Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains

Abstract

In this paper, we investigate a linear hyperbolic stochastic partial differential equation (SPDE) with rapidly oscillating ϵ-periodic coefficients in a domain with small holes (of size-ϵ) under Neumann conditions on the boundary of the holes and Dirichlet condition on the exterior boundary. When the number of these holes approach infinity, i.e. their sizes approach zero, the homogenized problem is a hyperbolic SPDE with constant coefficients in the domain without perforations. Moreover the convergence of the associated energy to that of the homogenized system is established.

Keywords

homogenization hyperbolic SPDEs Neumann problem perforated domains probabilistic compactness results

1. Introduction and setting of the problem

Homogenization is a mathematical theory aimed at understanding the behavior of processes that take place in heterogeneous media with highly oscillating heterogeneities. These heterogeneous materials consist of finely mixed different components like soil, paper, concrete for building, fibreglass, materials used in the manufacturing of high tech equipments such as planes, rockets and so on. This signifies that almost everything around us in real life is a heterogeneous material. The physical problems described on heterogeneous materials such as heat, mechanical constraints, flow of fluids in these media lead to the study of PDEs with highly oscillating coefficients depending on macroscopic scales or boundary value problems for PDEs in domain with fine grained boundaries. The main obstacle in solving these problems arises either from the character of the domain or the presence of high oscillations in the coefficients of the governing equation. To this end, it is expensive to compute solutions to these type of problems. Numerical methods have proved inefficient in solving such problems due to the fact that even the most advanced parallel computers are unable to simulate schemes related to the physically interesting such problems.

The study of homogenization for PDEs in periodic structures has been undertaken by many authors. It was originally based on the idea of asymptotic expansions in powers of the small perturbation parameter in the problem. This approach was fundamental in the celebrated work [8] of Bensoussan, Lions and Papanicolaou; we should also mention the monograph by Bakhvalov and Panacenko [5]. These authors studied wide range of partial differential equations, such as elliptic, parabolic and hyperbolic problems, mainly linear in structure. A great wealth of interesting results were obtained by many mathematicians, thanks to new methods such as Murat–Tartar’s H-convergence, Tartar’s method of oscillating test functions, Nguetseng–Allaire’s two scale convergence, Cioranescu–Damlamian–Griso’s unfolding method. It will not be possible to survey most of these results, some of which may be found for instance in [3,4,16–20,28,29,36,37,39,48,49,59,60]. In view of the prevalence of randomness in almost all natural phenomena, it was not long before homogenization of PDEs with random coefficients started to be investigated. Pioneers in this direction are certainly Kozlov [32], Papanicolaou and Varadhan [42]. Their work influenced many new research; see for instance [11,12,23,30,34,44,57,58]. We also note the closely related work [14,15,33] dealing with stochastic homogenization for SPDEs with small parameters.

It should be noted that the methods which enabled the study of homogenization in the periodic setting have recently been generalized to homogenization problems on nonperiodic algebras, see for instance [38,53–55,62].

As mentioned above, there was a need to consider homogenization of PDEs with random coefficients. However physical processes under random fluctuations are better modelled by stochastic partial differential equations (SPDEs). It was therefore natural to consider homogenization of this very important class of PDEs. Research in this direction is still at its infancy, despite the importance of such problems in both applied and fundamental sciences. Some relevant interesting work have recently been undertaken, mainly for parabolic SPDEs, see for instance [6,26,47,50,52].

The case of hyperbolic SPDEs is widely open till now. The first work done by means of hyperbolic SPDEs is [35], where we obtained homogenization result as well as corrector results by means of the two-scale convergence method. As far as the homogenization of deterministic hyperbolic (PDEs) is concerned, many work have been undertaken by several authors from different perspectives. We refer to [8] where the authors studied the homogenization of the hyperbolic equations based on asymptotic expansions. We also note the monograph of Cioranescu and Donato [19], where similar studies are carried through in the framework of Tartar’s method, which was introduced in [36,59]. Cioranescu and Donato also proved the convergence of the energy associated to the inhomogeneous wave equation to the energy associated to the homogenized problem; the corresponding corrector result was proved in [13], see also [24,25]. Recently, the new field of numerical homogenization is attracting a growing attention of researchers in applied mathematics. Some numerical works have considered wave equations in heterogeneous media using finite element heterogeneous multiscale method [1,2] and the upscaling method [10,31,41]. It would be of interest to investigate homogenization of hyperbolic SPDEs in the framework of these methods in our future work.

