0}(\tau))=\int\\!\\!f(v)\,\mu^{0}(dv)$ for each $\tau$ and
every $f$ as above. This implies the first assertion of the lemma.

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

$\int f(v)\mu^{0}(dv)=\mathbf{E}\langle
f\rangle(a^{0}(\tau))\quad\forall\,\tau.$

Using this relation with $f:=f\circ\Phi_{\theta\Lambda}$ and next with $f:=f$
we get that

$\int f\circ\Phi_{\theta\Lambda}(v)\mu^{0}(dv)=\mathbf{E}\langle
f\circ\Phi_{\theta\Lambda}\rangle(a^{0}(\tau))=\mathbf{E}\langle
f\rangle(a^{0}(\tau))=\int f(v)\mu^{0}(dv),$

for any $\theta\in\mathbb{R}$ and any $\tau$, for every bounded Lipschitz
function $f$. This implies the second assertion. ∎

If effective equation (4.5) is mixing, then it has a unique stationary
measure. In this case the measure $\mu^{0}$ in Proposition 5.4 does not depend
on a choice of the sequence ${\varepsilon}_{l}\to 0$, and so
$\mu^{\varepsilon}\rightharpoonup\mu^{0}$ as ${\varepsilon}\to 0$. Therefore,
we have

######  Theorem 5.5.

If in addition to Assumption 5.1 we assume that the effective equation is
mixing and $\mu^{0}$ is its unique stationary measure, then

$\mu^{\varepsilon}\rightharpoonup\mu^{0}\text{ in
}\mathcal{P}(\mathbb{C}^{n})\quad\text{as}\quad{\varepsilon}\to 0.$

Moreover, the measure $\mu^{0}$ is invariant for all operators
$\Phi_{\theta\Lambda}$, and the law of the stationary solution of equation
(2.6), written in the interaction presentation, converges to the law of the
stationary solution of effective equation (4.5).

We recall that Theorem 4.7 and Corollary 4.13 only assure that on finite time-
intervals $\tau\in[0,T]$ the actions of solutions for eq. (2.6), as
$\varepsilon\to 0$, converge in law to the actions of solutions for the
effective equation with the same initial data. In difference, when
${\varepsilon}\to 0$ all  of the stationary measure for eq. (2.6) converges to
that for the effective equation. This important fact was first observed in [9]
for a special class of equations (2.6).

######  Corollary 5.6.

Under the assumption of Theorem 5.5 , for any $v_{0}\in\mathbb{C}^{n}$ we have

$\lim_{{\varepsilon}\to
0}\lim_{\tau\to\infty}\mathcal{D}(v^{\varepsilon}(\tau;v_{0}))=\mu^{0}.$

######  Proof.

Since
$\lim_{\tau\to\infty}\mathcal{D}(v^{\varepsilon}(\tau))=\mu^{\varepsilon}$,
then the result follows from Theorem 5.5. ∎

######  Remark 5.7.

Let us decomplexify $\mathbb{C}^{n}$ to $\mathbb{R}^{2n}$ and write the
effective equation in the real coordinates $\\{x=(x_{1},\dots,x_{2n})\\}$:

$dx_{j}(\tau)-{\llangle
P\rrangle}_{j}(x)d\tau=\sum_{l=1}^{2n}\mathcal{B}_{jl}dW_{l}(\tau),\;j=1,\dots,2n,$

where $W_{l}$ are independent standard real Wiener processes. Then the
stationary measure $\mu^{0}\in\mathcal{P}(\mathbb{R}^{2n})$ satisfies the
stationary Fokker-Plank equation

$\frac{1}{2}\sum_{l=1}^{2n}\sum_{j=1}^{2n}\frac{\partial^{2}}{\partial
x_{l}\partial
x_{j}}(\mathcal{B}_{lj}\mu^{0})=\sum_{l=1}^{2n}\frac{\partial}{\partial
x_{l}}({\llangle P\rrangle}_{l}(x)\mu^{0})$

in the sense of distributions. If the dispersion matrix $\Psi$ is non-
degenerate, then the diffusion $\mathcal{B}$ also is, and since the drift
${\llangle P\rrangle}(x)$ is locally Lipschitz, then by the standard theory of
the Fokker-Plank equation $\mu^{0}=\varphi(x)dx$, where $\varphi\in
C^{1}(\mathbb{R}^{2n})$.888E.g. firstly Theorem 1.6.8 from [3] implies that
$\mu^{0}=\varphi(x)dx$, where $\varphi$ is a Hölder function, and then by the
usual elliptic regularity $\varphi\in C^{1}$.

##  6\. The non-resonant case

Assume that the frequency vector $\Lambda=(\lambda_{1},\dots,\lambda_{n})$ is
non-resonant (see (3.7)). In Section 3.1.2) we saw that in this case vector
field $\llangle P\rrangle$ may be calculated via the averaging (3.8) and
commutes with all rotations $\Phi_{w}$, $w\in\mathbb{R}^{n}$. For any
$j\in\\{1,\dots,n\\}$ let us denote $w^{j,t}:=(0,\dots 0,t,0,\dots,0)$ (only
the $j$-th entry is non-zero). Consider ${\llangle P\rrangle}_{1}(z)$ and
write it as ${\llangle P\rrangle}_{1}(z)=z_{1}R_{1}(z_{1},\dots,z_{n})$, for
some complex function $R_{1}$. Since for $w=w^{1,t}$ we have
$\Phi_{w}(z)=(e^{it}z_{1},z_{2},\dots,z_{n})$, then now the first component in
relation (3.10) reads
$e^{-it}z_{1}R_{1}(e^{-it}z,z_{1},\dots,z_{n})=e^{-it}z_{1}R_{1}(z_{1},\dots,z_{n}),$
for every $t$. So

$R_{1}(e^{-it}z_{1},z_{2},\dots,z_{n})\equiv R_{1}(z_{1},\dots,z_{n})$

and $R_{1}(z_{1},\dots,z_{n})$ depends not on $z_{1}$, but only on $|z_{1}|$.
Similarly we verify that $R_{1}(z_{1},\dots,z_{n})$ depends only on
$|z_{2}|,\dots,|z_{n}|$. Therefore ${\llangle
P\rrangle}_{1}(z)=z_{1}R_{1}(|z_{1}|,\dots,|z_{n}|)$. Same is true for any
${\llangle P\rrangle}_{j}(z)$. We then obtain the following statement:

######  Proposition 6.1.

If Assumption 2.1 holds and the frequency vector $\Lambda$ is non-resonant,
then $\llangle P\rrangle$ satisfies (3.8), and

  1. (1)

${\llangle P\rrangle}_{j}(a)=a_{j}R_{j}(|a_{1}|,\dots,|a_{n}|)$,
$j=1,\dots,n$.

  2. (2)

The effective equation reads

(6.1)
$da_{j}(\tau)-a_{j}R_{j}(|a_{1}|,\dots,|a_{n}|)d\tau=b_{j}d\beta^{c}_{j}(\tau),\quad\;j=1,\dots,n,$

where $b_{j}=(\sum_{l=1}^{n}|\Psi_{jl}|^{2})^{1/2}$ (and
$a\mapsto(a_{1}R_{1},\dots,a_{n}R_{n})$ is a locally Lipschitz vector-field).

  3. (3)

If $a(\tau)$ is a solution of (6.1), then the vector of its actions
$I(\tau)=(I_{1},\dots,I_{n})(\tau)\in\mathbb{R}_{+}^{n}$ is a weak solution of
equation

(6.2) $dI_{j}(\tau)-2I_{j}(\Re
R_{j})\big{(}\sqrt{2I_{1}},\dots,\sqrt{2I_{n}}\big{)}d\tau-
b_{j}^{2}d\tau=b_{j}\sqrt{2I_{j}}\,dW_{j}(\tau),\quad
I_{j}(0)=\tfrac{1}{2}|v_{0j}|^{2},$

$j=1,\dots,n$, where $\\{W_{j}\\}$ are independent standard real Wiener
processes.

  4. (4)

If in addition the assumptions of Theorem 5.5 are met and the matrix $\Psi$ is
non-degenerate, then the stationary measure $\mu^{0}$ reads
$d\mu^{0}=p(I)dI\,d\varphi,$ where $p$ is a continuous function on
$\mathbb{R}^{n}_{+}$, $C^{1}$-smooth outside the boundary
$\partial\mathbb{R}^{n}_{+}$.

######  Proof.

(1) is proved above, and (2) follows from it and (4.4).

