# Homogenization of a diffusion in a high-contrast random environment and
related Markov semigroups

B. Amaziane1, A. Piatnitski2,3,111The work of the second author was supported
by Leonhard Euler International Mathematical Institute in Saint Petersburg,
grant No. 075-15-2019-1619  , E. Zhizhina3,444The work of the third author was
supported by Leonhard Euler International Mathematical Institute in Saint
Petersburg, grant No. 075-15-2019-1620

(1 Laboratoire de Mathématiques et de leurs Applications de Pau - UMR CNRS
5142,
University of Pau and Pays de l’Adour Avenue de l’Université - BP 1155
64013 Pau, France
2 The Arctic university of Norway, campus Narvik,
P.O.Box 385, 8505 Narvik, Norway
3 The Institute for Information Transmission Problems of RAS,
19, Bolshoy Karetny per. 127051, Moscow, Russia
)

###### Abstract

The goal of the paper is to describe the large time behaviour of a Markov
process associated with a symmetric diffusion in a high-contrast random
environment and to characterize the limit semigroup and the limit process
under the diffusive scaling.

##  1 Introduction

The paper focuses on the large time behaviour of a diffusion in a high
contrast random statistically homogeneous environment. We also study the limit
behaviour of the corresponding semigroups. Equivalently, we consider the limit
behaviour of a diffusion defined in a high contrast environment with a random
microstructure on a finite time interval.

Elliptic and parabolic operators with high contrast rapidly oscillating
periodic coefficients have been widely studied in the homogenization theory.
The first rigorous results for parabolic operators of this type were obtained
in [10] and [3]. In particular, it was shown that, under proper choice of the
scaling coefficient, the homogenized problem contains a non-local in time
operator which reflects the so-called memory effect. Later on in [2], with the
help of the two-scale convergence technique, the limit problem was written as
a coupled system of parabolic PDEs in the space with higher number of
variables. In the works [16], [17] high contrast problems in domains with
singular or asymptotically singular periodic geometry were considered. At
present, there are many works in the existing mathematical literature that
describe the effective behaviour of high contrast periodic media. Under proper
scaling, in parabolic problems this usually results in the memory effect while
homogenization of spectral problems leads to a non-linear dependence on the
spectral parameter.

In this paper we deal with second order divergence form operators in
$\mathbb{R}^{d}$. Each such an operator is a generator of a Markov semigroup.
The corresponding Markov process (generalized diffusion) has continuous
trajectories. However, the presence of a non-local in temporal variable term
in the effective operator means that the limit dynamics of the coordinate
process is not Markov.

The goal of this work is to equip the coordinate process with additional
components in such a way that the dynamics of the enlarged process remains
Markovian in the limit. We show that it is sufficient to combine the
coordinate process in $\mathbb{R}^{d}$ with a position of the diffusion inside
the rescaled inclusions for the time intervals when the diffusion is trapped
by one of the inclusions.

It is interesting to observe that, although in the original processes the
additional components are functions of the coordinate process, in the limit
process these components are getting independent while the coordinate process
becomes coupled with them.

The explicit form of the limit operator on the extended space gives us a
possibility to use this operator as a generator of an approximation dynamics
for the processes in high contrast random stationary dispersive porous media.
The discrete version of such approximation process was constructed in the work
[13], where we considered a discrete diffusion in a high-contrast random
environment given by a jump random walk on the lattice $\mathbb{Z}^{d}$. The
crucial step in this construction is to describe the ”clock process” governing
transitions from the observable ”real” space to the supplementary ”astral”
spaces and back. The ”clock process” is a continuous time finite Markov chain
with transition rates depending on parameters of the limit operator.

In the paper we introduce proper functional spaces, construct the limit
semigroup, and prove the semigroup convergence.

In addition to proving the semigroup convergence, we study the spectrum of the
generator of the limit semigroup. Then the semigroup convergence in $L^{2}$
spaces allows us to provide some information about the limit behaviour of the
spectrum of the original operators.

To our best knowledge, the questions considered in this paper have not been
studied in the existing literature. In the discrete framework the results on
scaling limits of symmetric random walks in a high contrast periodic
environment were obtained in our previous work [13].

Our approach essentially relies on the approximation technique developed in
[5] and the technique of correctors in random media. In contrast with the
periodic framework, the auxiliary operators used to introduce correctors need
not be of Fredholm type in the case of random inclusions. The construction of
the first corrector can be found in the existing literature, see for instance
[9]. However, when defining the higher order correctors we face additional
difficulties.

##  2 Problem setup

Let $(\Omega,\mathcal{F},\mathbf{P})$ be a standard probability space.
Consider a symmetric diffusion operator in divergence form

$A^{\omega}_{\varepsilon}f(x)=\mathrm{div}\left(a^{\omega}_{\varepsilon}(x)\nabla
f(x)\right),$ (1)

where

$a^{\omega}_{\varepsilon}(x)=\left\\{\begin{array}[]{l}\mathbb{I},\quad
x\in\mathbb{R}^{d}\setminus\varepsilon
G^{\omega}=\big{(}G^{\omega}_{\varepsilon}\big{)}^{c},\\\
\varepsilon^{2}\mathbb{I},\quad x\in\varepsilon G^{\omega},\end{array}\right.$
(2)

$\mathbb{I}$ being the unit matrix. Here $G^{\omega}_{\varepsilon}=\varepsilon
G^{\omega}$ and $G^{\omega}\subset\mathbb{R}^{d},\,\omega\in\Omega,$ is a
classical disperse medium. That is $\mathbb{R}^{d}\setminus G^{\omega}$ is a
random statistically homogeneous set such that almost surely (a.s) it is
connected and unbounded and its complement $G^{\omega}\subset\mathbb{R}^{d}$
consists of a countable number of uniformly bounded simply connected domains
with uniformly Lipschitz boundary. Moreover, the distance between any two such
domains admits a uniform deterministic lower bound. The set $G^{\omega}$
corresponds to the matrix blocks (inclusions), and $\mathbb{R}^{d}\setminus
G^{\omega}\subset\mathbb{R}^{d}$ to the fractures system, see [4].

To be more specific in this work we consider a class of disperse media that
satisfy the following additional condition:
there is a finite collection of bounded domains in $\mathbb{R}^{d}$ such that
all these domains are uniformly $C^{2}$ regular and a.s. any connected
component of $G^{\omega}$ can be obtained by a proper translation and rotation
of one of these domains. The domains are denoted by ${\cal D}_{j}$,
$j=1,\ldots,N$, and the whole collection by ${\cal D}$,
$\mathcal{D}=\\{\mathcal{D}_{j}\\}_{j=1}^{N}$ with $N\in\mathbb{Z}^{+}$. We
assume without loss of generality that each domain $\mathcal{D}_{j}$, $j\geq
1$, contains the origin.

An example of such a disperse medium is associated with a Bernoulli site
percolation model on the lattice $\mathbb{Z}^{d}$ embedded in
$\mathbb{R}^{d}$. Let $\\{\xi_{j},\;j\in\mathbb{Z}^{d}\\},\
\xi_{j}\in\\{0,1\\}$ be a sequence of i.i.d. random variables having the
Bernoulli law: $\mathbb{P}(\xi_{j}=1)=p,\;\mathbb{P}(\xi_{j}=0)=1-p$. We then
define $\mathtt{B}_{j}=j+[-\frac{1}{2},\frac{1}{2}]^{d}$,
$j\in\mathbb{Z}^{d}$, and consider the set
$\mathtt{G}^{1,\omega}=\bigcup\limits_{\\{j\,:\,\xi_{j}=1\\}}\,\mathtt{B}_{j}$.
This set is a.s. a union of countable number of bounded connected sets
(components) and not more than one unbounded connected component, see [8]. We
replace each bounded connected component of $\mathtt{G}^{1,\omega}$ with the
minimal simply connected set that contains this component. Then we choose the
sets that have Lipschitz boundary and whose volumed does not exceed $M$, where
$M\geq 1$ is a given positive integer. We then smoothen these sets and obtain
a finite collection $\mathcal{D}$ of the reference sets . By the standard
arguments of percolation theory, $G^{\omega}$ is statistically homogeneous and
ergodic.

Denote by ${\cal G}^{\omega}_{j}$ the subset of $G^{\omega}$ that consists of
all the components which have the same geometry as $\mathcal{D}_{j}$, that is
each such a component can be obtained by a proper rotation and translation of
$\mathcal{D}_{j}$ in $\mathbb{R}^{d}$. The connected components of ${\cal
G}^{\omega}_{j}$ are denoted by $\\{{\cal
G}^{\omega,i}_{j}\\}_{i\in\mathbb{Z}^{+}}$. We also denote ${\cal
G}^{\omega}_{0}=\mathbb{R}^{d}\setminus G^{\omega}$.

Letting

$\alpha_{0}=\mathbf{P}\big{\\{}0\in(G^{\omega})^{c}\big{\\}},\quad\alpha^{o}_{j}=\mathbf{P}\big{\\{}0\in{\cal
G}^{\omega}_{j}\big{\\}},\quad\alpha_{0}+\sum_{\ j\geq 1}\alpha^{o}_{j}=1,$
(3)

we assume without loss of generality that $\alpha^{o}_{j}>0$ for all $j$. We
then introduce

$\alpha_{j}=|\mathcal{D}_{j}|^{-1}\alpha^{o}_{j}=|\mathcal{D}_{j}|^{-1}\mathbf{P}\big{\\{}0\in{\cal
G}^{\omega}_{j}\big{\\}}.$ (4)

For each $\varepsilon>0$ the operator $A^{\omega}_{\varepsilon}$ has random
statistically homogeneous coefficients in $\mathbb{R}^{d}$, where the
randomness is defined through the random geometry of $G^{\omega}$. These
operators are also called metrically transitive with respect to the unitary
group of the space translations in $\mathbb{R}^{d}$. In
$L^{2}(\mathbb{R}^{d})$ we introduce a domain of $A^{\omega}_{\varepsilon}$ by

$\begin{array}[]{rl}D(A^{\omega}_{\varepsilon})=\Big{\\{}f\in
H^{1}(\mathbb{R}^{d}),\;f\in H^{2}({G_{\varepsilon}^{\omega}})\cap
H^{2}({\mathbb{R}^{d}\setminus G_{\varepsilon}^{\omega}}),\\\\[4.2679pt]
\varepsilon^{2}\nabla f(x)\big{|}_{\partial G_{\varepsilon}^{\omega}}\cdot
n^{+}=-\nabla f(x)\big{|}_{\partial G_{\varepsilon}^{\omega}}\cdot
n^{-}\Big{\\}}\end{array}$ (5)

The last relation in (5) represents the continuity of flux
$a_{\varepsilon}\nabla f$ through the boundary $\partial
G_{\varepsilon}^{\omega}$. Here $n^{-},n^{+}$ are respectively the internal
and external normals on $\partial G_{\varepsilon}^{\omega}$.

######  Remark 2.1.

Notice that for any function $v\in D(A^{\omega}_{\varepsilon})$ its trace and
the trace of its flux on the interface $\partial G_{\varepsilon}^{\omega}$ is
a well-defined $L^{2}(\partial G_{\varepsilon}^{\omega})$ function.

Then $(A^{\omega}_{\varepsilon},D(A^{\omega}_{\varepsilon}))$ is almost surely
a self-adjoint operator in $L^{2}(\mathbb{R}^{d})$, and for any $\lambda>0$
the operator $(\lambda-A^{\omega}_{\varepsilon})$ is coercive. By the Hille-
Yosida theorem, $A^{\omega}_{\varepsilon}$ is a generator of a strongly
continuous, positive, contraction semigroup $T^{\omega}_{\varepsilon}(t)$ on
$L^{2}(\mathbb{R}^{d})$ for a.e. $\omega\in\Omega$.