In this work we will be concerned with establishing homogenization results for Neumann problem of linear hyperbolic equations with periodically oscillating coefficients in the framework of Tartar’s method, based on his ingenious construction of oscillating test functions. We study this problem in the cylindrical domain $Q^{ϵ} \times (0, T)$ where $T > 0$ and $Q^{ϵ}$ is a spatial domain with periodically distributed holes (with small size $ϵ > 0$ ), when $ϵ \to 0$ the number of these holes becomes infinite and $Q^{ϵ}$ converges to a domain Q without holes. One of the difficulties in obtaining the homogenization result is that, the spatial domain $Q^{ϵ}$ varies with ϵ. In order to avoid this difficulty we introduce extension-like operators, which enable us to reduce the problem to one in fixed domain. The other difficulty is that, in the deterministic case one needs some analytic compactness results in order to obtain the limit equation. However, these results are not enough in the probabilistic setting. As additional tools, we use the Prokhorov and Skorokhod’s probabilistic compactness results for which tightness of probability measures generated by the perturbed problems is needed. But for the tightness uniform estimates of various norms of the solutions of our original problems have to be established in appropriate probabilistic evolution spaces. Furthermore after the application of Prokhorov and Skorokhod’s processes the presence of the noise $g^{ϵ} (t, x) d W^{ϵ}$ prohibited direct adaptation of deterministic arguments due to the dependence of the noise on ϵ.

Fig. 1.

The structure of the domain.

Let Q be a sufficiently smooth bounded and open (at least Lipschitz) subset of $R^{n}$ , $Y = (0, l_{1}) \times \dots \times (0, l_{n})$ is the reference cell, S a smooth subset of Y, such that $\bar{S} \subset \bar{Y}$ and $G (ϵ \bar{S})$ is the set of all translated sets of $ϵ \bar{S}$ by $k l = (k_{1} l_{1}, \dots, k_{n} l_{n})$ , $k \in Z^{n}$ where $ϵ > 0$ . Assume that $G (ϵ \bar{S}) \cap \partial Q = \emptyset$ ; $\partial ∙$ denotes the boundary of the set ∙. Now we define the perforated domain $Q^{ϵ}$ with an ϵ-periodic structure by (see Fig. 1) $\begin{matrix} Q^{ϵ} = Q \cap {R^{n} ∖ G (ϵ \bar{S})} . \end{matrix}$ Let us define $S^{ϵ}$ by the set $S^{ϵ} = (Q \cap G (ϵ \bar{S}))$ , therefore, $\begin{matrix} \partial Q^{ϵ} = \partial Q \cup \partial S^{ϵ} . \end{matrix}$ We denote by $χ_{(O)}$ the characteristic function of any open set O in $R^{n}$ and by $| ∙ |$ the measure of any measurable set ∙ of $R^{n}$ . For $Y^{*} = Y ∖ \bar{S}$ , we define $θ = | Y^{*} | / | Y |$ . We also denote by $\tilde{f}$ the zero extension to the whole Q for any function f defined on $Q^{ϵ}$ . In this paper we study the asymptotic behaviour of solutions $u^{ϵ} = u^{ϵ} (ω, x, t)$ of the initial boundary value problem with oscillating data: $\begin{matrix} (1) & \{\begin{matrix} d u_{t}^{ϵ} = div A_{ϵ} \nabla u^{ϵ} d t + f^{ϵ} d t + g^{ϵ} d W in Q^{ϵ} \times (0, T), \\ u^{ϵ} = 0 on \partial Q \times (0, T), \\ \frac{\partial u^{ϵ}}{\partial ν_{A_{ϵ}}} = 0 on \partial S^{ϵ} \times (0, T), \\ u^{ϵ} (x, 0) = u_{0}^{ϵ}, u_{t}^{ϵ} (x, 0) = u_{1}^{ϵ} in Q^{ϵ}; \end{matrix} \end{matrix}$ where $W = (W (t))$ ( $t \in [0, T]$ ) an m-dimensional standard Wiener process defined on a given filtered complete probability space $(Ω, F, P, {(F_{t})}_{0 ⩽ t ⩽ T})$ ; $E$ denote the corresponding mathematical expectation, $f^{ϵ} (x, t) = f (\frac{x}{ϵ}, t)$ , $g^{ϵ} (x, t) = g (\frac{x}{ϵ}, t)$ , $u_{0}^{ϵ} (x) = u_{0} (\frac{x}{ϵ})$ , $u_{1}^{ϵ} (x) = u_{1} (\frac{x}{ϵ})$ and $A_{ϵ} (x) = A (\frac{x}{ϵ}) = {(a_{i, j} (\frac{x}{ϵ}))}_{1 ⩽ i, j ⩽ n}$ an $n \times n$ symmetric matrix and $\frac{\partial (\cdot)}{\partial ν_{A_{ϵ}}} = \sum_{i, j = 1}^{n} a_{i, j} (\frac{x}{ϵ}) \frac{\partial (\cdot)}{\partial x_{j}} n_{j}$ .