(3) Writing the diffusion in effective equation (4.5) as in eq. (6.1) and
applying Itô’s formula (see Appendix C) to $I_{j}=\frac{1}{2}|a_{j}|^{2}$ we
get that

(6.3) $dI_{j}(\tau)-\tfrac{1}{2}\big{(}\bar{a}_{j}\langle\\!\langle
P\rangle\\!\rangle_{j}+a_{j}\overline{\langle\\!\langle
P\rangle\\!\rangle}_{j}\big{)}d\tau-b_{j}^{2}d\tau=b_{j}\langle
a_{j}(\tau),d\beta_{j}(\tau)\rangle=:b_{j}|a_{j}|d\xi_{j}(\tau),$

where $d\xi_{j}(\tau)=\langle a_{j}/|a_{j}|,d\beta_{j}(\tau)\rangle$ (see
(2.2)) and for $a_{j}=0$ we define $a_{j}/|a_{j}|$ as 1. Using (1) we see that
the l.h.s. in (6.3) is the same as that in (6.2). Since
$\big{|}a_{j}/|a_{j}|\big{|}(\tau)\equiv 1$ for each $j$, then by Lévy’s
theorem (e.g. see [14, p. 157]), $\xi(\tau)=(\xi_{1},\dots,\xi_{n})(\tau)$ is
a standard $n$-dimensional Wiener process and (3) follows.

(4) By Theorem 5.5 the stationary measure $\mu^{0}$ is invariant for all
operators $\Phi_{\theta\Lambda}$, $\theta\in\mathbb{R}$. Since the curve
$\theta\mapsto\theta\Lambda\in\mathbb{T}^{n}$ is dense in $\mathbb{T}^{n}$,
then $\mu^{0}$ is invariant for all operators $\Phi_{w}$,
$w\in\mathbb{T}^{n}$. As the matrix $\Psi$ is non-degenerate, then by Remark
5.7 we have $d\mu^{0}=\tilde{p}(z)dz$, where $\tilde{p}$ is a $C^{1}$-function
($dz$ stands for the volume element in $\mathbb{C}^{n}\simeq\mathbb{R}^{2n}$).
Let us write $z_{j}=\sqrt{2I_{j}}e^{i\varphi_{j}}$. Then
$d\mu^{0}=p(I,\varphi)dz$. In the coordinates $(I,\varphi)$ the operators
$\Phi_{w}$ reads as $(I,\varphi)\mapsto(I,\varphi+w)$. Since $\mu^{0}$ is
invariant for all of them, then $p$ does not depend on $\varphi$. So
$d\mu^{0}=p(I)dz=p(I)\ dI\,d\varphi$ and (4) holds. ∎

By assertion (3) of the proposition, in the non-resonant case eq. (6.2)
describes asymptotic as $\varepsilon\to 0$ behaviour of actions $I_{j}$ of
solutions for eq. (2.6). But how regular is that equation? Let us denote by
$r_{j}=|a_{j}|=\sqrt{2I_{j}},\ 1\leq j\leq n,$ the radii of components of a
vector $a\in\mathbb{C}^{n}$, consider the smooth polar-coordinates mapping
$\mathbb{R}_{+}^{n}\times\mathbb{T}^{n}\to\mathbb{C}^{n}$,
$(r,\varphi)\mapsto(r_{1}e^{i\varphi_{1}},\dots,r_{n}e^{i\varphi_{n}}),$ and
extend it to a mapping

$\Phi:\mathbb{R}^{n}\times\mathbb{T}^{n}\to\mathbb{C}^{n},$

defined by the same formula. A $j$-th component of the drift in eq. (6.2),
written in the form (6.3) without the Itô term $b_{j}^{2}d\tau$, is
$\Re\big{(}a_{j}\overline{\llangle P\rrangle_{j}(a)}\big{)}$. Due to (3.8), in
the polar coordinates we write this as

$\begin{split}\frac{1}{(2\pi)^{n}}\int_{\mathbb{T}^{n}}\Re\Big{(}r_{j}e^{i\varphi_{j}}\overline{e^{iw_{j}}P_{j}(r,\varphi-w)}\Big{)}dw=\frac{1}{(2\pi)^{n}}\int_{\mathbb{T}^{n}}\Re\big{(}r_{j}e^{i\theta_{j}}\bar{P}_{j}(r,\theta)\Big{)}d\theta=:F_{j}(r),\quad
r\in\mathbb{R}^{n},\end{split}$

where $F_{j}(r)$ is a continuous function, vanishing with $r_{j}$. Since the
integrand in the second integral will not change if for some $l=1,\dots,n$ we
replace $r_{l}$ by $-r_{l}$ and replace $\theta_{l}$ by $\theta_{l}+\pi$, then
$F_{j}(r)$ is even in each its argument $r_{l}$. So it may be written as

$F_{j}(r_{1},\dots,r_{n})=f_{j}(r_{1}^{2},\dots,r_{n}^{2}),\qquad f_{j}\in
C(\mathbb{R}^{n}_{+}),$

where $f_{j}(x_{1},\dots,x_{n})$ vanishes with $x_{j}$. Now assume that the
vector field $P$ is $C^{2}$-smooth. In this case the integrand in the integral
for $F_{j}$ is $C^{2}$-smooth in
$(r,\theta)\in\mathbb{R}^{n}\times\mathbb{T}^{n}$ and $F_{j}$ is a
$C^{2}$-smooth function of $r$. Then by a result of H. Whitney (see in [27]
Theorem 1 with $s=1$ and the remark, concluding the paper), $f_{j}$ extends to
$\mathbb{R}^{n}$ in such a way that $f_{j}(x)$ is $C^{1}$-smooth in each its
argument $x_{l}$ and $(\partial/\partial x_{l})f_{j}(x)$ is continuous on
$\mathbb{R}^{n}$. So $f_{j}$ is $C^{1}$-smooth. Since $r_{j}^{2}=2I_{j}$, then
we have established the following result.

######  Proposition 6.2.

If the frequency vector $\Lambda$ is non-resonant and $P$ is $C^{2}$smooth,
then equation (6.2) may be written as

(6.4) $dI_{j}(\tau)-G_{j}(I_{1},\dots,I_{n})d\tau-
b_{j}^{2}d\tau=b_{j}\sqrt{2I_{j}}\,dW_{j}(\tau),\quad 1\leq j\leq n,$

where $G$ is a $C^{1}$-smooth vector-field such that $G_{j}(I)$ vanishes with
$I_{j}$, for each $j$.

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

##  7\. Uniform in time convergence.

In this section we study the uniform in time convergence in distribution of
solutions for eq. (2.11) to those for effective equation (4.5), with respect
to the dual-Lipschitz metric (see Definition 4.10). These results are finite-
dimensional versions of those, obtained in [11] for stochastic PDEs.
Throughout the section we assume

######  Assumption 7.1.

The first two items of Assumption 5.1 hold, and

  1. $(3^{\prime})$

Effective equation (4.5) is mixing with a stationary measure $\mu^{0}$. For
any its solution $a(\tau)$, $\tau\geq 0$, such that $\mathcal{D}(a(0))=:\mu$
and
$\langle|z|^{2m_{0}^{\prime}},\mu(dz)\rangle=\mathbf{E}|a(0)|^{2m_{0}^{\prime}}\leq
M^{\prime}$ for some $M^{\prime}>0$ (we recall notation (4.22)) we have

(7.1) $\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L,\mathbb{C}^{n}}^{*}\leqslant
g_{M^{\prime}}(\tau,d)\qquad\forall\tau\geqslant 0,\;\text{ if
}\|\mu-\mu^{0}\|_{L,\mathbb{C}^{n}}^{*}\leqslant d\leqslant 2.$

Here the function $g:\mathbb{R}_{+}^{3}\to\mathbb{R}_{+},\;(\tau,d,M)\mapsto
g_{M}(\tau,d),$ is continuous, vanishes with $d$, converges to zero when
$\tau\to\infty$ and is such that for each fixed $M\geq 0$ the function
$(\tau,d)\mapsto g_{M}(\tau,d)$ is uniformly continuous in $d$ for
$(\tau,d)\in[0,\infty)\times[0,2]$. 999So $g_{M}$ extends to a continuous
function on $[0,\infty]\times[0,2]$ which vanishes when $\tau=\infty$ or
$d=0$.

We emphasize that now we assume mixing for the effective equation, but not for
the original equation (2.6). Since Assumptions 7.1 imply Assumptions 2.1, then
the assertions of Section 4 with any $T>0$ hold to solutions of equations
(2.11) which we analyse in this section.