##  3 The limit operator.

In this section we describe the generator of the limit Markov semigroup.
Denote

$E=\mathbb{R}^{d}\times{\cal D}^{\star},\quad\mbox{ where }\;{\cal
D}^{\star}=\\{\star\\}\cup{\cal D}.$

We consider functions $F$ defined on $E$ of the following vector form

$F(x,\xi)=\left\\{\begin{array}[]{l}f_{0}(x),\quad\mbox{if
}\;x\in\mathbb{R}^{d},\,\xi=\star,\\\ f_{j}(x,\xi),\quad\mbox{if
}x\in\mathbb{R}^{d},\,\xi\in{\cal D}_{j},\;j=1,\,\ldots,N,\end{array}\right.$
(6)

with $f_{0}\in L^{2}(\mathbb{R}^{d}),\quad f_{j}\in
L^{2}(\mathbb{R}^{d}\times{\cal D}_{j})$. Equipped with the norm

$\|F\|^{2}=\alpha_{0}\int\limits_{\mathbb{R}^{d}}f^{2}_{0}(x)\,dx+\sum_{j=1}^{N}\alpha_{j}\int\limits_{\mathbb{R}^{d}}\int\limits_{{\cal
D}_{j}}(f_{j})^{2}(x,\xi)\,d\xi dx$ (7)

where $\alpha_{0},\ \alpha_{j}$ was defined in (4), $\alpha_{0}>0,\
\alpha_{j}>0$, it is a Hilbert space. We call this Hilbert space
$L^{2}(E,\alpha)$.

Let us consider in $L^{2}(E,\alpha)$ an operator of the following form

$(AF)(x,\xi)=\left(\begin{array}[]{c}\Theta\cdot\nabla\nabla
f_{0}(x)+\frac{1}{\alpha_{0}}\sum_{j\geq
1}\alpha_{j}\int\limits_{\partial{\cal D}_{j}}\frac{\partial
f_{j}(x,\xi)}{\partial n^{-}_{\xi}}d\sigma(\xi)\\\\[11.38109pt]
\triangle_{\xi}f_{1}(x,\xi)\\\ \ldots\\\
\triangle_{\xi}f_{N}(x,\xi)\end{array}\right)$ (8)

where a positive definite constant matrix $\Theta$ will be defined later on,
$\sigma(\xi)$ is the element of the surface volume on the Lipshitz boundary
$\partial{\cal D}_{j}^{k}$, $n_{\xi}^{-}$ is the (inner) normal to
$\partial{\cal D}_{j}^{k}$. Using the relation $n^{+}=-n^{-}$ and the Stokes
formula one can rewrite the operator (8) as follows:

$(AF)(x,\xi)=\left(\begin{array}[]{c}\Theta\cdot\nabla\nabla
f_{0}(x)-\frac{1}{\alpha_{0}}\sum\limits_{j\geq 1}\alpha_{j}\int\limits_{{\cal
D}_{j}}\triangle_{\xi}f_{j}(x,\xi)d\xi\\\\[11.38109pt]
\triangle_{\xi}f_{1}(x,\xi)\\\ \ldots\\\
\triangle_{\xi}f_{N}(x,\xi)\end{array}\right)$ (9)

We denote

$\Upsilon(x)=-\frac{1}{\alpha_{0}}\sum_{j\geq 1}\alpha_{j}\int\limits_{{\cal
D}_{j}}\triangle_{\xi}f_{j}(x,\xi)d\xi.$ (10)

For each set ${\cal D}_{j}$, $j\geq 1$, denote by $\mathbb{D}_{j}(\Delta)$ the
domain of a self-adjoint operator in $L^{2}({\cal D}_{j})$ that corresponds to
the Laplacian in ${\cal D}_{j}$ with homogeneous Dirichlet boundary
conditions. Since the boundary of ${\cal D}_{j}$ is $C^{2}$ regular, we have
$\mathbb{D}_{j}(\Delta)=H^{2}({\cal D}_{j})\cap H^{1}_{0}({\cal D}_{j})$.
Notice that this operator is positive. The space $\mathbb{D}_{j}(\Delta)$ is
equipped with the norm $\|g\|_{\mathbb{D}_{j}(\Delta)}=\|\Delta
g\|_{L^{2}({\cal D}_{j})}$.

Defining an operator $A$ in $L^{2}(E,\alpha)$ by formulas (8), (9), one can
easily check that, with a domain

$\begin{array}[]{c}\hat{D}(A)=\left\\{f_{0}\in H^{2}(\mathbb{R}^{d}),\
f_{j}-f_{0}\in L^{2}(\mathbb{R}^{d};{\mathbb{D}}_{j}(\Delta)),\
f_{j}(x,\xi)\Big{|}_{\xi\in\partial{\cal D}_{j}}=f_{0}(x),\right.\\\
\left.\Big{(}\Theta\cdot\nabla\nabla f_{0}(x)-\frac{1}{\alpha_{0}}\sum_{j\geq
1}\alpha_{j}\int\limits_{{\cal
D}_{j}}\triangle_{\xi}f_{j}(x,\xi)d\xi,\,\triangle_{\xi}f_{1}(x,\xi),\ldots,\triangle_{\xi}f_{N}(x,\xi)\Big{)}\in
L^{2}(E,\alpha)\right\\},\end{array}$ (11)

the operator $(A,\hat{D}(A))$ is a closed symmetric operator in
$L^{2}(E,\alpha)$, and $\hat{D}(A)$ is dense in $L^{2}(E,\alpha)$.

We introduce the following two spaces:

$H^{1}_{D}(E,\alpha)=\Big{\\{}f_{0}\in H^{1}(\mathbb{R}^{d}),\ f_{j}-f_{0}\in
L^{2}(\mathbb{R}^{d};H_{0}^{1}({\cal D}_{j}))\Big{\\}}$ (12)

and

$H^{2}_{D}(E,\alpha)=\Big{\\{}f_{0}\in H^{2}(\mathbb{R}^{d}),\ f_{j}-f_{0}\in
L^{2}(\mathbb{R}^{d};H^{2}({\cal D}_{j})\cap H_{0}^{1}({\cal
D}_{j}))\Big{\\}}.$ (13)

Notice that

$\sum\limits_{j=1}^{N}\alpha_{j}\int\limits_{\mathbb{R}^{d}}\int\limits_{{\cal
D}{j}}|\nabla_{\xi}f_{j}|^{2}(x,\xi)\,d\xi dx<\infty\quad\forall F\in
H^{1}_{D}(E,\alpha)$

and

$\sum\limits_{j=1}^{N}\alpha_{j}\|f_{j}\|^{2}_{L^{2}(\mathbb{R}^{d};H^{2}(\mathcal{D}_{j}))}<\infty\quad\forall
F\in H^{2}_{D}(E,\alpha).$

The space $H^{-1}(E,\alpha)$ is defined as the dual space to
$H^{1}_{D}(E,\alpha)$ in $L^{2}(E,\alpha)$.

######  Lemma 3.1.

For any $m>0$ the operator $(m-A,\hat{D}(A))$ is a coercive self-adjoint
operator in $L^{2}(E,\alpha)$.

###### Proof.

Consider the following quadratic form in $L^{2}(E,\alpha)$

$\Gamma(F,F)=\alpha_{0}\int\limits_{\mathbb{R}^{d}}\Theta\nabla
f_{0}(x)\cdot\nabla
f_{0}(x)\,dx+\sum_{j=1}^{N}\alpha_{j}\int\limits_{\mathbb{R}^{d}}\int\limits_{{\cal
D}{j}}|\nabla_{\xi}f_{j}|^{2}(x,\xi)\,d\xi dx+m\|F\|^{2}_{L^{2}(E,\alpha)}$
(14)

with a domain $D(\Gamma)=H^{1}_{c}(E,\alpha)$. Notice that
$f_{j}(x,\cdot)\big{|}_{\partial{\cal D}_{j}}=f_{0}(x)$ for any $F\in
D(\Gamma)$. According to [15, Theorem x.x] there exists a unique self-adjoint
operator $\tilde{A}_{m}$ that has the following properties:
\- its domain $D(\tilde{A}_{m})$ is dense in $L^{2}(E,\alpha)$;
\- $D(\tilde{A}_{m})$ belongs to $D(\Gamma)$;
\- $(\tilde{A}_{m}F,F)_{L^{2}(E,\alpha)}=\Gamma(F,F)$ for any $F\in
D(\tilde{A}_{m})$.

We are going to show that $\tilde{A}_{m}$ coincides with $m-A$. First we prove
that $D(\tilde{A}_{m})\subset\hat{D}(A)$. Separating the first component
$f_{0}$ in (6) we will use the notation $F=(f_{0},V)$. Taking $F\in
D(\tilde{A}_{m})$ and $U=(0,U_{1})\in D(\Gamma)$ with $U_{1}\in
C_{0}^{\infty}(\mathbb{R}^{d}\,;\,C_{0}^{\infty}({\cal D}))$, and using the
relation $(\tilde{A}_{m}F,U)_{L^{2}(E,\alpha)}=\Gamma(F,U)$, we obtain

$(\tilde{A}_{m}F,U)_{L^{2}(E,\alpha)}=\Gamma(F,U)=\sum\limits_{j\geq
1}\alpha_{j}((m-\Delta_{\xi})V_{j},U_{1,j}),$

where the terms $(-\Delta_{\xi}V_{j},U_{1,j})$ on the right-hand side are
understood as a pairing between $L^{2}(\mathbb{R}^{d}\,;\,H^{-1}({\cal
D}_{j}))$ and $L^{2}(\mathbb{R}^{d}\,;\,H^{1}_{0}({\cal D}_{j}))$. This
implies that $(m-\Delta_{\xi})V_{j}\in L^{2}(\mathbb{R}^{d}\times{\cal
D}_{j})$ and $\big{(}0,\\{(m-\Delta_{\xi})V_{j}\\}_{j\geq 1}\big{)}\in
L^{2}(E,\alpha)$. Therefore, $(0,V)\in H^{2}_{c}(E,\alpha)$. Choosing now
$U=(u_{0}(x),0)$ with $u_{0}\in C_{0}^{\infty}(\mathbb{R}^{d})$ and
considering the fact that $\sum\limits_{j\geq 1}\alpha_{j}\int\limits_{{\cal
D}_{j}}\triangle_{\xi}V_{j}(\cdot,\xi)d\xi\in L^{2}(\mathbb{R}^{d})$, we get
$mf_{0}-\mathrm{div}(\Theta\nabla f_{0})\in L^{2}(\mathbb{R}^{d})$. Therefore,
$f_{0}\in H^{2}(\mathbb{R}^{d})$, and $D(\tilde{A}_{m})\subset\hat{D}(A)$.

Moreover, $\tilde{A}_{m}F=(m-A)F$ for any $F\in D(\tilde{A}_{m})$. Since
$\tilde{A}_{m}$ is self-adjoint, $D(\tilde{A}_{m})=\hat{D}(A)$. This yields
the desired statement. ∎

We define the following set of functions:

$D_{A}=\big{\\{}f_{0}(x)\in
C_{0}^{\infty}(\mathbb{R}^{d}),\;f_{j}(x,\xi)-f_{0}(x)\in
C_{0}^{\infty}(\mathbb{R}^{d},{\mathbb{D}}_{j}(\Delta))\big{\\}}.$ (15)

Notice that $f_{j}(x,\xi)\big{|}_{\xi\in\partial{\cal D}_{j}}=f_{0}(x)$ for
any $F=\\{f_{j}\\}_{j\geq 0}\in D_{A}$.

######  Corollary 1.

The set $D_{A}\subset L^{2}(E,\alpha)$ defined in (15) is a core of $A$, i.e.
$D_{A}$ is a dense subset of $L^{2}(E,\alpha)$ and
$A=\overline{A\big{|}_{D_{A}}}$, see [5] for the details.

###### Proof.

Clearly, $D_{A}$ is a dense subset in $L^{2}(E,\alpha)$. In order to show that
$D_{A}$ is a core of $A$ we should also check that for some $m>0$ the set
$\\{(m-A)F,\ F\in D_{A}\\}$ is dense in $L^{2}(E,\alpha)$. Denote
$J^{\infty}=\\{(u_{0},U)=(u_{0}(x),U_{j}(x,\xi))\,:\,u_{0}\in
C_{0}^{\infty}(\mathbb{R}^{d}),\ U_{j}\in
C_{0}^{\infty}(\mathbb{R}^{d}\,;\,C_{0}^{\infty}({\cal D}_{j}))\\}$. Observe
that $J^{\infty}$ is dense in $L^{2}(E,\alpha)$. By Lemma 3.1 for an arbitrary
$U\in J^{\infty}$ and for any $m>0$ the equation

$mF-AF=U$ (16)

has a unique solution $F=(f_{0},\,V)\in\hat{D}(A)$. Then the equation for $V$
can be rewritten as

$(m-\Delta_{\xi})(V(x,\xi)-f_{0}(x))=U(x,\xi)-mf_{0}(x)\ \ \hbox{in
}\mathcal{D},\quad(V(x,\xi)-f_{0}(x))\big{|}_{\xi\in\partial{\cal D}}=0,$ (17)

or, in the coordinate form,

$(m-\Delta_{\xi})(V_{j}(x,\xi)-f_{0}(x))=U_{j}(x,\xi)-mf_{0}(x)\ \ \hbox{in
}\mathcal{D}_{j},\quad(V_{j}(x,\xi)-f_{0}(x))\big{|}_{\xi\in\partial{\cal
D}_{j}}=0,\ \ j\geq 1.$ (18)

From this equation we derive the following relation:

$V_{j}(x,\xi)=V^{I}_{j}(x,\xi)+mf_{0}(x)V^{0}_{j}(\xi)+f_{0}(x)$ (19)

with $V^{I}_{j}=(m-\Delta_{\xi})^{-1}U_{j}\in
C_{0}^{\infty}(\mathbb{R}^{d};\;{\mathbb{D}}_{j}(\Delta))$ and
$V^{0}_{j}=(m-\Delta_{\xi})^{-1}1\in{\mathbb{D}}_{j}(\Delta)$.