In order to state some facts we need to introduce the following spaces $L^{2} (Q)$ , $H^{1} (Q)$ , $H_{0}^{1} (Q)$ , $C_{per}^{\infty} (Y)$ the subspace of $C^{\infty} (R^{n})$ of Y-periodic functions where $Y = (0, l_{1}) \times \dots \times (0, l_{n})$ . Let $H_{per}^{1} (Y)$ be the closure of $C_{per}^{\infty} (Y)$ in the $H^{1}$ -norm, and $H_{per} (Y)$ the subspace of $H_{per}^{1} (Y)$ with zero mean on Y. We also define $H_{ϵ} (Q^{ϵ}) = {u \in H^{1} (Q^{ϵ}) : u |_{\partial Q} = 0}$ with the norm $‖ u ‖_{H_{ϵ} (Q^{ϵ})} = ‖ \nabla u ‖_{{[L^{2} (Q^{ϵ})]}^{n}}$ in the $H^{1}$ -norm. We denote by $⟨ \cdot, \cdot ⟩$ the duality pairing between $H^{- 1} (Q)$ and $H_{0}^{1} (Q)$ as well as between $H^{- 1} (Q^{ϵ})$ and $H_{ϵ} (Q^{ϵ})$ ; $H^{- 1} (Q^{ϵ})$ denotes the dual of $H_{ϵ} (Q^{ϵ})$ .

For a Banach space X, and $1 ⩽ p, q ⩽ \infty$ , we denote by $L^{p} (0, T; X)$ the space of measurable functions $ϕ : t \in [0, T] ⟶ ϕ (t) \in X$ such that $‖ ϕ (t) ‖_{X} \in L^{p} (0, T)$ and by $L^{q} (Ω, F, P; L^{p} (0, T; X))$ we denote the space of functions $u : (ω, t) \in Ω \times [0, T] ⟶ u (ω, t, \cdot) \in X$ such that $u (ω, t, x)$ is measurable with respect to $(ω, t)$ and for each t is $F_{t}$ -measurable in ω, we endow the later space with the norm $\begin{matrix} ‖ u ‖_{L^{q} (Ω, F, P; L^{p} (0, T; X))} = {(E ‖ u ‖_{L^{p} (0, T; X)}^{q})}^{1 / q} . \end{matrix}$ When $p = \infty$ , the space $\begin{matrix} L^{\infty} (0, T; X) = {ϕ : [0, T] ⟶ X such that \underset{X}{ess sup} ‖ ϕ ‖_{X} < \infty}, \end{matrix}$ where ${ess sup}_{X} ‖ ϕ ‖_{X} = ‖ ϕ ‖_{L^{\infty} (0, T; X)}$ . When $p = \infty$ , we endow $L^{q} (Ω, F, P, L^{\infty} (0, T; X))$ with the following norm $\begin{matrix} ‖ ϕ ‖_{L^{q} (Ω, F, P, L^{\infty} (0, T; X))} = {(E ‖ ϕ ‖_{L^{\infty} (0, T; X)}^{q})}^{1 / q} . \end{matrix}$ It is well known that, under the above norm, $L^{q} (Ω, F, P, L^{p} (0, T; X))$ is a Banach space. We shall often omit ω in the notation of $u_{ϵ}$ . Our main assumptions are