######  Proposition 7.2.

Assume that the first two items of Assumption 5.1 hold, eq. (4.5) is mixing
with a stationary measure $\mu^{0}$, and for each $M>0$ and any
$v^{1},v^{2}\in\bar{B}_{M}(\mathbb{C}^{n})$ we have

(7.2)
$\|\mathcal{D}a(\tau;v^{1})-\mathcal{D}a(\tau;v^{2})\|_{L,\mathbb{C}^{n}}^{*}\leq{\mathfrak{g}}_{M}(\tau),$

where ${\mathfrak{g}}$ is a continuous function of $(M,\tau)$ which goes to
zero when $\tau\to\infty$ and is a non-decreasing function of $M$. Then
condition $(3^{\prime})$ holds with some function $g$.

The proposition is proved below in Subsection 7.1.

Note that (7.2) holds (with ${\mathfrak{g}}$ replaced by $2{\mathfrak{g}}$) if

(7.3)
$\|\mathcal{D}a(\tau;v^{1})-\mu^{0}\|_{L,\mathbb{C}^{n}}^{*}\leq{\mathfrak{g}}_{M}(\tau)\quad\forall\,v^{1}\in\bar{B}_{M}(\mathbb{C}^{n}).$

Usually a proof of mixing for eq. (4.5) in fact establishes (7.3). So
condition $(3^{\prime})$ is a rather mild restriction.

######  Example 7.3.

If the assumption of Proposition 9.4 below is fulfilled, then (7.2) is
satisfied since in this case (7.3) holds with
${\mathfrak{g}}_{M}(\tau)=\bar{V}(M)e^{-c\tau}$. Here $c>0$ is a constant and
$\bar{V}(M)=\max\\{V(x):x\in\bar{B}_{M}(\mathbb{C}^{n})\\}$, where $V(x)$ is
the Lyapunov function as in Proposition 9.3. See e.g. [22, Theorem 2.5] and
[20, Section 3.3].

######  Theorem 7.4.

Under Assumption 7.1, for any $v_{0}\in\mathbb{C}^{n}$

$\lim_{\varepsilon\to 0}\,\sup_{\tau\geqslant
0}\|\mathcal{D}(a^{\varepsilon}(\tau;v_{0}))-\mathcal{D}(a^{0}(\tau;v_{0}))\|_{L,\mathbb{C}^{n}}^{*}=0,$

where $a^{\varepsilon}(\tau;v_{0})$ and $a^{0}(\tau;v_{0})$ solve respectively
(2.11) and (4.5) with the same initial condition
$a^{\varepsilon}(0;v_{0})=a^{0}(0;v_{0})=v_{0}$.

######  Proof.

Since $v_{0}$ is fixed, we abbreviate $a^{\varepsilon}(\tau;v_{0})$ to
$a^{\varepsilon}(\tau)$. Due to (5.1)

(7.4) ${\mathbf{E}}|a^{\varepsilon}(\tau)|^{2m^{\prime}_{0}}\leqslant
C_{m^{\prime}_{0}}(|v_{0}|)=:M^{*}\quad\forall\,\tau\geq 0.$

By (7.4) and (4.19) 101010Indeed, for any $N>0$ the estimate with
$|a|^{2m^{\prime}_{0}}$ replaced by $|a|^{2m^{\prime}_{0}}\wedge N$ follows
from the convergence
$\mathcal{D}a^{\varepsilon}({\tau};v_{0})\rightharpoonup\mathcal{D}a^{0}({\tau};v_{0})$.
Then the required estimate follows from Fatou’s lemma as $N\to\infty$.

(7.5)
${\mathbf{E}}|a^{0}(\tau;v_{0})|^{2m^{\prime}_{0}}=\langle|a|^{2m^{\prime}_{0}},\mathcal{D}a^{0}({\tau};v_{0})\rangle\leq
M^{*}\quad\forall\,\tau\geq 0.$

Since $\mathcal{D}a^{0}({\tau};0)\rightharpoonup\mu^{0}$ as ${\tau}\to\infty$,
then from the estimate above with $v_{0}=0$ we get that

(7.6) $\langle|a|^{2m^{\prime}_{0}},\mu^{0}\rangle\leq
C_{m^{\prime}_{0}}(0)=:C_{m^{\prime}_{0}}.$

For later usage we note that since to derive estimates (7.5) and (7.6) we only
used Assumptions 5.1.(1), 5.1.(2) and the fact that eq. (4.5) is mixing, then
the two estimates hold under the assumptions of Proposition 7.2.

The constants in estimates below depend on $M^{*}$, but usually this
dependence is not indicated. For any $T\geq 0$ we denote by $a_{T}^{0}(\tau)$
a weak solution of effective equation (4.5) such that

$\mathcal{D}a^{0}_{T}(0)=\mathcal{D}a^{\varepsilon}(T).$

Note that $a^{0}_{T}(\tau)$ depends on $\varepsilon$ and that
$a^{0}_{0}(\tau)=a^{0}(\tau;v_{0})$.

######  Lemma 7.5.

Take any $\delta>0$. Then

  1. (1)

for any $T>0$ there exists $\varepsilon_{1}=\varepsilon_{1}(\delta,T)>0$ such
that if $\varepsilon\leq\varepsilon_{1}$, then

(7.7)
$\sup_{\tau\in[0,T]}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+\tau))-\mathcal{D}(a_{T^{\prime}}^{0}(\tau))\|_{L,\mathbb{C}^{n}}^{*}\leqslant\delta/2\quad\forall\,T^{\prime}\geqslant
0.$

  2. (2)

Let us choose a $T^{*}=T^{*}(\delta)>0$, satisfying
$g_{M^{*}}(T,2)\leq\delta/4$ for any $T\geq T^{*}$. Then there exists
$\varepsilon_{2}=\varepsilon_{2}(\delta)>0$ such that if
$\varepsilon\leq\varepsilon_{2}$ and
$\|\mathcal{D}(a^{\varepsilon}(T^{\prime}))-\mu^{0}\|_{L,\mathbb{C}^{n}}^{*}\leqslant\delta$
for some $T^{\prime}\geq 0$, then also

(7.8)
$\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+T^{*}))-\mu^{0}\|_{L,\mathbb{C}^{n}}^{*}\leqslant{\delta},$

and

(7.9)
$\sup_{\tau\in[T^{\prime},T^{\prime}+T^{*}]}\|\mathcal{D}(a^{\varepsilon}(\tau))-\mu^{0}\|_{L,\mathbb{C}^{n}}^{*}\leqslant\tfrac{\delta}{2}+\sup_{\tau\geq
0}g_{M^{*}}(\tau,\delta).$

Below we abbreviate $\|\cdot\|^{*}_{L,\mathbb{C}^{n}}$ to $\|\cdot\|^{*}_{L}$,
for a measure $\nu\in\mathcal{P}(\mathbb{C}^{n})$ denote by
$a^{\varepsilon}(\tau;\nu)$ a weak solution of eq. (2.11) such that
$\mathcal{D}(a^{\varepsilon}(0))=\nu$, and define $a^{0}(\tau;\nu)$ similarly.
Since eq. (2.11) defines a Markov process in $\mathbb{C}^{n}$ (e.g. see [14,
Section 5.4.C] and [17, Section 3.3]), then

$\mathcal{D}a^{\varepsilon}(\tau;\nu)=\int_{\mathbb{C}^{n}}\mathcal{D}a^{\varepsilon}(\tau;v)\,\nu(dv),$

and a similar relation holds for $\mathcal{D}a^{0}(\tau;\nu)$.

######  Proof.

Denote $\nu^{\varepsilon}=\mathcal{D}(a^{\varepsilon}(T^{\prime}))$. Then

(7.10)
$\mathcal{D}(a^{\varepsilon}(T^{\prime}+\tau))=\mathcal{D}(a^{\varepsilon}(\tau;\nu^{\varepsilon})),\quad\mathcal{D}(a^{0}_{T^{\prime}}(\tau))=\mathcal{D}(a^{0}(\tau;\nu^{\varepsilon})).$

By (7.4), for any $\delta>0$ there exists $K_{\delta}>0$ such that for each
$\varepsilon$,
$\nu^{\varepsilon}(\mathbb{C}^{n}\setminus\bar{B}_{K_{\delta}})\leq\delta/8$,
where $\bar{B}_{K_{\delta}}:=\bar{B}_{K_{\delta}}(\mathbb{C}^{n})$. So

$\nu^{\varepsilon}=A^{\varepsilon}\nu^{\varepsilon}_{\delta}+\bar{A}^{\varepsilon}\bar{\nu}^{\varepsilon}_{\delta},\quad
A^{\varepsilon}=\nu^{\varepsilon}(\bar{B}_{K_{\delta}}),\;\bar{A}^{\varepsilon}=\nu^{\varepsilon}(\mathbb{C}^{n}\setminus\bar{B}_{K_{\delta}}),$

where $\nu^{\varepsilon}_{\delta}$ and $\bar{\nu}^{\varepsilon}_{\delta}$ are
the conditional probabilities
$\nu^{\varepsilon}(\cdot\mid\bar{B}_{K_{\delta}})$ and
$\nu^{\varepsilon}(\cdot\mid\mathbb{C}^{n}\setminus\bar{B}_{K_{\delta}})$.
Accordingly,