Substituting the right-hand side of (19) for $V$ into the first equation in
(16) yields

$mf_{0}-\Theta\cdot\nabla\nabla f_{0}-cmf_{0}=w_{0},$

where

$w_{0}=u_{0}-\sum\limits_{j\geq
1}\alpha_{j}\int\limits_{\mathcal{D}_{j}}\Delta_{\xi}(m-\Delta_{\xi})^{-1}U_{j}(\cdot,\xi)\,d\xi,\quad
c=-\sum\limits_{j\geq
1}\alpha_{j}\int\limits_{\mathcal{D}_{j}}\Delta_{\xi}(m-\Delta_{\xi})^{-1}1\,d\xi.$

Under our assumptions on $U$ we have $w_{0}\in
C_{0}^{\infty}(\mathbb{R}^{d})$. Also, it is straightforward to check that
$c<1$ for any $m>0$. Consequently, $f_{0}$ is a Schwartz class function in
$\mathbb{R}^{d}$. Taking a proper sequence of smooth cut-off functions
$\varphi_{n}$ we conclude that $(m-A)(\varphi_{n}f_{0},V+\varphi_{n}f_{0})$
converges in $L^{2}(E,\alpha)$ to $U$. Since
$(\varphi_{n}f_{0},V+\varphi_{n}f_{0})\in D_{A}$, this yields the desired
statement. ∎

##  4 The semigroup convergence.

Applying the Hille-Yosida theorem, we conclude that $A$ is a generator of a
strongly continuous, positive, contraction semigroup $T(t)$ on
$L^{2}(E,\alpha)$.

Define a bounded linear transformation
$\pi_{\varepsilon}^{\omega}:\,L^{2}(E,\alpha)\to L^{2}(\mathbb{R}^{d})$ for
every $\varepsilon\in(0,1)$ and every $\omega\in\Omega$ as follows:

$(\pi^{\omega}_{\varepsilon}F)(x)\ =\
\left\\{\begin{array}[]{ll}f_{0}(x),&\mbox{if}\;x\in\mathbb{R}^{d}\backslash
G_{\varepsilon}^{\omega};\\\\[5.69054pt]
\hat{f}_{j}(\hat{x}_{j}^{\omega,i},\,\frac{x-\hat{x}_{j}^{\omega,i}}{\varepsilon}),&\mbox{if}\;x\in\varepsilon{\cal
G}^{\omega,i}_{j},\end{array}\right.$ (20)

where $\varepsilon^{-1}\hat{x}^{\omega,i}_{j}$ is the vector that defines the
translation which maps $\mathcal{D}_{j}$ to ${\cal G}^{\omega,i}_{j}$, and

$\hat{f}_{j}(\hat{x}^{\omega,i}_{j},\xi)=\frac{1}{\varepsilon^{d}|{\cal
D}_{j}|}\int_{\varepsilon{\cal
D}{j}}f_{j}(\hat{x}^{\omega,i}_{j}+\eta,\xi)\,d\eta.$ (21)

######  Lemma 4.1.

Almost surely the linear operators $\pi_{\varepsilon}^{\omega}$ are uniformly
bounded in the operator norm for all $\varepsilon\in(0,1)$, that is

$\|\pi_{\varepsilon}^{\omega}F\|_{L^{2}(\mathbb{R}^{d})}\leq
C\|F\|_{L^{2}(E,\alpha)}$ (22)

for any $F\in L^{2}(E,\alpha)$; the constant $C$ is deterministic and does not
depend on $\varepsilon$. Moreover, for each $F\in L^{2}(E,\alpha)$ the
following relation holds a.e.

$\|\pi^{\omega}_{\varepsilon}F||^{2}_{L^{2}(\mathbb{R}^{d})}\ \to\
\|F\|^{2}_{L^{2}(E,\alpha)}\qquad\mbox{as }\;\varepsilon\to 0.$ (23)

###### Proof.

For every $x\in\mathbb{R}^{d}$ and every $\omega$ we have

$\sum_{j\geq 0}\chi\big{.}_{{\cal G}^{\omega}_{j}}(x)\ =\ 1,$

where ${\cal G}_{0}^{\omega}={\mathbb{R}^{d}\backslash
G^{\omega}_{\varepsilon}}$. Then we get

$\begin{array}[]{l}\displaystyle\|\pi^{\omega}_{\varepsilon}F\|^{2}_{L^{2}(\mathbb{R}^{d})}=\int\limits_{\mathbb{R}^{d}}\big{(}\pi^{\omega}_{\varepsilon}F(x)\big{)}^{2}dx=\sum\limits_{j\geq
0}\
\int\limits_{\mathbb{R}^{d}}\big{(}\pi^{\omega}_{\varepsilon}F(x)\big{)}^{2}\,\chi\big{.}_{{\cal
G}^{\omega}_{j}}(\frac{x}{\varepsilon})dx\\\\[5.69054pt]
\displaystyle=\int\limits_{\mathbb{R}^{d}}f_{0}^{2}(x)\,\chi\big{.}_{{\cal
G}_{0}^{\omega}}(\frac{x}{\varepsilon})dx+\sum\limits_{j\geq
1}\sum\limits_{i}\
\int\limits_{\mathbb{R}^{d}}\big{(}\hat{f}_{j}(\varepsilon\hat{x}_{j}^{\omega,i},\frac{x}{\varepsilon}-\hat{x}^{\omega,i}_{j})\big{)}^{2}\,\chi_{{\cal
G}^{\omega,i}_{j}}(\frac{x}{\varepsilon})dx.\end{array}$ (24)

By the Jenssen inequality and the definition of $\hat{f}_{j}$ in (21) this
implies that

$\|\pi^{\omega}_{\varepsilon}F\|^{2}_{L^{2}(\mathbb{R}^{d})}\leq\int\limits_{\mathbb{R}^{d}}f_{0}^{2}(x)\,dx+\sum_{j>0}\frac{1}{|{\cal
D}^{j}|}\int\limits_{\mathbb{R}^{d}}\int\limits_{{\cal
D}_{j}}\big{(}f_{j}(x,\xi)\big{)}^{2}\,d\xi\,dx\leq\check{C}\|F\|_{L^{2}(E,\alpha)}$

with $\check{C}=\max\limits_{j}\\{(\alpha_{j}^{o})^{-1}\\}$. This yields (22).

We turn to (23) and consider the set of functions in $L^{2}(E,\alpha)$ which
are piece-wise constant and compactly supported with respect to the first
variable $x$. We denote this set by ${\cal E}$ and notice that it is dense in
$L^{2}(E,\alpha)$. If $F\in{\cal E}$ then (23) holds by the Birkhoff ergodic
theorem. Then, taking into account (22) we conclude that (23) holds for any
$F\in L^{2}(E,\alpha)$. ∎

Now we are ready to formulate the main result of the work.

######  Theorem 4.1 (Main theorem).

For every $F\in L^{2}(E,\alpha)$ a.e.

$T^{\omega}_{\varepsilon}(t)\pi^{\omega}_{\varepsilon}F\to\ T(t)F,\quad\mbox{
i.e.
}\;\|T^{\omega}_{\varepsilon}(t)\pi^{\omega}_{\varepsilon}F-\pi^{\omega}_{\varepsilon}\
T(t)F\|_{L^{2}(\mathbb{R}^{d})}\to 0\quad\mbox{for all}\quad t\geq 0$ (25)

as $\varepsilon\to 0$.

The proof of the semigroup convergence in (25) relies on the following
approximation theorem [5, Theorem 6.1, Ch.1].
Theorem (see [5]). For $n=1,2,\ldots$, let $T_{n}(t)$ and $T(t)$ be strongly
continuous contraction semigroups on Banach space ${\cal L}_{n}$ and $\cal L$,
with generators $A_{n}$ and $A$. Let $D$ be a core for $A$. Then the following
are equivalent:

a) For each $f\in{\cal L}$, $T_{n}(t)\pi_{n}f\ \to\ T(t)f\;$ for all $t\geq
0$.

b) For each $f\in D$, there exists $f_{n}\in{\cal L}_{n}$ for each $n\geq 1$
such that $f_{n}\to f$ and $A_{n}f_{n}\to Af$.

According to this theorem the semigroups convergence (25) is a consequence of
the following statement:

######  Theorem 4.2.

Let the generators $A$ and $A^{\omega}_{\varepsilon}$ of the strongly
continuous, positive, contraction semigroups $T(t)$ and
$T^{\omega}_{\varepsilon}(t)$ be defined by (8) and (1), (2), (5),
respectively, and assume that a core $D_{A}\subset L^{2}(E,\alpha)$ for the
generator $A$ is defined by (15), and that a bounded linear transformation
$\pi^{\omega}_{\varepsilon}:\ L^{2}(E,\alpha)\to L^{2}(\mathbb{R}^{d})$ is
defined by (20) for every $\varepsilon\in(0,1)$.

Then there exists a positive definite symmetric constant matrix $\Theta$ such
that a.s. for every $F\in D_{A}$, there exists $F^{\omega}_{\varepsilon}\in
D(A^{\omega}_{\varepsilon})$ such that

$\|F^{\omega}_{\varepsilon}-\pi^{\omega}_{\varepsilon}F\|_{L^{2}(\mathbb{R}^{d})}\to
0\quad\mbox{ and
}\quad\|A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon}-\pi^{\omega}_{\varepsilon}AF\|_{L^{2}(\mathbb{R}^{d})}\to
0\quad\mbox{as }\;\varepsilon\to 0.$ (26)

###### Proof.

The proof relies on the correctors technique. For any $F\in D_{A},\
F=(f_{0},\\{f_{j}\\})$, where

$f_{0}(x)\in C_{0}^{\infty}(\mathbb{R}^{d}),\quad f_{j}(x,\xi)\in
C_{0}^{\infty}\big{(}\mathbb{R}^{d};\,C^{2}\big{(}\overline{{\cal
D}_{j}}\big{)}\big{)},$

with

$f_{j}(x,\xi)\big{|}_{\xi\in\partial{\cal D}_{j}}=f_{0}(x),\quad
x\in\mathbb{R}^{d},\;\;\forall\ j=1,\ldots,N,$ (27)

we construct the following family of functions $F_{\varepsilon}^{\omega}$
depending on the realization $\omega$ of random environment:

$F^{\omega}_{\varepsilon}(x)\ =\
\left\\{\begin{array}[]{l}f_{0}(x)+\varepsilon(\nabla
f_{0}(x),h^{\omega}_{\varepsilon}(\frac{x}{\varepsilon}))+\varepsilon^{2}(\nabla\nabla
f_{0}(x),g_{\varepsilon}^{\omega}(\frac{x}{\varepsilon}))+\varepsilon^{2}q^{\omega}_{\varepsilon}(x,\frac{x}{\varepsilon}),\;\;x\in\mathbb{R}^{d}\backslash
G_{\varepsilon}^{\omega},\\\\[5.69054pt]
f_{1}(x,\frac{x}{\varepsilon})+\varepsilon\phi^{\omega}_{1}(x,\frac{x}{\varepsilon}),\;\;x\in\varepsilon{\cal
G}_{1}^{\omega},\\\\[5.69054pt] \ldots\\\\[5.69054pt]
f_{N}(x,\frac{x}{\varepsilon})+\varepsilon\phi_{N}^{\omega}(x,\frac{x}{\varepsilon}),\;\;x\in\varepsilon{\cal
G}^{\omega}_{N}.\end{array}\right.$ (28)

Here
$h^{\omega}_{\varepsilon}(\xi),\,g_{\varepsilon}^{\omega}(\xi),\,q^{\omega}_{\varepsilon}(x,\xi)$
are random functions of $\xi$ (so-called correctors) that also depend on
$\varepsilon$; $h^{\omega}_{\varepsilon}(\xi)$ is the random vector function
whose gradient does not depend on $\varepsilon$,
$g_{\varepsilon}^{\omega}(\xi)$ is the random matrix function. In what follows
we drop both indices $\omega$ and $\varepsilon$ when refer to these functions.
Correctors $\phi_{j}^{\omega}(x,\xi)$ has been introduced in order to ensure
the continuity of the function $F^{\omega}_{\varepsilon}$ and the fluxes on
the boundary $\partial\big{(}\varepsilon{\cal G}^{\omega}_{j}\big{)}$ of the
corresponding inclusion.