(7.11)
$\mathcal{D}(a^{\kappa}(\tau;\nu^{\varepsilon}))=A^{\varepsilon}\mathcal{D}(a^{\kappa}(\tau;\nu^{\varepsilon}_{\delta}))+\bar{A}^{\varepsilon}\mathcal{D}(a^{\kappa}(\tau;\bar{\nu}^{\varepsilon}_{\delta})),$

where $\kappa=\varepsilon$ or $\kappa=0$. Therefore,

$\|\mathcal{D}(a^{\varepsilon}(\tau;\nu^{\varepsilon}))-\mathcal{D}(a^{0}(\tau;\nu^{\varepsilon}))\|_{L}^{*}\leq
A^{\varepsilon}\|\mathcal{D}(a^{\varepsilon}(\tau;\nu^{\varepsilon}_{\delta}))-\mathcal{D}(a^{0}(\tau;\nu^{\varepsilon}_{\delta}))\|_{L}^{*}+\bar{A}^{\varepsilon}\|\mathcal{D}(a^{\varepsilon}(\tau;\bar{\nu}^{\varepsilon}_{\delta}))-\mathcal{D}(a^{0}(\tau;\bar{\nu}^{\varepsilon}_{\delta}))\|_{L}^{*}.$

The second term on the r.h.s obviously is bounded by
$2\bar{A}^{\varepsilon}\leqslant\frac{\delta}{4}$. While by Proposition 4.12
and (4.23) there exists $\varepsilon_{1}>0$, depending only on $K_{\delta}$
and $T$, such that for $0\leq\tau\leq T$ and
$\varepsilon\in(0,\varepsilon_{1}]$ the first term in the r.h.s. is
$\leqslant\frac{\delta}{4}$. Due to (7.10) this proves the first assertion.

To prove the second assertion let us choose
$\varepsilon_{2}=\varepsilon_{1}(\delta/2,T^{*}(\delta))$. Then we get from
(7.7), (7.4), (7.1) and the definition of $T^{*}$ that for
$\varepsilon\leq\varepsilon_{2}$,

$\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+T^{*}))-\mu^{0}\|_{L}^{*}\leqslant\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+T^{*}))-\mathcal{D}(a^{0}_{T^{\prime}}(T^{*}))\|_{L}^{*}+\|\mathcal{D}(a^{0}_{T^{\prime}}(T^{*}))-\mu^{0}\|_{L}^{*}\leqslant{\delta}.$

This proves (7.8). Next, in view of (7.7) and (7.1), (7.5),

$\begin{split}\sup_{\theta\in[0,T^{*}]}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+\theta))-\mu^{0}\|_{L}^{*}&\leqslant\sup_{\theta\in[0,T^{*}]}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+\theta))-\mathcal{D}(a_{T^{\prime}}^{0}(\theta))\|_{L}^{*}\\\
+\sup_{\theta\in[0,T^{*}]}\|\mathcal{D}(a_{T^{\prime}}^{0}(\theta))-\mu^{0}\|_{L}^{*}&\leqslant\frac{\delta}{2}+\max_{\theta\in[0,T^{*}]}g_{M^{*}}(\theta,\delta).\end{split}$

This implies (7.9). ∎

Now we continue to prove the theorem. Let us fix arbitrary $\delta>0$ and take
some $0<\delta_{1}\leq\delta/4$. Below in the proof the functions
$\varepsilon_{1}$, $\varepsilon_{2}$ and $T^{*}$ are as in Lemma 7.5.

i) By the definition of $T^{*}$, (7.1) and (7.4),

(7.12)
$\|\mathcal{D}\big{(}a^{0}_{T^{\prime}}(\tau))-\mu^{0}\|_{L}^{*}\leq\delta_{1}\quad\forall\,\tau\geq
T^{*}(\delta_{1}),$

for any $T^{\prime}\geq 0$. We will abbreviate $T^{*}(\delta_{1})$ to $T^{*}$.

ii) By (7.7), if
$\varepsilon\leq\varepsilon_{1}=\varepsilon_{1}(\delta_{1},T^{*})>0$, then

(7.13) $\sup_{0\leq\tau\leq
T^{*}}\|\mathcal{D}\big{(}a^{\varepsilon}(\tau))-\mathcal{D}\big{(}a^{0}(\tau;v_{0})\big{)}\|_{L}^{*}\leq\tfrac{\delta_{1}}{2}.$

In particular, in view of (7.12) with $T^{\prime}=0$,

(7.14)
$\|\mathcal{D}\big{(}a^{\varepsilon}(T^{*}))-\mu^{0}\|_{L}^{*}<2\delta_{1}.$

iii) By (7.14) and (7.8) with $\delta:=2\delta_{1}$ and with
$T^{\prime}=nT^{*}$, $n=1,2,\dots$ we get inductively that

(7.15) $\|\mathcal{D}\big{(}a^{\varepsilon}(nT^{*}))-\mu^{0}\|_{L}^{*}\leq
2\delta_{1}\quad\forall\,n\in\mathbb{N},$

if $\varepsilon\leq\varepsilon_{2}=\varepsilon_{2}(2\delta_{1})$.

iv) Now by (7.15) and (7.9) with $\delta:=2\delta_{1}$, for any
$n\in\mathbb{N}$ and $0\leq\theta\leq T^{*}$,

(7.16)
$\|\mathcal{D}\big{(}a^{\varepsilon}(nT^{*}+\theta))-\mu^{0}\|_{L}^{*}\leq\delta_{1}+\sup_{\theta\geq
0}g_{M^{*}}(\theta,2\delta_{1}),$

if $\varepsilon\leq\varepsilon_{2}(2\delta_{1})$.

v) Finally, if
$\varepsilon\leq\varepsilon_{\\#}(\delta_{1})=\min\big{(}\varepsilon_{1}(\delta_{1},T^{*}),\varepsilon_{2}(2\delta_{1})\big{)}$,
then by (7.13) if $\tau\leq T^{*}$ and by (7.12)+(7.16) if $\tau\geq T^{*}$ we
have that

$\|\mathcal{D}\big{(}a^{\varepsilon}(\tau))-\mathcal{D}\big{(}a^{0}(\tau;v_{0})\big{)}\|_{L}^{*}\leq
2\delta_{1}+\sup_{\theta\geq
0}g_{M^{*}}(\theta,2\delta_{1})\qquad\forall\,\tau\geq 0.$

By the assumption, imposed in $(3^{\prime})$ on function $g_{M}$, $g_{M}(t,d)$
is uniformly continuous in $d$ and vanishes at $d=0$. So there exists
$\delta^{*}=\delta^{*}(\delta)$, which we may assume to be $\leq\delta/4$,
such that if $\delta_{1}=\delta^{*}$, then
$g_{M^{*}}(\theta,2\delta_{1})\leq\delta/2$ for every $\theta\geqslant 0$.
Then by the estimate above,

$\|\mathcal{D}\big{(}a^{\varepsilon}(\tau))-\mathcal{D}\big{(}a^{0}(\tau;v_{0})\big{)}\|_{L}^{*}\leq\delta\quad\text{if}\quad\varepsilon\leq\varepsilon_{*}(\delta):=\varepsilon_{\\#}(\delta^{*}\big{(}\delta)\big{)}>0,$

for every positive $\delta$. This proves the theorem’s assertion. ∎

Since the interaction representation does not change actions, then for the
action variables of solutions for the original equations (2.6) we have

######  Corollary 7.6.

Under the assumptions of Theorem 7.4 the actions of a solution
$v^{\varepsilon}(\tau;v_{0})$ for eq. (2.6) which equals $v_{0}$ at $\tau=0$
satisfy

$\lim_{\varepsilon\to 0}\,\sup_{\tau\geqslant
0}\|\mathcal{D}(I\big{(}v^{\varepsilon}(\tau;v_{0}))\big{)}-\mathcal{D}((I\big{(}a^{0}(\tau;v_{0}))\big{)}\|_{L,\mathbb{C}^{n}}^{*}=0.$

In [9, Theorem 2.9] the assertion of the corollary is proved for a class of
systems (2.6). The proof in [9] is based on the observation that the mixing
rate in the corresponding equation (2.6) is uniform in $0<\varepsilon\leq 1$.
This is a delicate property which is harder to establish than $(3^{\prime})$
in Assumption 7.1. We also note that Theorem 7.4 immediately implies that if
equations (2.11) are mixing with stationary measures $\mu^{\varepsilon}$, then
$\mu^{\varepsilon}\rightharpoonup\mu^{0}$. Cf. Theorem 5.5.

###  7.1. Proof of Proposition 7.2

In this subsection we write solutions $a^{0}(\tau;v)$ of effective equation
(4.5) as $a(\tau;v)$. We will prove the assertion of Proposition 7.2 in four
steps.

i) At this step, for any non-random
$v^{1},v^{2}\in\bar{B}_{M}(\mathbb{C}^{n})$ we denote by
$a_{j}(\tau):=a(\tau;v^{j})$, $j=1,2$, and examine the distance
$\|\mathcal{D}(a_{1}(\tau))-\mathcal{D}(a_{2}(\tau))\|_{L}^{*}$ as a function
of $\tau$ and $|v^{1}-v^{2}|$. Let us set $w(\tau)=a_{1}(\tau)-a_{2}(\tau)$
and assume that $|v^{1}-v^{2}|\leq\bar{d}$ for some $\bar{d}\geq 0$. Then we
have

$\dot{w}={\llangle P\rrangle}(a_{1})-{\llangle P\rrangle}(a_{2}).$

Since by Lemma 3.2 and Assumption 2.1.(1)

$|{\llangle P\rrangle}(a_{1}(\tau))-{\llangle P\rrangle}(a_{2}(\tau))|\leq
C|w(\tau)|\,X(\tau),\quad
X(\tau)=1+|a_{1}(\tau)|^{m_{0}}\vee|a_{2}(\tau)|^{m_{0}},$

then $(d/d\tau)|w|^{2}\leq 2CX(\tau)|w|^{2}$ with $|w(0)|\leq\bar{d}.$ So

(7.17) $|w(\tau)|\leq\bar{d}\exp\big{(}C\int_{0}^{\tau}X(l)\,dl\big{)}.$

Denote $Y(T)=\sup_{0\leq\tau\leq T}|X(\tau)|$. By (5.1) estimate (2.7) holds
with $C_{m_{0}^{\prime}}(|v_{0}|,T)=C_{m_{0}^{\prime}}(M)(T+1).$ So by Remark
4.8.2) $\mathbf{E}Y(T)\leq(C_{m^{\prime}_{0}}(M)+1)(T+1)$ (since
$m_{0}^{\prime}>(m_{0}\vee 1)$).

For $K>0$ denote by $\Omega_{K}(T)$ the event $\\{Y(T)\geq K\\}$. Then
$\mathbf{P}(\Omega_{K}(T))\leq(C_{m^{\prime}_{0}}(M)+1)(T+1)K^{-1},$ and
$\int_{0}^{\tau}X(l)\,dl\leq\tau K$ for $\omega\notin\Omega_{K}(T)$. From here
and (7.17) we see that if $f$ is such that $|f|\leq 1$ and Lip$\,f\leq 1$,
then

(7.18)
$\begin{split}\mathbf{E}\big{(}f(a_{1}(\tau))-f(a_{2}(\tau))\big{)}&\leq
2\mathbf{P}(\Omega_{K}(\tau))+\bar{d}\exp\big{(}C\tau K)\\\
&=2(C_{m^{\prime}_{0}}(M)+1)(\tau+1)K^{-1}+\bar{d}\exp\big{(}C\tau
K)\quad\forall\,K>0.\end{split}$

Let us denote by $g^{1}_{M}(\tau,\bar{d})$ the function in the r.h.s. with
$K=\ln\ln\big{(}\bar{d}^{-1}\vee 3\big{)}$. This is a continuous function of
$(\tau,\bar{d},M)\in\mathbb{R}_{+}^{3}$, vanishing when $\bar{d}=0$. Due to
(7.2) and (7.18),

(7.19)
$\begin{split}\|\mathcal{D}(a(\tau;v^{1}))-\mathcal{D}(a(\tau;v^{2}))\|_{L}^{*}&=\|\mathcal{D}(a_{1}(\tau))-\mathcal{D}(a_{2}(\tau))\|_{L}^{*}\\\
&\leq{\mathfrak{g}}_{M}(\tau)\wedge g_{M}^{1}(\tau,\bar{d})\wedge
2=:g_{M}^{2}(\tau,\bar{d})\quad\text{if}\;\;|v^{1}-v^{2}|\leq\bar{d}.\end{split}$

The function $g_{M}^{2}$ is continuous in the variables $(\tau,\bar{d},M)$,
vanishes with $\bar{d}$ and goes to zero when $\tau\to\infty$ since
$\mathfrak{g}_{M}(\tau)$ does.

ii) At this step we consider a solution $a^{0}(\tau;\mu)=:a(\tau;\mu)$ of
(4.5) with $\mathcal{D}(a(0))=\mu$ as in Assumption 7.1.$(3^{\prime})$ and
examine the l.h.s of (7.1) as a function of $\tau$. For any $M>0$ consider the
conditional probabilities
$\mu_{M}=\mathbf{P}(\cdot\mid\bar{B}_{M}(\mathbb{C}^{n}))$ and
$\bar{\mu}_{M}=\mathbf{P}(\cdot\mid\mathbb{C}^{n}\setminus\bar{B}_{M}(\mathbb{C}^{n})).$
Then

(7.20)
$\mathcal{D}(a(\tau;\mu))=A_{M}\mathcal{D}(a(\tau;\mu_{M}))+\bar{A}_{M}\mathcal{D}(a(\tau;\bar{\mu}_{M})),$

where $A_{M}=\mu(\bar{B}_{M}(\mathbb{C}^{n}))$ and
$\bar{A}_{M}=\mu(\mathbb{C}^{n}\setminus\bar{B}_{M}(\mathbb{C}^{n}))$ (cf.
(7.11)). As $\mathbf{E}|a(0)|^{2m_{0}^{\prime}}\leq M^{\prime}$, then
$\bar{A}_{M}=\mathbf{P}\\{a(0)>M\\}\leqslant M^{\prime}/M^{2m_{0}^{\prime}}$.
Since eq. (4.5) is assumed to be mixing, then
$\|\mathcal{D}(a(\tau;0))-\mu^{0}\|_{L}^{*}\leq\bar{g}(\tau),$ where
$\bar{g}\geq 0$ is a continuous function, going to 0 as $\tau\to\infty$. So in
view of (7.2),

$\|\mathcal{D}(a(\tau;v))-\mu^{0}\|_{L}^{*}\leq{\mathfrak{g}}_{M}(\tau)+\bar{g}(\tau)=:\tilde{g}_{M}(\tau),\qquad\forall\,v\in\bar{B}_{M}(\mathbb{C}^{n}).$

From here,

$\begin{split}\|\mathcal{D}(a(\tau;\mu_{M}))-\mu^{0}\|_{L}^{*}=\|\int\big{[}\mathcal{D}(a(\tau;v))\big{]}\mu_{M}(dv)-\mu^{0}\|_{L}^{*}\leq\int\|\mathcal{D}(a(\tau;v))-\mu^{0}\|_{L}^{*}\mu_{M}(dv)\leq\tilde{g}_{M}(\tau).\end{split}$

Therefore due to (7.20),

$\begin{split}\|\mathcal{D}(a(\tau;\mu))-&\mu^{0}\|_{L}^{*}\leqslant
A_{M}\|\mathcal{D}(a(\tau;\mu_{M}))-\mu^{0}\|_{L}^{*}+\bar{A}_{M}\|\mathcal{D}(a(\tau;\bar{\mu}_{M}))-\mu^{0}\|_{L}^{*}\\\
&\leq\|\mathcal{D}(a(\tau;\mu_{M}))-\mu^{0}\|_{L}^{*}+2\bar{A}_{M}\leq\tilde{g}_{M}(\tau)+2\frac{M^{\prime}}{M^{2m^{\prime}_{0}}}\quad\text{for
any $M>0$ and $\tau\geq 0$. }\end{split}$

Let $M_{1}(\tau)>0$ be a continuous non-decreasing function, growing to
infinity with $\tau$, and such that $\tilde{g}_{M_{1}(\tau)}(\tau)\to 0$ as
$\tau\to\infty$ (it exists since $\tilde{g}_{M}(\tau)$ is a continuous
function of $(M,\tau)$, going to 0 as $\tau\to\infty$ for each fixed $M$).
Then

(7.21) $\|\mathcal{D}(a(\tau;\mu))-\mu^{0}\|_{L}^{*}\leqslant
2\frac{M^{\prime}}{M_{1}(\tau)^{2m_{0}^{\prime}}}+\tilde{g}_{M_{1}(\tau)}(\tau)=:{\hat{g}}_{M^{\prime}}(\tau).$