Observe that for any $F\in D_{A}$ as well as for any $F\in C(E)$ and any
$x\in\varepsilon{\cal G}^{\omega,i}_{j}$ we have:

$(\pi^{\omega}_{\varepsilon}F)(x)=\hat{f}_{j}(\hat{x}^{\omega,i}_{j},\frac{x-\hat{x}^{\omega,i}_{j}}{\varepsilon})=f_{j}(x,\frac{x-\hat{x}^{\omega,i}_{j}}{\varepsilon})+O(\varepsilon),$
(29)

where the $L^{\infty}$ norm of $O(\varepsilon)$ does not exceed
$C\varepsilon$. Our goal is to choose the correctors in such a way that the
function $F^{\omega}_{\varepsilon}$ defined in (28) belongs to
$D(A_{\varepsilon}^{\omega})$, and both relations in (26) are fulfilled.
Denote by $B_{0}$ the ball in $\mathbb{R}^{d}$ centered at $0$ that contains
the supports in $x$ of all the functions $f_{j}$, $j=0,1,\ldots,N$.

In order to introduce the correctors in (28) we substitute for
$F^{\omega}_{\varepsilon}$ in the expression
$A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon}-\pi^{\omega}_{\varepsilon}AF$
the right-hand side of (28). Using repeatedly the formula

$\frac{\partial}{\partial
x}f(x,\frac{x}{\varepsilon})=\Big{(}\frac{\partial}{\partial
x}f(x,\xi)+\frac{1}{\varepsilon}\frac{\partial}{\partial\xi}f(x,\xi)\Big{)}\Big{|}_{\xi=\frac{x}{\varepsilon}},$
(30)

for $x\in\mathbb{R}^{d}\backslash G_{\varepsilon}^{\omega}$ after
straightforward computation we obtain

$\begin{array}[]{c}\displaystyle(A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon})(x)=\triangle_{x}\Big{(}f_{0}(x)+\varepsilon\nabla
f_{0}(x)\cdot h(\frac{x}{\varepsilon})+\varepsilon^{2}\nabla\nabla
f_{0}(x)\cdot
g(\frac{x}{\varepsilon})+\varepsilon^{2}q(x,\frac{x}{\varepsilon})\Big{)}\\\\[5.69054pt]
\displaystyle=\Big{(}\triangle f_{0}(x)+2\nabla\nabla
f_{0}(x)\nabla_{\xi}h(\xi)+\frac{1}{\varepsilon}\nabla f_{0}(x)\
\triangle_{\xi}h(\xi)+\nabla\nabla
f_{0}(x)\triangle_{\xi}g(\xi)\\\\[5.69054pt]
\displaystyle+\varepsilon^{2}\triangle_{x}q(x,\xi)+\Xi^{\omega}_{\varepsilon}(x,\xi)\Big{)}\big{|}_{\xi=\frac{x}{\varepsilon}}\
,\end{array}$ (31)

with

$\Xi^{\omega}_{\varepsilon}(x,\xi)=\Delta\nabla f_{0}(x)\cdot\varepsilon
h(\xi)+2\nabla\nabla\nabla
f_{0}(x)\cdot\varepsilon\nabla_{\xi}g(\xi)+\Delta\nabla\nabla
f_{0}(x)\cdot\varepsilon^{2}g(\xi)$

In a similar way for $x\in\varepsilon{\cal G}^{\omega}_{j}$ we have

$(A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon})(x)=\varepsilon^{2}\triangle_{x}\left(f_{j}(x,\frac{x}{\varepsilon})+\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\right)=\big{(}\triangle_{\xi}f_{j}(x,\xi)+\Psi^{\omega}_{j}(x,\xi)\big{)}\big{|}_{\xi=\frac{x}{\varepsilon}}\
.$ (32)

with

$\begin{array}[]{l}\displaystyle\Psi^{\omega}_{j}(x,\xi)=\varepsilon^{2}\Delta_{x}f_{j}(x,\xi)+2\varepsilon\nabla_{x}\cdot\nabla_{\xi}f_{j}(x,\xi)\\\\[5.69054pt]
\displaystyle+\varepsilon^{3}\Delta_{x}\phi^{\omega}_{j}(x,\xi)+2\varepsilon^{2}\nabla_{x}\cdot\nabla_{\xi}\phi^{\omega}_{j}(x,\xi)+\varepsilon\Delta_{\xi}\phi^{\omega}_{j}(x,\xi).\end{array}$

In order to make $F^{\omega}_{\varepsilon}$ belong to
$D(A^{\omega}_{\varepsilon})$ we should design it in such a way that the
following conditions are fulfilled on $\partial G_{\varepsilon}^{\omega}$:
1) continuity condition on $\partial(\varepsilon{\cal G}^{\omega}_{j})$

$\begin{array}[]{cc}\displaystyle\Big{(}f_{0}(x)+\varepsilon\nabla
f_{0}(x)\cdot h(\frac{x}{\varepsilon})+\varepsilon^{2}\nabla\nabla
f_{0}(x)\cdot
g(\frac{x}{\varepsilon})+\varepsilon^{2}q(x,\frac{x}{\varepsilon})\Big{)}\Big{|}_{x\in\partial(\varepsilon{\cal
G}^{\omega}_{j})}\\\\[5.69054pt]
\displaystyle=\Big{(}f_{j}\left(x,\frac{x}{\varepsilon}\right)+\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\Big{)}\Big{|}_{x\in\partial(\varepsilon{\cal
G}^{\omega}_{j})};\end{array}$ (33)

2) continuity of fluxes condition

$\begin{array}[]{cc}\displaystyle\nabla_{x}\left(f_{0}(x)+\varepsilon\nabla
f_{0}(x)\cdot h\big{(}\frac{x}{\varepsilon}\big{)}+\varepsilon^{2}\nabla\nabla
f_{0}(x)\cdot
g(\frac{x}{\varepsilon})+\varepsilon^{2}q\big{(}x,\frac{x}{\varepsilon}\big{)}\right)\Big{|}_{x\in\partial(\varepsilon{\cal
G}^{\omega}_{j})}\cdot n^{-}\\\\[8.53581pt]
\displaystyle=-\varepsilon^{2}\nabla_{x}\Big{(}f_{j}\left(x,\frac{x}{\varepsilon}\right)+\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\Big{)}\Big{|}_{x\in\partial(\varepsilon{\cal
G}^{\omega}_{j})}\cdot n^{+},\end{array}$ (34)

The main purpose of the functions $\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})$
is to compensate the discrepancy between the inner and outer expansions for
the function $F^{\omega}_{\varepsilon}$ at the boundary
$\partial{G}_{\varepsilon}^{\omega}$, see Proposition 4.2 below. It follows
from (27) that continuity condition (33) leads to the relation

$\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\Big{|}_{x\in\partial(\varepsilon{\cal
G}^{\omega}_{j})}=\Big{(}\nabla f_{0}(x)\cdot
h(\frac{x}{\varepsilon})+\varepsilon\nabla\nabla f_{0}(x)\cdot
g(\frac{x}{\varepsilon})+\varepsilon
q(x,\frac{x}{\varepsilon})\Big{)}\Big{|}_{x\in\partial(\varepsilon{\cal
D}^{\omega}_{j})}$ (35)

Notice that equality (35) defines the functions
$\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})$ only for
$x\in\partial(\varepsilon{\cal G}^{\omega}_{j})$.

With the help of (30) the relation (34) can be rewritten as

$\begin{array}[]{cc}\displaystyle\Big{(}\nabla
f_{0}(x)+\varepsilon\nabla\nabla f_{0}(x)h(\xi)+\nabla_{\xi}(\nabla
f_{0}(x)h(\xi))+\varepsilon^{2}\nabla\nabla\nabla
f_{0}(x)g(\xi)\\\\[7.11317pt]
\displaystyle+\varepsilon\nabla_{\xi}(\nabla\nabla
f_{0}(x)g(\xi))+\varepsilon^{2}\nabla_{x}q(x,\xi)+\varepsilon\nabla_{\xi}q(x,\xi)\Big{)}\Big{|}_{\xi=\frac{x}{\varepsilon}\in\partial\,{\cal
G}^{\omega}_{j}}\cdot n^{-}=\\\\[9.95845pt]
\displaystyle-\big{(}\varepsilon^{2}\nabla_{x}f_{j}(x,\xi)+\varepsilon\nabla_{\xi}f_{j}(x,\xi)+\varepsilon^{2}\nabla_{\xi}\phi_{j}^{\omega}(x,\xi)+\varepsilon^{3}\nabla_{x}\phi^{\omega}_{j}(x,\xi)\big{)}\Big{|}_{\xi=\frac{x}{\varepsilon}\in\partial\,{\cal
G}^{\omega}_{j}}\cdot n^{+}.\end{array}$ (36)

We first consider the ansatz in (28) in the set $\mathbb{R}^{d}\backslash
G_{\varepsilon}^{\omega}$. Collecting power-like terms in (31) and (36) and
considering the terms of order $\varepsilon^{-1}$ in (31) and of order
$\varepsilon^{0}$ in (36), we conclude that $h(\cdot)$ should satisfy the
equation

$\nabla f_{0}(x)\
\triangle_{\xi}h(\xi)=0,\;\xi\in\mathcal{G}^{\omega}_{0},\qquad\big{(}\nabla
f_{0}(x)+\nabla f_{0}(x)\nabla_{\xi}h(\xi)\big{)}\cdot
n^{-}_{\xi}=0,\;\xi\in\partial\mathcal{G}_{0}^{\omega};$

here $x$ is a parameter. Since $f_{0}$ does not depend on $\xi$, this problem
can be rewritten as follows:

$\triangle_{\xi}h(\xi)=0,\;\;\xi\in\mathcal{G}^{\omega}_{0},\qquad\nabla_{\xi}h(\xi)\cdot
n^{-}_{\xi}=-n^{-}_{\xi},\;\;\xi\in\partial\mathcal{G}^{\omega}_{0}.$ (37)

This suggests the choice of $h$, it should coincide with the standard
corrector used for homogenization of the Neumann problem in a random
perforated domain, see [9]. We recall that the gradient of $h(\xi)$ is a
statistically homogeneous matrix function that does not depend on
$\varepsilon$, and $h$ satisfies equation (37). Moreover, $h$ shows a
sublinear growth in $L^{2}$. Namely, assuming that
$\int_{B_{0}}h(\frac{x}{\varepsilon})dx=0$, we have

$\big{\|}\varepsilon
h\big{(}\frac{\cdot}{\varepsilon}\big{)}\big{\|}_{L^{2}(B_{0})}\longrightarrow
0,\quad\hbox{a.s. as }\varepsilon\to 0.$ (38)

We also have

$\big{\|}\nabla_{\xi}h\big{(}\frac{\cdot}{\varepsilon}\big{)}\big{\|}_{L^{2}(B_{0})}\leq
C$

a.s. with a constant $C$ that does not depend on $\varepsilon$, see [9].

The matrix $\Theta$ in (8) is then defined by

$\Theta\ =\
\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h(\xi))\,\chi_{\mathcal{G}_{0}^{\omega}}(\xi)\big{]},\quad\mbox{i.e.
}\;\Theta^{ij}\ =\
\mathbb{E}\big{[}(\delta_{ij}+\nabla^{i}_{\xi}h^{j}(\xi))\,\chi_{\mathcal{G}_{0}^{\omega}}(\xi)\big{]},$
(39)

where $\chi_{\mathcal{G}_{0}^{\omega}}(\cdot)$ is the characteristic function
of ${\mathcal{G}_{0}^{\omega}}=\mathbb{R}^{d}\setminus G^{\omega}$. It is
proved in [9] that $\Theta$ is positive definite.