Obviously ${\hat{g}}_{M^{\prime}}(\tau)\geq 0$ is a continuous function on
$\mathbb{R}_{+}^{2}$ , converging to $0$ as $\tau\to\infty$.

iii) Now we examine the l.h.s of (7.1) as a function of $\tau$ and $d$. Recall
that the Kantorovich distance between measures $\nu_{1},\nu_{2}$ on
$\mathbb{C}^{n}$ is

$\|\nu_{1}-\nu_{2}\|_{K}=\sup_{\text{Lip}\,f\leq 1}\langle
f,\nu_{1}\rangle-\langle f,\nu_{2}\rangle\leq\infty.$

Obviously $\|\nu_{1}-\nu_{2}\|_{L}^{*}\leq\|\nu_{1}-\nu_{2}\|_{K}$. Due to
(7.6) and the assumption on $\mu$, the $2m_{0}^{\prime}$-moments of $\mu$ and
$\mu^{0}$ are bounded by $M^{\prime}\vee C_{m_{0}^{\prime}}$. Thus

(7.22) $\|\mu-\mu^{0}\|_{K}\leq\tilde{C}(M^{\prime}\vee
C_{m_{0}^{\prime}})^{\gamma_{1}}d^{\gamma_{2}}:=D,\quad\gamma_{1}=\tfrac{1}{2m^{\prime}_{0}},\;\gamma_{2}=\tfrac{2m^{\prime}_{0}-1}{2m_{0}^{\prime}},$

see [5, Section 11.4] and [26, Chapter 7]. So by the Kantorovich–Rubinstein
theorem (see [26, 5]) there exist r.v.’s $\xi$ and $\xi_{0}$, defined on a new
probability space
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbf{P}^{\prime})$, such that
$\mathcal{D}(\xi)=\mu$, $\mathcal{D}(\xi_{0})=\mu^{0}$ and

(7.23) $\mathbf{E}\,|\xi_{1}-\xi_{0}|=\|\mu-\mu^{0}\|_{K}.$

Then using (7.19) and denoting by $a_{st}(\tau)$ a stationary solution of eq.
(4.5), $\mathcal{D}(a_{st}(\tau))\equiv\mu^{0}$, we have:

$\begin{split}\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}=\|\mathcal{D}(a(\tau;a(0)))-\mathcal{D}(a_{st}(\tau))\|_{L}^{*}\leq\mathbf{E}^{\omega^{\prime}}\|\mathcal{D}(a(\tau;\xi^{\omega^{\prime}}))-\mathcal{D}(a(\tau;\xi_{0}^{\omega^{\prime}}))\|_{L}^{*}\\\
\leq\mathbf{E}^{\omega^{\prime}}g_{\bar{M}}^{2}(\tau,|\xi^{\omega^{\prime}}-\xi_{0}^{\omega^{\prime}}|),\qquad{\bar{M}}={\bar{M}}^{\omega^{\prime}}=|\xi^{\omega^{\prime}}|\vee|\xi_{0}^{\omega^{\prime}}|.\end{split}$

As $\mathbf{E}^{\omega^{\prime}}{\bar{M}}^{2m^{\prime}_{0}}\leq
2(M^{\prime}\vee C_{m_{0}^{\prime}})$ by (7.5) and the assumption on $\mu$,
then denoting $Q^{\prime}_{K}=\\{\bar{M}\geq K\\}\subset\Omega^{\prime}$, for
any $K>0$ we have

$\mathbf{P}^{\omega^{\prime}}(Q^{\prime}_{K})\leq
2K^{-2m_{0}^{\prime}}(M^{\prime}\vee C_{m_{0}^{\prime}}).$

Since $g_{M}^{2}\leq 2$ and for $\omega^{\prime}\notin Q^{\prime}_{K}$ we have
$|\xi^{\omega^{\prime}}|,|\xi_{0}^{\omega^{\prime}}|\leq K$, then

$\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq
4K^{-2m_{0}^{\prime}}(M^{\prime}\vee
C_{m_{0}^{\prime}})+\mathbf{E}^{\omega^{\prime}}g_{K}^{2}(\tau,|\xi^{\omega^{\prime}}-\xi_{0}^{\omega^{\prime}}|).$

Now let
$\Omega^{\prime}_{r}=\\{|\xi^{\omega^{\prime}}-\xi_{0}^{\omega^{\prime}}|\geq
r\\}$. Then by (7.23) and (7.22),
$\mathbf{P}^{\omega^{\prime}}\Omega^{\prime}_{r}\leq Dr^{-1}$. So

(7.24) $\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq
4K^{-2m^{\prime}_{0}}(M^{\prime}\vee
C_{m^{\prime}_{0}})+2Dr^{-1}+g_{K}^{2}(\tau,r)\qquad\forall\,\tau\geq
0,\;\forall\,K,r>0.$

iv) End of the proof. Let $g_{0}(s)$ be a positive continuous function on
$\mathbb{R}_{+}$ such that $g_{0}(s)\to\infty$ as $s\to+\infty$ and
$|C_{m_{0}^{\prime}}\big{(}g_{0}(s)\big{)}(\ln\ln s)^{-1/2}|\leqslant
2C_{m_{0}^{\prime}}(0)$ for $s\geqslant 3$. Taking $r=D^{1/2}$ and choosing in
the r.h.s. of (7.24) $K=g_{0}(r^{-1})$, we denote the r.h.s as
$g_{M^{\prime}}^{3}(\tau,r)$ (so we substitute in (7.24) $D=r^{2}$ and
$K=g_{0}(r^{-1})$). By (7.24) and the definition of $g_{M}^{2}$ (see (7.18)
and (7.19)), we have

$\begin{split}g_{M^{\prime}}^{3}(\tau,r)&\leqslant
4(g_{0}(r^{-1}))^{-2m^{\prime}_{0}}(M^{\prime}\vee C_{m^{\prime}_{0}})+2r\\\
&+2(C_{m^{\prime}_{0}}(g(r^{-1}))+1)\big{(}\ln\ln(r^{-1}\vee
3)\big{)}^{-1}+r\exp\big{(}C\tau\ln\ln(r^{-1}\vee 3)\big{)}.\end{split}$

By the choice of $g_{0}$, as $r\to 0$ the first, the second and the fourth
terms converge $0$. The third term is $\leqslant
4(C_{m^{\prime}_{0}}(0)+1)(\ln\ln(r^{-1}))^{-1/2}$ for
$r\leqslant\frac{1}{3}$, so it also converges to zero with $r$. Hence
$g_{M^{\prime}}^{3}(\tau,r)$ defines a continuous function on
$\mathbb{R}_{+}^{3}$, vanishing with $r$. Using the expression for $D$ in
(7.22) let us write $r=D^{1/2}$ as $r=R_{M^{\prime}}(d)$, where $R$ is a
continuous function $\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}$, non-decreasing in
$d$ and vanishing with $d$. Setting
$g^{4}_{M^{\prime}}(\tau,d)=g^{3}_{M^{\prime}}(\tau,R_{M^{\prime}}(d\wedge
2))$ we get from the above that

$\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq
g^{4}_{M^{\prime}}(\tau,d)\quad\text{if}\;\;\|\mu-\mu^{0}\|_{L}^{*}\leq d\leq
2.$

Finally, evoking (7.21) we arrive at (7.1) with $g=g^{5}$, where

$g^{5}_{M^{\prime}}(\tau,d)=g^{4}_{M^{\prime}}(\tau,d)\wedge{\hat{g}}_{M^{\prime}}(\tau)\wedge
2.$

The function $g^{5}$ is continuous, vanishes with $d$ and converges to zero as
$\tau\to\infty$. For any fixed ${M^{\prime}}>0$ this convergence is uniform in
$d$ due to the term $\hat{g}_{M^{\prime}}(\tau)$. So for a fixed
${M^{\prime}}>0$ the function $(\tau,d)\mapsto g_{M^{\prime}}^{5}(\tau,d)$
extends to a continuous function on the compact set $[0,\infty]\times[0,2]$,
where it vanishes when $\tau=\infty$. Thus $g_{M^{\prime}}^{5}$ is uniformly
continuous in $d$, and the assertion of the proposition is proved.