At the next step we collect the terms of order $\varepsilon^{0}$ on the right-
hand side of (31) and equate them to $\Theta\cdot\nabla\nabla
f_{0}(x)+\Upsilon(x)$ in order to make the difference
$(A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon}-\pi^{\omega}_{\varepsilon}AF)=\big{(}A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon}(x)-\left(\Theta\cdot\nabla\nabla
f_{0}(x)+\Upsilon(x)\right)\big{)}$ small in
$L^{2}(\varepsilon\mathcal{G}_{0}^{\omega})$ norm. This yields

$\Big{(}\triangle f_{0}(x)+2\nabla\nabla
f_{0}(x)\cdot\nabla_{\xi}h(\xi)+\nabla\nabla
f_{0}(x)\cdot\triangle_{\xi}g(\xi)+\varepsilon^{2}\triangle_{x}q(x,\xi)\Big{)}\big{|}_{\xi=\frac{x}{\varepsilon}}=\Theta\cdot\nabla\nabla
f_{0}(x)+\Upsilon(x),$ (40)

where $x\in(\varepsilon\mathcal{G}_{0}^{\omega}\cap B_{0})$; the function
$\Upsilon(x)$ is defined in (10). We also collect the terms of order
$\varepsilon^{1}$ in (36):

$\begin{array}[]{c}\displaystyle\varepsilon\Big{(}\nabla\nabla
f_{0}(x)h(\xi)+\nabla_{\xi}(\nabla\nabla f_{0}(x)\cdot
g(\xi))+\nabla_{\xi}q(x,\xi)\Big{)}\big{|}_{\xi=\frac{x}{\varepsilon}\in\partial\,{\cal
G}^{\omega}_{j}}\cdot n^{-}\\\\[5.69054pt]
\displaystyle=-\varepsilon\nabla_{\xi}f_{j}(x,\xi)\big{|}_{\xi=\frac{x}{\varepsilon}\in\partial\,{\cal
G}^{\omega}_{j}}\cdot n^{+}.\end{array}$ (41)

Selecting all the terms in (40)-(41) that contain the second order derivatives
of $f_{0}$, we arrive at the following problem for the random matrix valued
function $g(\frac{x}{\varepsilon})=\\{g_{ij}(\frac{x}{\varepsilon})\\}$:

$\begin{array}[]{c}\displaystyle\Big{(}\triangle f_{0}(x)+2\nabla\nabla
f_{0}(x)\cdot\nabla_{\xi}h(\xi)+\nabla\nabla
f_{0}(x)\cdot\triangle_{\xi}g(\xi)\Big{)}\big{|}_{\xi=\frac{x}{\varepsilon}}=\Theta\cdot\nabla\nabla
f_{0}(x),\\\\[5.69054pt] \displaystyle
x\in\varepsilon\mathcal{G}_{0}^{\omega}\cap B_{0},\\\\[5.69054pt]
\displaystyle\nabla_{\xi}g(\xi)\cdot
n^{-}\big{|}_{\xi=\frac{x}{\varepsilon}}=-h(\xi)\otimes
n^{-}\big{|}_{\xi=\frac{x}{\varepsilon}},\quad
x\in\varepsilon\partial\mathcal{G}_{0}^{\omega}\cap B_{0}.\end{array}$ (42)

In addition to these two equations we impose the homogeneous Dirichlet
boundary condition on the boundary of $B_{0}$

$g\Big{(}\frac{x}{\varepsilon}\Big{)}=0\quad\hbox{on }\partial B_{0}.$

Finally, $g(\frac{x}{\varepsilon})$ is introduced as a solution to the
following problem:

$\begin{array}[]{c}\displaystyle\varepsilon^{2}\triangle_{x}g(\frac{x}{\varepsilon})\
=\
\mathbb{E}\big{[}(\mathbf{I}+\nabla_{\xi}h(\xi))\,\chi^{\omega}(\xi)\big{]}-\mathbf{I}-2\varepsilon\nabla_{x}h(\frac{x}{\varepsilon}),\quad
x\in V^{\varepsilon}:=\varepsilon\mathcal{G}_{0}^{\omega}\cap
B_{0},\\\\[5.69054pt]
\displaystyle\varepsilon\nabla_{x}g(\frac{x}{\varepsilon})\cdot n^{-}\ =\
-h(\frac{x}{\varepsilon})\otimes n^{-},\quad
x\in\varepsilon\partial\mathcal{G}_{0}^{\omega}\cap B_{0},\\\\[5.69054pt]
\displaystyle g(\frac{x}{\varepsilon})\ =\ 0,\quad x\in\partial
B_{0};\end{array}$ (43)

here ${\bf I}$ stands for the unit $d\times d$ matrix.

######  Lemma 4.2.

Problem (43) has a unique solution. Moreover, a.s.

$\lim\limits_{\varepsilon\to 0}\|\varepsilon^{2}{\textstyle
g\big{(}\frac{\cdot}{\varepsilon}\big{)}}\|_{H^{1}(\varepsilon\mathcal{G}_{0}^{\omega}\cap
B_{0})}=0.$ (44)

The proof of this lemma is provided in Appendix 1, Section 6.

Next, collecting the remaining terms in (40) and (41), we arrive at the
following problem for the function $q(x,\frac{x}{\varepsilon})$:

$\begin{array}[]{c}\displaystyle\varepsilon^{2}\triangle_{x}q(x,\frac{x}{\varepsilon})=\Upsilon(x),\quad
x\in\varepsilon\mathcal{G}_{0}^{\omega}\cap B_{0},\\\\[5.69054pt]
\displaystyle\nabla_{\xi}q(x,\xi)\cdot
n^{-}\big{|}_{\xi=\frac{x}{\varepsilon}}=-\nabla_{\xi}f_{j}(x,\xi)\cdot
n^{+}\big{|}_{\xi=\frac{x}{\varepsilon}},\quad
x\in\varepsilon\partial\mathcal{G}_{0}^{\omega}\cap B_{0},\end{array}$ (45)

where the function $\Upsilon(x)\in C_{0}^{\infty}(\mathbb{R}^{d})$ is defined
in (10). We then equip system (45) with the homogeneous Dirichlet boundary
condition at $\partial B_{0}$:

$q(x,\frac{x}{\varepsilon})=0\quad\hbox{for }x\in\partial B_{0}.$ (46)

Denote $\Phi_{\varepsilon}(x)=\varepsilon^{2}q(x,\frac{x}{\varepsilon})$. Let
$\phi_{\Phi}(\cdot)\in C_{0}^{\infty}(B_{0})$ be a function such that

$\phi_{\Phi}\geq 0\quad\mbox{and}\quad\phi_{\Phi}=1\hbox{ for all
}x\in\\{x\in\mathbb{R}^{d}\,:\,\hbox{ there exist }j\hbox{ and }\xi\hbox{ such
that }f_{j}(x,\xi)\not=0\\}.$ (47)

######  Proposition 4.1.

The following limit relations hold a.s.:

$\lim\limits_{\varepsilon\to
0}\|\Phi_{\varepsilon}\|_{H^{1}(\varepsilon\mathcal{G}_{0}^{\omega}\cap
B_{0})}=0,$ (48) $\lim\limits_{\varepsilon\to
0}\|\phi_{\Phi}\Phi_{\varepsilon}\|_{H^{1}(\varepsilon\mathcal{G}_{0}^{\omega}\cap
B_{0})}=0.$ (49)

Moreover,

$\lim\limits_{\varepsilon\to
0}\|\Delta_{x}\big{(}\phi_{\Phi}\Phi_{\varepsilon}\big{)}-\Upsilon\|_{L^{2}(\varepsilon\mathcal{G}_{0}^{\omega}\omega)}=0.$
(50)

The proof of this statement is given in Appendix 2.

We now turn to the correctors
$\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})$, $j=1,\ldots,N$. Our
goal is to define them in such a way that

$\begin{array}[]{rl}\displaystyle\hat{f}_{j}(\hat{x}^{\omega,i}_{j},\frac{x-\hat{x}^{\omega,i}_{j}}{\varepsilon})+\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})=\\!\\!&\\!\\!f_{0}(x)+\varepsilon(\nabla
f_{0}(x),h(\frac{x}{\varepsilon}))\\\\[5.69054pt]
\displaystyle\hfil&\\!\\!+\varepsilon^{2}(\nabla\nabla
f_{0}(x),g(\frac{x}{\varepsilon}))+\varepsilon^{2}q(x,\frac{x}{\varepsilon})\qquad\hbox{on
}\varepsilon\partial\mathcal{G}_{j}^{\omega},\end{array}$ (51)
$\begin{array}[]{rl}\displaystyle\varepsilon^{2}\nabla\Big{[}\hat{f}_{j}(\hat{x}^{\omega,i}_{j},\frac{x-\hat{x}^{\omega,i}_{j\omega}}{\varepsilon})+\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\Big{]}\cdot
n\\!\\!&\\!\\!=\nabla\Big{[}f_{0}(x)+\varepsilon(\nabla
f_{0}(x),h(\frac{x}{\varepsilon}))\\\\[5.69054pt]
\displaystyle\hfil&\\!\\!+\varepsilon^{2}(\nabla\nabla
f_{0}(x),g(\frac{x}{\varepsilon}))+\varepsilon^{2}q(x,\frac{x}{\varepsilon})\Big{]}\cdot
n\qquad\hbox{on }\varepsilon\partial\mathcal{G}_{j}^{\omega},\end{array}$ (52)
$\|\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\|_{L^{2}(\varepsilon\mathcal{G}_{j}^{\omega})}+\varepsilon^{2}\|\Delta_{x}\big{(}\varepsilon\phi^{\omega}_{j}(x,\frac{x}{\varepsilon})\big{)}\|_{L^{2}(\varepsilon\mathcal{G}_{j}^{\omega})}\longrightarrow
0,\quad\hbox{as }\varepsilon\to 0.$ (53)

######  Proposition 4.2.

There exists a family of functions $\phi^{\omega}_{j}$ with $j=1,\ldots,N$ and
$\varepsilon\in(0,1)$ such that the relations in (51)–(53) are fulfilled.

For the proof, see Appendix 2.

We turn back to the Proof of Theorem 4.2. The statement of this Theorem is now
a straightforward consequence of (37), (38), Lemma 4.2 and Propositions 4.1 \-
4.2. Indeed, due to (51) and (52), we have $F_{\varepsilon}^{\omega}\in
D(A_{\varepsilon}^{\omega})$. Then the convergence

$\|F^{\omega}_{\varepsilon}-\pi^{\omega}_{\varepsilon}F\|_{L^{2}(\mathbb{R}^{d})}\to
0\quad\hbox{as }\varepsilon\to 0$

follows from (29), (38), (44), (49) and (53). Finally, by (42), (50) and (53)
we obtain

$\|A^{\omega}_{\varepsilon}F^{\omega}_{\varepsilon}-\pi^{\omega}_{\varepsilon}AF\|_{L^{2}(\mathbb{R}^{d})}\to
0\quad\mbox{as }\;\varepsilon\to 0.$

This completes the proof of Theorem 4.2. ∎

##  5 Spectrum of the limit operator

We proceed with the description of the spectrum of the limit operator $A$
given by (9), and then using the strong convergence of Markov semigroup
$T^{\omega}_{\varepsilon}(t)$ in $L^{2}(E)$ we describe the limit behaviour of
the spectra of operators $A^{\omega}_{\varepsilon}$, as $\varepsilon\to 0$
almost surely.