##  8\. Averaging for systems with general noises

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

Let us consider $v$-equation (2.6) with a general noise (possibly non-
additive) and decomplexify it by writing the components $v_{k}(\tau)$ as
$(\tilde{v}_{2k-1}(\tau),\tilde{v}_{2k}(\tau))\in\mathbb{R}^{2}$,
$k=1,\dots,n$. Now a solution $v(\tau)$ is a vector in $\mathbb{R}^{2n}$ and
the equation reads

(8.1)
$dv(\tau)+\varepsilon^{-1}Av(\tau)\,d\tau=P(v(\tau))\,d\tau+\mathcal{B}(v(\tau))\,d\beta(\tau),\qquad
v(0)=v_{0}\in\mathbb{R}^{2n}.$

Here $A$ is the block-diagonal matrix as in Section 2, $\mathcal{B}(v)$ is a
real $2n\times n_{2}$-matrix and
$\beta(\tau)=(\beta_{1}(\tau),\dots,\beta_{n_{2}}(\tau))$, where
$\\{\beta_{j}(\tau)\\}$ are independent standard real Wiener processes. Note
that in the real coordinates in $\mathbb{R}^{2n}\simeq\mathbb{C}^{n}$ the
operator $\Phi_{w}$ in (2.10) with some vector $w\in\mathbb{R}^{n}$ is given
by a block-diagonal matrix, where for $j=1,\dots,n$ its $j$-th diagonal block
is the $2\times 2$-matrix of rotation by the angle $w_{j}$.

In this section we assume that

######  Assumption 8.1.

The dirft $P\in$ Lip${}_{m_{0}}(\mathbb{R}^{2n},\mathbb{R}^{2n})$, the matrix-
function $\mathcal{B}(v)$ belongs to
Lip${}_{m_{0}}\big{(}\mathbb{R}^{2n},M(2n\times n_{2})\big{)}$, equation (8.1)
is well-posed and its solutions satisfy (2.7).

Passing to the interaction representation
$v(\tau)=\Phi_{\tau\varepsilon^{-1}\Lambda}a(\tau)$ we rewrite the equation as

(8.2)
$da(\tau)=\Phi_{\tau\varepsilon^{-1}\Lambda}P(v(\tau))\,d\tau+\Phi_{\tau\varepsilon^{-1}\Lambda}\mathcal{B}(v(\tau))\,d\beta(\tau),\qquad
a(0)=v_{0}.$

As in Section 4 we will see that as $\varepsilon\to 0$, the asymptotic
behaviour of distributions of the equation’s solutions is described by an
effective equation. As before, the effective drift is $\llangle P\rrangle(a)$.
To calculate the effective dispersion, as in the proof of Lemma 4.6 we
consider the martingale

$N^{Y,\varepsilon}:=a^{\varepsilon}(\tau)-\int_{0}^{\tau}Y(a^{\varepsilon}(s),s\varepsilon^{-1})ds=v_{0}+\int_{0}^{\tau}\mathcal{B}^{\Lambda}(a^{\varepsilon}(s);s\varepsilon^{-1})d\beta(s),$

where $Y$ is defined in (4.7) and
$\mathcal{B}^{\Lambda}(a;t)=\Phi_{t\Lambda}\mathcal{B}(\Phi_{-t\Lambda}a)$. By
Itô’s formula, for $i,j=1,\dots,n$ the process

$N_{i}^{Y,\varepsilon}(\tau)N_{j}^{Y,\varepsilon}(\tau)-\int_{0}^{\tau}\mathcal{A}_{ij}^{\Lambda}(a^{\varepsilon}(s);s\varepsilon^{-1})ds,\qquad(\mathcal{A}_{ij}^{\Lambda}(a;t))=\mathcal{B}^{\Lambda}(a;t)\mathcal{B}^{\Lambda*}(a;t),$

where $\mathcal{B}^{\Lambda*}$ is the transpose of $\mathcal{B}^{\Lambda}$,
also is a martingale. By a straightforward analogy of Lemma 3.2, the limit

$\mathcal{A}^{0}(a):=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\mathcal{A}^{\Lambda}(a;t)dt$

exists and belongs to $\text{Lip}_{2m_{0}}(\mathbb{R}^{2n},M(2n\times 2n))$.
Then, by analogy with Lemma 4.3,

$\mathbf{E}\Big{|}\int_{0}^{\tau}\mathcal{A}^{\Lambda}(a^{\varepsilon}(s);s\varepsilon^{-1})ds-\int_{0}^{\tau}\mathcal{A}^{0}(a^{\varepsilon}(s))ds\Big{|}\to
0,\;\text{ as }\varepsilon\to 0,$

for any $\tau\geqslant 0$. From here, as in Section 4, we conclude that now
for the effective diffusion we should take $\mathcal{A}^{0}(a)$, which is a
non-negative symmetric matrix. Denoting by
$\mathcal{B}^{0}(a)=\mathcal{A}^{0}(a)^{1/2}$ its principal square root, as in
Section 4 we verify that any limiting measure $Q_{0}$ as in (2.14) is a
solution for the martingale problem for the effective equation

(8.3) $da(\tau)-{\llangle
P\rrangle}(a(\tau))d\tau=\mathcal{B}^{0}(a(\tau))d\beta(\tau),\qquad
a(0)=v_{0},$

and so is a weak solution of the equation. If the noise in equation (8.1) is
additive, then $\mathcal{B}^{0}$ is a constant matrix, eq. (8.3) has a unique
solution and Theorem 4.7 with (modified) effective equation (8.3) remains true
for solutions of eq. (8.2). In particular, the theorem applies to equation
(2.5) with general additive random forces (2.4) (but then the effective
dispersion matrix is given by a more complicated formula than in Section 4).

Similarly, if the diffusion in (8.1) is non-degenerate in the sense that

(8.4)
$|\mathcal{B}(v)\mathcal{B}^{*}(v)\xi|\geqslant\alpha|\xi|,\qquad\forall\,v,\;\forall\,\xi\in\mathbb{R}^{2n},$

for some $\alpha>0$, then the matrix $\mathcal{B}^{\Lambda}(a,\tau)$ also
satisfies (8.4) for all $a$ and $\tau$, i.e.
$\langle\mathcal{A}^{\Lambda}(a;s\varepsilon^{-1})\xi,\xi\rangle\geqslant\alpha|\xi|^{2}$.
Thus $\mathcal{A}^{0}(a)\geqslant\alpha\mathbb{I}$, and so
$\mathcal{B}^{0}(a)=\mathcal{A}^{0}(a)^{1/2}$ is a locally Lipschitz matrix
function of $a$ (e.g. see [24, Theorem 5.2.2]). So again eq. (8.3) has a
unique solution and Theorem 4.7 remains true for eq. (8.1) (with the effective
equation of the form (8.3)).

To treat equations (8.1) with degenerate non-additive noises we write the
matrix $\mathcal{A}^{0}(a)$ as

$\mathcal{A}^{0}(a)=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\big{(}\Phi_{t\Lambda}\mathcal{B}(\Phi_{-t\Lambda}a)\big{)}\cdot\big{(}\Phi_{t\Lambda}\mathcal{B}(\Phi_{-t\Lambda}a)\big{)}^{*}dt.$

By the same reason as in Proposition 3.4 we find that

$|\mathcal{A}^{0}|_{C^{2}(B_{R})}\leqslant
C|\mathcal{B}|_{C^{2}(B_{R})}^{2},\quad\forall R>0.$

Now using [24, Theorem 5.2.3], we get that

(8.5) $\text{Lip}(\mathcal{B}^{0}(a)|_{\bar{B}_{R}})\leqslant
C|\mathcal{A}^{0}|_{C^{2}(B_{R+1})}^{1/2}\leqslant
C_{1}|\mathcal{B}|_{C^{2}(B_{R+1})},\quad\forall R>0.$

So the matrix-function $\mathcal{B}^{0}(a)$ is locally Lipschitz continuous,
eq. (8.3) has a unique solution and the assertion of Theorem 4.7 remains true
for eq. (8.1). We have obtained

######  Theorem 8.2.

Suppose that Assumption 8.1 holds and for the matrix function $\mathcal{B}(v)$
in (8.1) one of the following three options is true:

  1. (1)

it is $v$-independent;

  2. (2)

it satisfies the non-degeneracy condition (8.4);

  3. (3)

it is a $C^{2}$-smooth matrix-function of $v$.

Then for any $v_{0}\in\mathbb{R}^{2n}$ a solution
$a^{\varepsilon}(\tau;v_{0})$ of eq. (8.2) satisfies

$\mathcal{D}(a^{{\varepsilon}}(\cdot;v_{0}))\rightharpoonup Q_{0}\text{ in
}\mathcal{P}(C([0,T],\mathbb{C}^{n}))\quad\text{ as }{\varepsilon}\to 0,$

where $Q_{0}$ is the law of a unique weak solution of the effective equation
(8.3).

An obvious analogy of Corollary 4.13 holds for solutions of eq. (8.1).

##  9\. A sufficient condition for Assumptions 2.1, 5.1 and 7.1

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

Consider a stochastic differential equation on $\mathbb{R}^{l}$,

(9.1)
$dx=b(x)d\tau+\sigma(x)d\beta(\tau),\;\;\;x\in\mathbb{R}^{l},\;\tau\geqslant
0,$

where $\sigma(x)$ is an $l\times k$-matrix and $\beta(\tau)$ is a standard
Wiener processes in $\mathbb{R}^{k}$. We assume

######  Assumption 9.1.

The drift $b(x)$ and dispersion $\sigma(x)$ are locally Lipschitz in $x$, and
$\mathcal{C}^{m}(b)$, $\mathcal{C}^{m}(\sigma)\leq C<\infty$ for some $m\geq
0$.

The diffusion $a(x)=\sigma(x)\cdot\sigma^{t}(x)$ is a nonnegative symmetric
$l\times l$-matrix. Consider the differential operator

$\mathscr{L}\big{(}v(x)\big{)}=\sum_{j=1}^{l}b_{j}(x)\frac{\partial
v}{\partial
x_{j}}+\frac{1}{2}\sum_{i=1}^{l}\sum_{j=1}^{l}a_{ij}(x)\frac{\partial^{2}v}{\partial
x_{i}\partial x_{j}}.$

We have the following result concerning the well-posedness of eq. (9.1) from
[17, Theorem 3.5]:

######  Theorem 9.2.

Assume Assumption 9.1 and suppose that there exists a nonnegative function
$V(x)\in C^{2}(\mathbb{R}^{l})$ such that for some constant $c>0$ we have

$\mathscr{L}(V(x))\leqslant cV(x)\quad\forall\tau\geqslant
0,\;x\in\mathbb{R}^{l},$

and

$\inf_{|x|>R}V(x)\to\infty\;\text{ as }R\to\infty.$

Then for any $x_{0}\in\mathbb{R}^{l}$ equation (9.1) has a unique strong
solution $X(\tau)$ with the initial condition $X(0)=x_{0}$. Furthermore, the
process $X(\tau)$ satisfies

${\mathbf{E}}V(X(\tau))\leqslant e^{c\tau}V(x_{0})\quad\forall\tau\geqslant
0.$

The function $V$ is called a Lyapunov function for equation (9.1). In its
terms a sufficient condition for the mixing in equation (9.1) is given by the
following statement:

######  Proposition 9.3.

Assume that in addition to Assumption 9.1 we have

  1. (1)

the drift $b$ satisfies

(9.2) $\langle
b(x),x\rangle\leqslant-{\alpha_{1}}|x|+{\alpha_{2}}\qquad\forall
x\in\mathbb{R}^{l},$

for some constants ${\alpha_{1}}>0$ and ${\alpha_{2}}\geqslant 0$, where
$\langle\cdot,\cdot\rangle$ is the standard inner product in $\mathbb{R}^{l}$.

  2. (2)

The diffusion matrix $a(x)=\sigma(x)\sigma(x)^{t}$ is uniformly non-
degenerate, that is

(9.3) $\gamma_{2}\mathbb{I}\leqslant
a(x)\leqslant\gamma_{1}\mathbb{I}\quad\forall x\in\mathbb{R}^{l},$

for some $\gamma_{1}\geqslant\gamma_{2}>0$.

Then for any $c^{\prime}>0$ equation (9.1) has a smooth Lyapunov function
$V(x)$, equal $\exp(c^{\prime}|x|)$ for $|x|\geq 1$, estimate (5.1) holds true
for its solutions for every $m\in\mathbb{N}$, and the equation is mixing.

In Appendix A it is shown how to derive the proposition from abstract results
in [17]. Moreover, it can be proved that under the assumptions of the
proposition the equation is exponentially mixing and (7.3) holds, see Example
7.3.

Let us decomplexify $\mathbb{C}^{n}$ to $\mathbb{R}^{2n}$ and identify
equation (2.6) with a real equation (9.1), where $l=2n$ (and $x=v$). Then
$b(v)\cong(b_{j}(v)=(-i{\varepsilon}^{-1}\lambda_{j}v_{j}+P_{j}(v),\;j=1,\dots,n)$,
where $b_{j}\in\mathbb{C}\cong\mathbb{R}^{2}\subset\mathbb{R}^{2n}$. Since in
complex terms the real inner product reads as $\langle v,w\rangle=Re\sum
v_{j}\bar{w}_{j}$, then

$\langle b(v),v\rangle=\langle P(v),v\rangle.$

So for equation (2.6) condition (9.2) is equivalent to

(9.4) $\langle
P(v),v\rangle\leqslant-{\alpha_{1}}|v|+{\alpha_{2}},\quad\forall
v\in\mathbb{C}^{n},$

for some constants ${\alpha_{1}}>0$ and ${\alpha_{2}}\geqslant 0$.

Now consider effective equation (4.5). Since in eq. (2.11) the drift is

$Y(a,\tau{\varepsilon}^{-1})=(\Phi_{\tau{\varepsilon}^{-1}\Lambda})_{*}P(a),$

then under the assumption (9.4) we have

$\langle Y(a,\tau{\varepsilon}^{-1}),a\rangle=\langle
P(\Phi_{\tau{\varepsilon}^{-1}\Lambda}a),\Phi_{\tau{\varepsilon}^{-1}\Lambda}a\rangle\leqslant-\alpha_{1}|\Phi_{-\tau{\varepsilon}^{-1}\Lambda}a|+{\alpha_{2}}=-{\alpha_{1}}|a|+{\alpha_{2}},$

for all ${\varepsilon}$. Therefore $\llangle P\rrangle$ satisfies

$\langle{\llangle
P\rrangle}(a),a\rangle=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\langle
Y(a,\tau{\varepsilon}^{-1}),a\rangle
d\tau\leqslant-{\alpha_{1}}|a|+{\alpha_{2}}.$

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

As it was pointed out, if the dispersion matrix $\Psi$ is non-degenerate, then
the dispersion $B$ in the effective equation is non-degenerate as well. The
corresponding diffusion matrix also is non-degenerate and condition (9.3)
holds for it. Thus we obtained the following statement:

######  Proposition 9.4.

If in equation (2.6) the dispersion matrix $\Psi$ is non-degenerate, the drift
$P\in\text{Lip}_{m_{0}}$ for some $m_{0}\in\mathbb{N}$ and (9.4) holds for
some constants ${\alpha_{1}}>0$ and ${\alpha_{2}}\geqslant 0$, then the
assumption of Theorem 5.5 holds true, and so does the assumption of Theorems
4.7 and 7.4.

##  Appendix A Proof of Proposition 9.3

By condition (9.3) the diffusion $a$ is uniformly bounded. So there exist
constants $k_{1},k_{2}>0$ such that

(A.1) $\text{Tr}(a(x))\leqslant k_{1},\;\|a(x)\|\leqslant k_{2}\quad\forall
x\in\mathbb{R}^{l}.$

Take $V(x)=e^{{c^{\prime}}f(x)}$, where ${c^{\prime}}>0$ is a positive
constant and $f(x)$ is a non-negative smooth function, equal $|x|$ for
$|x|\geq 1$, and such that its first and second derivatives are bounded by 3.
Then

$\frac{\partial V(x)}{\partial
x_{j}}={c^{\prime}}V(x)\partial_{x_{j}}f(x),\quad\frac{\partial^{2}V(x)}{\partial
x_{j}\partial
x_{j}}={c^{\prime}}V(x)\partial_{x_{i}x_{j}}f(x)+{c^{\prime}}^{2}V(x)\partial_{x_{i}}f(x)\partial_{x_{j}}f(x).$

Therefore we have that

$\mathscr{L}\big{(}V(x)\big{)}={c^{\prime}}V(x)\mathcal{K}(c^{\prime},x),$

where

$\mathcal{K}(c^{\prime},x)=\sum_{j=1}^{l}b_{j}(x)\partial_{x_{j}}f(x)+\frac{1}{2}\sum_{ij}a_{ij}(x)\partial_{x_{i}x_{j}}f(x)+\frac{1}{2}c^{\prime}\sum_{ij}a_{ij}(x)\partial_{x_{i}}f(x)\partial_{x_{j}}f(x).$

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

(A.2)
$\begin{cases}|\mathcal{K}(c^{\prime},x)|\leqslant(c^{\prime}+1)C,&\text{if
}|x|<1,\\\
\mathcal{K}(c^{\prime},x)\leqslant(-\alpha_{1}+\frac{\alpha_{2}}{|x|}+\frac{C}{|x|}+c^{\prime}C),&\text{if