Remind that each component $f_{j}(x,\xi)$ of $F\in D_{A}$ can be written as
the sum

$f_{j}(x,\xi)=f_{0}(x)+r_{j}(x,\xi)\quad\mbox{with
}\;r_{j}(x,\xi)\big{|}_{\xi\in\partial{\cal
D}_{j}}=0\quad\forall\;j=1,\ldots,N.$

Then (9) takes the form

$(-AF)(x,\xi)=\left(\begin{array}[]{c}-\Theta\cdot\nabla\nabla
f_{0}(x)+\frac{1}{\alpha_{0}}\sum_{j=1}^{N}\alpha_{j}\int\limits_{{\cal
D}_{j}}\triangle_{\xi}r_{j}(x,\xi)d\xi\\\ \\\ -\triangle_{\xi}r_{1}(x,\xi)\\\
\ldots\\\ -\triangle_{\xi}r_{N}(x,\xi)\end{array}\right)$ (54)

For each $j$ the operator $-\triangle_{\xi}$ on ${\cal D}_{j}$ with
homogeneous Dirichlet boundary condition has a discrete positive spectrum
$\\{\beta^{j}_{m}\\}_{m\in\mathbb{N}}$, $\;\beta^{j}_{m}>0$,
$\;\beta^{j}_{m}\to\infty$. We denote by
$\varkappa^{j}_{m}(\xi),\;m=1,2,\ldots,$ the corresponding normalized
eigenfunctions and by $\mathbb{M}$ the set of all indices $(j,m)$. We
introduce the set $\mathbb{M}^{*}\subset\mathbb{M}$ of indices $(j,m)$ such
that $\int\limits_{{\cal
D}_{j}}\varkappa^{j}_{m}(\xi)d\xi=\langle\varkappa^{j}_{m}\rangle\not=0$. Let
$\mathbb{B}$ be a (countable) set of all $\beta^{j}_{m}$:

$\mathbb{B}=\bigcup_{(j,m)\in\mathbb{M}}\beta^{j}_{m},$

and

$b_{1}=\min\limits_{(j,m)\in\mathbb{M}}\beta^{j}_{m}=\min\limits_{(j,m)\in\mathbb{M}^{*}}\beta^{j}_{m},\quad
b_{1}>0.$

######  Lemma 5.1.

The continuous spectrum $\sigma_{cont}(-A)$ of the operator $-A$ is a
countable set of non-overlapping segments

$\sigma_{cont}(-A)\ =\
\bigcup\limits_{(j,m)\in\mathbb{M}^{*}}[\hat{\lambda}^{j}_{m},\beta^{j}_{m}],$

where $\hat{\lambda}_{1}=0$, and $\hat{\lambda}^{j}_{m}<\beta^{j}_{m}$ is the
nearest to $\beta^{j}_{m}$ solution of equation

$\frac{1}{\alpha_{0}}\sum_{j}\alpha_{j}\sum\limits_{m}\frac{(u^{j}_{m})^{2}\
\beta^{j}_{m}}{\beta^{j}_{m}-\lambda}+1=0\quad\mbox{\rm with }\quad
u^{j}_{m}=\langle\varkappa^{j}_{m}\rangle.$

The point spectrum of the operator $-A$ is the union of eigenvalues
$\beta^{j}_{m}$ with $(j,m)\in\mathbb{M}\setminus\mathbb{M}^{*}$:

$\sigma_{p}(-A)\ =\
\bigcup\limits_{(j,m)\in\mathbb{M}\setminus\mathbb{M}^{*}}\beta^{j}_{m}.$

Each eigenvalue $\beta^{j}_{m}\in\sigma_{p}(-A)$ has infinite multiplicity, so
that $\sigma_{p}(-A)$ belongs to the essential spectrum of $-A$.

###### Proof.

Each line in the equation $-AF=\lambda F$ except of the first one reads

$-A(f_{0}(x)+r_{j}(x,\xi))=-\triangle_{\xi}r_{j}(x,\xi)=\lambda(f_{0}(x)+r_{j}(x,\xi)),\quad\xi\in{\cal
D}_{j}.$ (55)

The function $f_{0}(x)$ does not depend on $\xi$, its Fourier series w.r.t.
$\\{\varkappa^{j}_{m}(\xi)\\}$ for every $j$ takes the form

$f_{0}(x)\cdot
1=f_{0}(x)\sum\limits_{m}u^{j}_{m}\varkappa^{j}_{m}(\xi),\quad\mbox{with
}\;u^{j}_{m}=\int\limits_{{\cal D}_{j}}\varkappa^{j}_{m}(\xi)\,d\xi.$ (56)

Denoting by $\gamma^{j}_{m}=\gamma^{j}_{m}(x)$ the Fourier coefficients of
$r_{j}$, from (55) - (56) we get

$-\triangle_{\xi}r_{j}(x,\xi)=\sum\limits_{m}\beta^{j}_{m}\gamma^{j}_{m}\varkappa^{j}_{m}(\xi)=\lambda
f_{0}(x)\sum\limits_{m}u^{j}_{m}\varkappa^{j}_{m}(\xi)+\lambda\sum\limits_{m}\gamma^{j}_{m}\varkappa^{j}_{m}(\xi).$

Consequently, for any $\lambda\not\in\mathbb{B}$ we have
$\gamma^{j}_{m}=\lambda f_{0}(x)\,\frac{u^{j}_{m}}{\beta^{j}_{m}-\lambda}$,
and thus the function

$r_{j}(x,\xi)=\sum\limits_{m}\gamma^{j}_{m}\varkappa^{j}_{m}(\xi)=\lambda
f_{0}(x)\sum\limits_{m}\frac{u^{j}_{m}}{\beta^{j}_{m}-\lambda}\varkappa^{j}_{m}(\xi),$
(57)

is a solution of equation
$-A(f_{0}+r_{j})=-\triangle_{\xi}r_{j}=\lambda(f_{0}+r_{j})$ for any $j$ and
any $\lambda\not\in\mathbb{B}$.

Inserting (57) in the first line of the equation $-AF=\lambda F$ with $-A$
given by (54) yields

$-\Theta\cdot\nabla\nabla f_{0}(x)-\lambda
f_{0}(x)\frac{1}{\alpha_{0}}\sum_{j}\alpha_{j}\sum\limits_{m}\frac{u^{j}_{m}\
\beta^{j}_{m}}{\beta^{j}_{m}-\lambda}\int\limits_{{\cal
D}_{j}}\varkappa^{j}_{m}(\xi)d\xi=\lambda f_{0}(x).$

Consequently

$-\Theta\cdot\nabla\nabla f_{0}(x)=\lambda
f_{0}(x)\left(\frac{1}{\alpha_{0}}\sum_{j}\alpha_{j}\sum\limits_{m}\frac{(u^{j}_{m})^{2}\
\beta^{j}_{m}}{\beta^{j}_{m}-\lambda}+1\right).$ (58)

Since the spectrum of the operator $-\Theta\cdot\nabla\nabla$ fills up the
positive half-line, we obtain that all $\lambda>0$ such that

$\frac{1}{\alpha_{0}}\sum_{j}\alpha_{j}\sum\limits_{m}\frac{(u^{j}_{m})^{2}\
\beta^{j}_{m}}{\beta^{j}_{m}-\lambda}+1\geq 0$

belong to the spectrum of the operator $-A$. One can easily check that the
segment $[0,\beta_{1}]$, $b_{1}=\min_{(j,m)\in\mathbb{M}^{*}}\beta^{j}_{m}>0$
belongs to the continuous spectrum of $-A$. This implies the desired statement
on $\sigma_{cont}(-A)$.

It is straightforward to check that the functions
$F^{(j,m)}=(0,\ldots,\varphi(x)\,\varkappa^{j}_{m}(\xi),0,\ldots,0)$ with
$\varphi(x)\in L^{2}(\mathbb{R}^{d})$ for all
$(j,m)\in\mathbb{M}\setminus\mathbb{M}^{*}$, i.e. such that
$\langle\varkappa^{j}_{m}\rangle=0$, are the eigenfunctions of $-A$ with
corresponding eigenvalue $\beta^{j}_{m}$. This completes the proof. ∎

Notice that the operators $A^{\omega}_{\varepsilon}$ for every $\varepsilon$
have statistically homogeneous coefficients, i.e. they are metrically
transitive with respect to the unitary group of the space translations in
$\mathbb{R}^{d}$. Then from the general results, see e.g. [12], it follows
that the spectra of the operators $A^{\omega}_{\varepsilon}$ are non-random
for a.e. $\omega$.

######  Proposition 5.1.

For any $\lambda\in\sigma(-A)$ a.s. there exists a sequence
$\lambda_{\varepsilon}$,
$\lambda_{\varepsilon}\in\sigma(A^{\omega}_{\varepsilon})$, that converges to
$\lambda$ as $\varepsilon\to 0$.

###### Proof.

Since $\lambda\in\sigma(-A)$, there exist functions $F_{n}\in D_{A}$,
$\|F_{n}\|_{L^{2}(E,\alpha)}=1$ such that
$\|(A+\lambda)F_{n}\|_{L^{2}(E,\alpha)}\to 0$ as $n\to\infty$. Using Theorem
4.2 we additionally have that for any $F_{n}\in D_{A}$ there exists
$F^{\omega}_{n,\varepsilon}\in D(A^{\omega}_{\varepsilon})$ for a.e. $\omega$
such that

$\|F^{\omega}_{n,\varepsilon}-\pi^{\omega}_{\varepsilon}F_{n}\|_{L^{2}(\mathbb{R}^{d})}\to
0\quad\mbox{ and
}\quad\|A^{\omega}_{\varepsilon}F^{\omega}_{n,\varepsilon}-\pi^{\omega}_{\varepsilon}AF_{n}\|_{L^{2}(\mathbb{R}^{d})}\to
0\quad\mbox{ as }\;\varepsilon\to 0.$

Thus using Lemma 4.1 we obtain that for any (small) $\delta>0$ there exists
$\varepsilon_{0}=\varepsilon_{0}(\lambda,\delta)>0$ such that for all
$\varepsilon<\varepsilon_{0}$ there exists $F^{\omega}_{n,\varepsilon}\in
L^{2}(\mathbb{R}^{d})$ with $\|F^{\omega}_{n,\varepsilon}\|=1$, and

$\|A^{\omega}_{\varepsilon}F^{\omega}_{n,\varepsilon}+\lambda
F^{\omega}_{n,\varepsilon}\|_{L^{2}(\mathbb{R}^{d})}<\delta.$ (59)

This implies that there is a point of the spectrum of
$-A^{\omega}_{\varepsilon}$ in the $\delta$-neighbourhood of $\lambda$. ∎

##  6 Appendix 1. The second corrector $g$. Proof of Lemma 4.2.

Recall the matrix valued function $g(\frac{x}{\varepsilon})$ was defined as a
solution to the following problem:

$\begin{array}[]{c}\displaystyle\varepsilon^{2}\triangle_{x}g(\frac{x}{\varepsilon})\
=\
\mathbb{E}\big{[}(\mathbf{I}+\nabla_{\xi}h(\xi))\,\chi^{\omega}(\xi)\big{]}-\mathbf{I}-2\varepsilon\nabla_{x}h(\frac{x}{\varepsilon}),\quad
x\in V^{\varepsilon}:=\varepsilon\mathcal{G}_{0}^{\omega}\cap
B_{0},\\\\[5.69054pt]
\displaystyle\varepsilon\nabla_{x}g(\frac{x}{\varepsilon})\cdot n^{-}\ =\
-h(\frac{x}{\varepsilon})\otimes n^{-},\quad
x\in\varepsilon\partial\mathcal{G}_{0}^{\omega}\cap B_{0},\\\\[5.69054pt]
\displaystyle g_{\varepsilon}^{\omega}(\frac{x}{\varepsilon})\ =\ 0,\quad
x\in\partial B_{0};\end{array}$ (60)

in this section we will use notation
$\chi^{\omega}(\cdot)=\chi_{\mathcal{G}_{0}^{\omega}}(\cdot)$ for the
characteristic function of the random set $\mathcal{G}_{0}^{\omega}$. We also
recall that the matrix $\Theta$ defined by (39) is positive definite, see [9].
In the coordinate form the above problem reads

$\begin{array}[]{c}\displaystyle\varepsilon^{2}\triangle_{x}g^{ij}(\frac{x}{\varepsilon})\
=\
\mathbb{E}\big{[}(\delta_{ij}+\nabla^{i}_{\xi}h^{j}(\xi))\,\chi^{\omega}(\xi)\big{]}-\delta_{ij}-2\varepsilon\nabla^{j}_{x}h^{i}(\frac{x}{\varepsilon}),\quad
x\in V^{\varepsilon},\\\\[5.69054pt]
\displaystyle\varepsilon\nabla_{x}g^{ij}(\frac{x}{\varepsilon})\cdot n^{-}\ =\
-h^{i}(\frac{x}{\varepsilon})\,(n^{-})^{j},\quad x\in\partial
V^{\varepsilon}\cap B_{0},\\\\[5.69054pt] \displaystyle
g^{ij}(\frac{x}{\varepsilon})\ =\ 0,\quad x\in\partial B_{0}.\end{array}$ (61)

Each component of
$g(\frac{x}{\varepsilon})=\\{g^{ij}(\frac{x}{\varepsilon})\\}$ can be
considered separately and in what follows we omit a super index $ij$.

Denote
$\Psi_{\varepsilon}(\frac{x}{\varepsilon})=\varepsilon^{2}g(\frac{x}{\varepsilon})$.
Our first goal is to prove that the set of functions
$\Psi_{\varepsilon}(\frac{x}{\varepsilon})$ is bounded in
$H^{1}(V^{\varepsilon})$. Integrating by parts and using the second equality
in (61) we get

$\begin{array}[]{l}\displaystyle\int\limits_{V^{\varepsilon}}\varepsilon^{2}\triangle_{x}g(\frac{x}{\varepsilon})\,\varepsilon^{2}g(\frac{x}{\varepsilon})dx=-\varepsilon^{4}\int\limits_{V^{\varepsilon}}\big{|}\nabla_{x}g(\frac{x}{\varepsilon})\big{|}^{2}dx+\varepsilon^{4}\int\limits_{\partial\,V^{\varepsilon}}\frac{\partial
g(\frac{x}{\varepsilon})}{\partial
n}\,g(\frac{x}{\varepsilon})\,d\sigma(x)\\\\[5.69054pt]
\displaystyle=-\varepsilon^{4}\int\limits_{V^{\varepsilon}}\big{|}\nabla_{x}g(\frac{x}{\varepsilon})\big{|}^{2}dx-\varepsilon^{3}\int\limits_{\partial\,V^{\varepsilon}}h(\frac{x}{\varepsilon})\,n^{-}\,g(\frac{x}{\varepsilon})\,d\sigma(x).\end{array}$
(62)

On the other hand, using the first equality in (61) and integrating by parts
we transform the left hand side of (62) as follows:

$\begin{array}[]{l}\displaystyle\int\limits_{V^{\varepsilon}}\varepsilon^{2}\triangle_{x}g(\frac{x}{\varepsilon})\,\varepsilon^{2}g(\frac{x}{\varepsilon})dx\\\\[5.69054pt]
\displaystyle=\varepsilon^{2}\int\limits_{V^{\varepsilon}}\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h)\chi^{\omega}\big{]}\,g(\frac{x}{\varepsilon})dx-\varepsilon^{2}\int\limits_{V^{\varepsilon}}\big{(}\mathbb{I}+\nabla_{\xi}h(\frac{x}{\varepsilon})\big{)}g(\frac{x}{\varepsilon})dx-\varepsilon^{3}\int\limits_{V^{\varepsilon}}\nabla_{x}h(\frac{x}{\varepsilon})g(\frac{x}{\varepsilon})dx\\\\[5.69054pt]
\displaystyle=\varepsilon^{2}\int\limits_{V^{\varepsilon}}\left(\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h)\chi^{\omega}\big{]}-(\mathbb{I}+\nabla_{\xi}h(\frac{x}{\varepsilon}))\right)g(\frac{x}{\varepsilon})dx\\\\[5.69054pt]
\displaystyle-\varepsilon^{3}\int\limits_{\partial
V^{\varepsilon}}h(\frac{x}{\varepsilon})n^{-}g(\frac{x}{\varepsilon})d\sigma(x)+\varepsilon^{3}\int\limits_{V^{\varepsilon}}h(\frac{x}{\varepsilon})\nabla_{x}g(\frac{x}{\varepsilon})dx.\end{array}$
(63)

Thus, (62) - (63) imply

$\varepsilon^{4}\int\limits_{V^{\varepsilon}}\big{|}\nabla_{x}g(\frac{x}{\varepsilon})\big{|}^{2}dx=-\varepsilon^{2}\int\limits_{V^{\varepsilon}}\Big{(}\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h)\chi^{\omega}\big{]}-(\mathbb{I}+\nabla_{\xi}h(\frac{x}{\varepsilon}))\Big{)}g(\frac{x}{\varepsilon})dx-\varepsilon^{3}\int\limits_{V^{\varepsilon}}h(\frac{x}{\varepsilon})\nabla_{x}g(\frac{x}{\varepsilon})dx.$
(64)

We get from (64) that

$\|\nabla_{x}\Psi_{\varepsilon}\|^{2}_{L^{2}(V^{\varepsilon})}\leq
A\|\Psi_{\varepsilon}\|_{L^{2}(V^{\varepsilon})}+\|\varepsilon
h\|_{L^{2}(V^{\varepsilon})}\,\|\nabla_{x}\Psi_{\varepsilon}\|_{L^{2}(V^{\varepsilon})}.$
(65)

We have used here the fact that
$\Lambda^{\omega}_{\varepsilon}=\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h^{\omega})\chi^{\omega}\big{]}-(\mathbb{I}+\nabla_{\xi}h^{\omega}(\frac{x}{\varepsilon}))\chi^{\omega}(\frac{x}{\varepsilon})$
is a stationary random field with finite second moment
$\mathbb{E}(\Lambda^{\omega}_{\varepsilon})^{2}<+\infty$. Moreover, by the
Birkhoff’ theorem

$\lim_{\varepsilon\to 0}\mathbb{E}(\Lambda^{\omega}_{\varepsilon})=0,$ (66)

and thus $\Lambda^{\omega}_{\varepsilon}$ a.s. weakly converges to zero in
$L^{2}(B_{0})$ as $\varepsilon\to 0$.

Next we apply the results on extensions in random perforated domains, see [1],
[9]. According to these results there exists a liner extension operator
$L:H^{1}(V^{\varepsilon})\to H^{1}(B_{0})$ such that for any $f\in
H^{1}(V^{\varepsilon})$

$Lf|_{H^{1}(V^{\varepsilon})}=f,\qquad\|Lf\|_{L^{2}(B_{0})}\leq
C\,\|f\|_{L^{2}(V^{\varepsilon})},\qquad\|\nabla
Lf\|_{L^{2}(B_{0})}\leq\tilde{C}\,\|\nabla f\|_{L^{2}(V^{\varepsilon})},$

where the constants $C$ and $\tilde{C}$ do not depend on $\varepsilon$.
Keeping for the extended function $L\Psi_{\varepsilon}$ the same notation
$\Psi_{\varepsilon}$ and considering the Dirichlet boundary condition on
$\partial B_{0}$ in (61), by the Friedrichs inequality we obtain

$\|\Psi_{\varepsilon}\|^{2}_{L^{2}(V^{\varepsilon})}\leq\|\Psi_{\varepsilon}\|^{2}_{L^{2}(B_{0})}\leq
c_{1}\|\nabla_{x}\Psi_{\varepsilon}\|^{2}_{L^{2}(B_{0})}\leq
C\|\nabla_{x}\Psi_{\varepsilon}\|^{2}_{L^{2}(V^{\varepsilon})}.$ (67)

Combining this with (65) yields

$\|\nabla_{x}\Psi_{\varepsilon}\|_{L^{2}(V^{\varepsilon})}\leq
A_{1},\qquad\|\Psi_{\varepsilon}\|_{L^{2}(V^{\varepsilon})}\leq A_{2}$ (68)

with the constants $A_{1}$ and $A$ that do not depend on $\varepsilon$. Thus
a.s. the family of functions $\\{\Psi_{\varepsilon}\\}$ is bounded in
$H^{1}(V^{\varepsilon})$ and in $H^{1}(B_{0})$. Due to the compactness of
embedding of $H^{1}(B_{0})$ in $L^{2}(B_{0})$ we can pass to the limit in the
product $\Lambda^{\omega}_{\varepsilon}\,\Psi_{\varepsilon}$ as
$\varepsilon\to 0$. Thus the integral

$\varepsilon^{2}\int\limits_{V^{\varepsilon}}\Big{(}\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h)\chi^{\omega}\big{]}-(\mathbb{I}+\nabla_{\xi}h(\frac{x}{\varepsilon}))\Big{)}g(\frac{x}{\varepsilon})dx=\int\limits_{B_{0}}\Big{(}\mathbb{E}\big{[}(\mathbb{I}+\nabla_{\xi}h)\chi^{\omega}\big{]}-(\mathbb{I}+\nabla_{\xi}h(\frac{x}{\varepsilon}))\Big{)}\chi^{\omega}\Big{(}\frac{x}{\varepsilon}\Big{)}\Psi_{\varepsilon}(\frac{x}{\varepsilon})dx$

tends to zero as $\varepsilon\to 0$ a.s. Taking into account relations (38) we
derive from (64) that

$\|\nabla_{x}\Psi_{\varepsilon}\|_{L^{2}(B_{0})}\ \to\ 0$ (69)

and, by the Friedrichs inequality,

$\|\Psi_{\varepsilon}\|_{L^{2}(B_{0})}\ \to\ 0.$ (70)

##  7 Appendix 2. Proofs of Propositions 4.1 and 4.2

We begin this section by proving Proposition 4.1. Denote
$\Phi_{\varepsilon}(x):=\varepsilon^{2}q(x,\frac{x}{\varepsilon})$. Then
$\Phi_{\varepsilon}(x)$ is a solution of the following problem:

$\begin{array}[]{c}\displaystyle\triangle_{x}\Phi_{\varepsilon}(x)\ =\
\Upsilon(x),\quad x\in V^{\varepsilon}=\varepsilon\mathcal{G}_{0}^{\omega}\cap
B_{0},\\\\[5.69054pt] \displaystyle\nabla_{x}\Phi_{\varepsilon}(x)\cdot n^{-}\
=\ -\varepsilon^{2}\nabla_{x}f_{j}(x,\frac{x}{\varepsilon})\cdot n^{+},\quad
x\in\partial(\varepsilon\mathcal{G}_{j})\cap B_{0},\\\\[5.69054pt]
\displaystyle\Phi_{\varepsilon}(x)\ =\ 0,\quad x\in\partial B_{0}.\end{array}$
(71)

In what follows we will use the following notations:

$\nabla
f(x,\frac{x}{\varepsilon})=\nabla_{x}f(x,\frac{x}{\varepsilon}),\quad\nabla
h(\xi)=\nabla_{\xi}h(\xi),\quad\nabla\Phi_{\varepsilon}(x)=\nabla_{x}\Phi_{\varepsilon}(x).$

In order to show that the functions $\Phi_{\varepsilon}(x)$ are bounded in
$H^{1}(V^{\varepsilon})$ we follow the line of the proof in the previous
sectiuon. Multiplying the equation in (71) by $\Phi_{\varepsilon}(x)$ and
integrating the resulting relation over $V^{\varepsilon}$ after integration by
paths we obtain

$\begin{array}[]{l}\displaystyle\int\limits_{V^{\varepsilon}}\Upsilon(x)\,\Phi_{\varepsilon}(x)dx=\int\limits_{V^{\varepsilon}}\triangle\Phi_{\varepsilon}(x)\,\Phi_{\varepsilon}(x)dx=\int\limits_{\partial\,V^{\varepsilon}}\Phi_{\varepsilon}(x)\,\nabla\Phi_{\varepsilon}(x)\cdot
n^{-}\,d\sigma(x)\\\\[5.69054pt]
\displaystyle-\int\limits_{V^{\varepsilon}}\big{|}\nabla\Phi_{\varepsilon}(x)\big{|}^{2}dx=-\int\limits_{V^{\varepsilon}}\big{|}\nabla\Phi_{\varepsilon}(x)\big{|}^{2}dx-\sum\limits_{j}\,\int\limits_{\varepsilon\partial{\mathcal{G}}^{\omega}_{j}}\Phi_{\varepsilon}(x)\,\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot n^{+}\,d\sigma(x).\end{array}$ (72)

By the Friedrichs inequality

$\Big{|}\int\limits_{V^{\varepsilon}}\triangle\Phi_{\varepsilon}(x)\,\Phi_{\varepsilon}(x)dx\Big{|}\leq
C_{1}\|\Upsilon(x)\|_{L^{2}(B_{0})}\,\|\nabla\Phi_{\varepsilon}(x)\|_{L^{2}(V^{\varepsilon})}.$
(73)

with a constant $C_{1}$ that does not depend on $\varepsilon$.

To estimate the second integral on the right-hand side of (72) we extend the
functions $\Phi_{\varepsilon}$ on $B_{0}$, denote the extended functions by
$\bar{\Phi}_{\varepsilon}(x)$ and apply the Stokes formula. This yields

$\begin{array}[]{l}\displaystyle\int\limits_{\varepsilon\partial{\mathcal{G}^{\omega}_{j}}}\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot
n^{+}\Phi_{\varepsilon}(x)\,d\sigma(x)\\\\[5.69054pt]
\displaystyle=\int\limits_{\varepsilon\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\triangle
f_{j}(x,\frac{x}{\varepsilon})\,\bar{\Phi}_{\varepsilon}(x)\,dx+\int\limits_{\varepsilon\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot\nabla\bar{\Phi}_{\varepsilon}(x)\,dx.\end{array}$
(74)

From this relation by the Friedrichs inequality we derive the following upper
bound:

$\sum\limits_{j}\,\Big{|}\int\limits_{\varepsilon\partial\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot
n^{+}\,\Phi_{\varepsilon}(x)\,d\sigma(x)\Big{|}\leq
C\,\|\nabla\Phi_{\varepsilon}\|_{L^{2}(V^{\varepsilon})}.$ (75)

Finally, (72), (73) and (75) imply the desired upper bound:

$\|\nabla\Phi_{\varepsilon}\|_{L^{2}(V^{\varepsilon})}\leq\tilde{C},$ (76)

i.e. the functions $\Phi_{\varepsilon}(x)$ are bounded in
$H^{1}(V^{\varepsilon})$. Consequently, the extensions
$\bar{\Phi}_{\varepsilon}(x)$ are also bounded in $H^{1}(B_{0})$ and form a
compact set in $L^{2}(B_{0})$. Thus, there exists $\Phi_{0}\in H^{1}(B_{0})$
such that, for a subsequence,

$\|\bar{\Phi}_{\varepsilon}-\Phi_{0}\|_{L^{2}(B_{0})}\to 0.$

Our goal is to prove that $\Phi_{0}\equiv 0$, or equivalently

$\|\Phi_{\varepsilon}\|_{L^{2}(B_{0})}\to 0\quad\mbox{as }\;\varepsilon\to 0.$
(77)

For an arbitrary $\hat{\psi}\in C^{\infty}(G_{\varepsilon}^{\omega})$ with a
compact support in $B_{0}$ we have

$\begin{array}[]{c}\displaystyle\int\limits_{V^{\varepsilon}}\triangle\Phi_{\varepsilon}(x)\,\hat{\psi}(x)\,dx\\\\[5.69054pt]
\displaystyle=-\int\limits_{V^{\varepsilon}}\nabla\Phi_{\varepsilon}(x)\cdot\nabla\hat{\psi}(x)\,dx\
+\ \sum\limits_{j}\
\int\limits_{\varepsilon\partial\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot
n^{+}\,\hat{\psi}(x)\,d\sigma(x).\end{array}$ (78)

On the other hand,

$\int\limits_{V^{\varepsilon}}\triangle_{x}\Phi_{\varepsilon}(x)\,\hat{\psi}(x)dx=\int\limits_{V^{\varepsilon}}\Upsilon(x)\hat{\psi}(x)\,dx=\int\limits_{B_{0}}\Upsilon(x)\,\chi_{\\{\varepsilon\mathcal{G}^{\omega}_{0}\\}}(x)\,\hat{\psi}(x)\,dx.$
(79)

Therefore,

$\begin{array}[]{l}\displaystyle\int\limits_{B_{0}}\nabla\Phi_{\varepsilon}(x)\cdot\nabla\hat{\psi}(x)\,\chi_{\\{\varepsilon\mathcal{G}^{\omega}_{0}\\}}(x)\,dx\\\\[5.69054pt]
\displaystyle=\sum\limits_{j}\
\int\limits_{\varepsilon\partial\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot
n^{+}\,\hat{\psi}(x)\,d\sigma(x)-\int\limits_{V^{\varepsilon}}\Upsilon(x)\hat{\psi}(x)\,dx.\end{array}$
(80)

For an arbitrary $\psi\in C_{0}^{\infty}(B_{0})$, substituting in the last
relation the function $\psi(x)+\varepsilon
h\big{(}\frac{x}{\varepsilon}\big{)}\nabla\psi(x)$ for $\hat{\psi}$ we obtain

$\begin{array}[]{l}\displaystyle\int\limits_{B_{0}}\nabla\Phi_{\varepsilon}(x)\cdot\big{(}\nabla\psi(x)+\nabla
h\big{(}\frac{x}{\varepsilon}\big{)}\nabla\psi(x)\big{)}\,\chi_{\\{\varepsilon\mathcal{G}^{\omega}_{0}\\}}(x)\,dx+o(1)\\\\[5.69054pt]
\displaystyle=\sum\limits_{j}\
\int\limits_{\varepsilon\partial\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\nabla
f_{j}(x,\frac{x}{\varepsilon})\cdot
n^{+}\,\hat{\psi}(x)\,d\sigma(x)-\int\limits_{V^{\varepsilon}}\Upsilon(x)\hat{\psi}(x)\,dx\\\\[5.69054pt]
\displaystyle=-\sum\limits_{j}\
\int\limits_{\varepsilon\mathcal{G}_{j}^{\omega}}\varepsilon^{2}\triangle
f_{j}(x,\frac{x}{\varepsilon})\psi(x)\,dx-\int\limits_{V^{\varepsilon}}\Upsilon(x)\psi(x)\,dx+o(1),\end{array}$
(81)

where $o(1)$ a.s. tends to zero as $\varepsilon\to 0$; here we have used the
inequality $\|\varepsilon
h\big{(}\frac{\cdot}{\varepsilon}\big{)}\|_{H^{1}(\varepsilon\mathcal{G}_{0}^{\omega})}\leq
C$ and the fact that $\|\varepsilon
h\big{(}\frac{\cdot}{\varepsilon}\big{)}\|_{L^{2}(\varepsilon\mathcal{G}_{0}^{\omega})}$
vanishes as $\varepsilon\to 0$. Using representation (10) of the function
$\Upsilon(x)$, the Stokes formula and the Birkhoff ergodic theorem we conclude
that the right-hand side in (81) tends to 0 as $\varepsilon\to 0$ for any
$\psi\in C_{0}^{\infty}(B_{0})$ and thus

$\begin{array}[]{l}\displaystyle\lim\limits_{\varepsilon\to
0}\,\int\limits_{B_{0}}\nabla\Phi_{\varepsilon}(x)\cdot\Big{(}\nabla\psi(x)+\nabla
h\big{(}\frac{x}{\varepsilon}\big{)}\nabla\psi(x)\Big{)}\,\chi_{\\{\varepsilon\mathcal{G}_{0}^{\omega}\\}}(x)\,dx=0\\\\[14.22636pt]
\displaystyle\Phi_{\varepsilon}(x)\big{|}_{\partial B_{0}}=0.\end{array}$ (82)

The subsequence of $\nabla\Phi_{\varepsilon}$ converges weakly in
$(L^{2}(B_{0}))^{d}$ to $\nabla\Phi_{0}$, as $\varepsilon\to 0$. By the
definition of matrix $\Theta$ the sequence $\big{(}\nabla\psi+\nabla
h\big{(}\frac{\cdot}{\varepsilon}\big{)}\nabla\psi\big{)}\,\chi_{\\{\varepsilon\mathcal{G}_{0}^{\omega}\\}}$
converges weakly in $(L^{2}(B_{0}))^{d}$ to $\Theta\nabla\psi$. Since the
function $h(\cdot)$ satisfies equation (37), we have

$\mathrm{div}\big{[}\big{(}\nabla\psi(x)+\nabla
h\big{(}\frac{x}{\varepsilon}\big{)}\nabla\psi(x)\big{)}\,\chi_{\\{\varepsilon\mathcal{G}_{0}^{\omega}\\}}(x)\big{]}=\big{(}\Delta\psi(x)+\nabla
h\big{(}\frac{x}{\varepsilon}\big{)}\nabla\nabla\psi(x)\big{)}\,\chi_{\\{\varepsilon\mathcal{G}_{0}^{\omega}\\}}(x).$

The right-hand side here is bounded in $L^{2}(B_{0})$ and thus compact in
$H^{-1}(B_{0})$. By the compensated compactness theorem, see [11], we obtain

$0=\lim\limits_{\varepsilon\to
0}\,\int\limits_{B_{0}}\nabla\Phi_{\varepsilon}(x)\cdot\Big{(}\nabla\psi(x)+\nabla
h\big{(}\frac{x}{\varepsilon}\big{)}\nabla\psi\Big{)}\,\chi_{\\{\varepsilon\mathcal{G}_{0}^{\omega}\\}}(x)\,dx=\int_{B_{0}}\nabla\Psi_{0}\cdot\Theta\nabla\psi\,dx.$

Since $\Phi_{0}=0$ on $\partial B_{0}$, this implies that $\Phi_{0}=0$, and
(77) follows. This convergence to $\Phi_{0}=0$ holds for the whole family
$\Phi_{\varepsilon}^{\omega}$.

The proof of other statements of Proposition 4.1 is now straightforward.

We turn to the proof of Proposition 4.2. Denote

$\tilde{\Xi}^{\varepsilon}\big{(}x\big{)}=f_{0}(x)+\varepsilon(\nabla
f_{0}(x),h(\frac{x}{\varepsilon}))+\varepsilon^{2}(\nabla\nabla
f_{0}(x),g(\frac{x}{\varepsilon}))+\varepsilon^{2}q(x,\frac{x}{\varepsilon}).$

For each $(j,i)$ with $j\in\\{1,\ldots,N\\}$ and $i\in\mathbb{Z}$ we define an
open set
$\varepsilon\mathcal{Q}^{\varkappa}_{\omega,ji}=\\{x\in\mathbb{R}^{d}\,:\,\mathrm{dist}(x,\varepsilon\partial\mathcal{G}^{\omega,i}_{j})<\varepsilon\varkappa\\}$
and introduce in this set coordinates $y$ such that
$y^{\prime}=(y_{2},\ldots,y_{d})$ are smooth coordinates on
$\varepsilon\partial\mathcal{G}^{\omega,i}_{j}$, and $y_{1}$ is directed along
the exterior normal,
$y_{1}=-\mathrm{dist}(x,\varepsilon\partial\mathcal{G}^{\omega,i}_{j})$ if
$x\in\varepsilon\mathcal{G}^{\omega,i}_{j}$ and
$y_{1}=\mathrm{dist}(x,\varepsilon\partial\mathcal{G}^{\omega,i}_{j})$ if
$x\in\mathbb{R}^{d}\setminus\varepsilon\mathcal{G}^{\omega,i}_{j}$. Under our
assumptions on the geometry of $\mathcal{G}^{\omega,i}_{j}$ there exists
$\varkappa>0$ such that

  * •

$\varepsilon\mathcal{Q}^{\varkappa}_{\omega,ji}$ do not intersect with
$\varepsilon\mathcal{Q}^{\varkappa}_{\omega,km}$, if $(j,i)\not=(k,m)$.

  * •

Coordinates $y=y(x)$ are well defined in
$\varepsilon\mathcal{Q}^{\varkappa}_{\omega,ji}$, that is $y=y(x)$ is an
invertible diffeomorphism.

Letting
$\Xi^{\varepsilon}\big{(}y\big{)}=\tilde{\Xi}^{\varepsilon}\big{(}x(y)\big{)}$
and
$\check{f}^{\varepsilon}(y)=\hat{f}_{j}(\hat{x}^{\omega,i}_{j},\frac{x(y)-\hat{x}^{\omega,i}_{j}}{\varepsilon})$,
in
$\varepsilon\mathcal{Q}^{\varkappa}_{\omega,ji}\cap\varepsilon\mathcal{G}^{\omega,i}_{j}$
we then define

$\varepsilon\phi^{\omega}_{j}(y)=\Big{(}\Xi^{\varepsilon}\big{(}0,y^{\prime}\big{)}-\check{f}^{\varepsilon}_{j}(0,y^{\prime})+y_{1}\big{[}\frac{\partial}{\partial
y_{1}}\Xi^{\varepsilon}\big{(}0,y^{\prime}\big{)}-\frac{\partial}{\partial
y_{1}}\check{f}^{\varepsilon}_{j}\big{(}0,y^{\prime}\big{)}\big{]}-\overline{\Xi}\Big{)}\theta\big{(}-\frac{y_{1}}{\varepsilon}\big{)}+\overline{\Xi},$

where $\theta(s)$ is a $C^{\infty}$ cut-off function such that
$0\leq\theta\leq 1$, $\theta=1$ for $s<\frac{\varkappa}{3}$ and $\theta=0$ for
$s>\frac{2\varkappa}{3}$; $\overline{\Xi}$ is the mean value of
$\Xi^{\varepsilon}$ over
$\varepsilon\mathcal{Q}^{\varkappa}_{\omega,ji}\cap(\mathbb{R}^{d}\setminus\varepsilon\mathcal{G}^{\omega,i}_{j})$.

By construction (51) and (52) are fulfilled. Relation (53) follows from the
properties of the correctors and elliptic estimates, see [7, Chapter X].

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