$\sum_{i, j = 1}^{n} a_{i, j} ξ_{i} ξ_{j} ⩾ α \sum_{i = 1}^{n} ξ_{i}^{2}$ for all, $ξ \in R^{n}$ and α is a positive constant,

$a_{i, j} \in L^{\infty} (R^{n})$ , $i, j = 1, \dots, n$ ,

$a_{i, j}$ , are Y-periodic, $\forall i, j = 1, \dots, n$ ,

$u_{0}^{ϵ} \in H_{ϵ} (Q^{ϵ})$ , $u_{1}^{ϵ} \in L^{2} (Q^{ϵ})$ ,

$f^{ϵ} \in L^{2} (Q \times (0, T))$ , $g^{ϵ} \in L^{2} (Q^{ϵ} \times (0, T))$ .

In the following we introduce the notion of strong probabilistic solution to our problem

Definition 1.

We define the strong probabilistic solution of the problem (1) by $\begin{matrix} u^{ϵ} : Ω \times [0, T] ⟶ H_{ϵ} (Q^{ϵ}), \end{matrix}$ such that

$u^{ϵ}$ , $u_{t}^{ϵ}$ are continuous with respect to time in $H_{ϵ} (Q^{ϵ})$ , $L^{2} (Q^{ϵ})$ , respectively,

$u^{ϵ}$ , $u_{t}^{ϵ}$ are adapted to the filtration $F_{t}$ ,

$u^{ϵ} \in L^{2} (Ω, F, P; L^{\infty} (0, T; H_{ϵ} (Q^{ϵ})))$ , $u_{t}^{ϵ} \in L^{2} (Ω, F, P; L^{\infty} (0, T; L^{2} (Q^{ϵ})))$ ,

$\forall t \in [0, T]$ , $u^{ϵ} (t, \cdot)$ satisfies $\begin{array}{l} - \int_{0}^{t} (u_{t}^{ϵ} (t, \cdot), ϕ_{t}) d s + \int_{0}^{t} (A_{ϵ} \nabla u^{ϵ} (s, \cdot), \nabla ϕ) d s \\ (2) & = \int_{0}^{t} (f^{ϵ} (s, \cdot) ϕ) d s + (\int_{0}^{t} g^{ϵ} (s, \cdot) d W (s), ϕ), \end{array}$ $\forall ϕ \in D ((0, T) \times Q)$ .

Analogously to the existence and uniqueness result for the Dirichlet problem by Pardoux [43], we have the following result

Theorem 1.

Suppose that the assumptions (A1)–(A5) hold. Then for fixed $ϵ > 0$ , the problem (1) has a unique strong probabilistic solution $\begin{matrix} u^{ϵ} \in L^{2} (Ω, F, P; C ([0, T]; H_{ϵ} (Q^{ϵ}))), u_{t}^{ϵ} \in L^{2} (Ω, F, P; C ([0, T]; L^{2} (Q^{ϵ}))), \end{matrix}$ in the sense of Definition 1 .

Our goals are described as follows: First, we show that the sequence of solutions $u^{ϵ}$ converges as $ϵ \to 0$ to a solution u of the following stochastic partial differential equation (SPDE) $\begin{matrix} (3) & \{\begin{matrix} θ d u_{t} = div A_{0} \nabla u d t + f d t + g d \hat{W} in Q \times (0, T), \\ u (x, t) = 0 on \partial Q \times (0, T), \\ u (x, 0) = \frac{u_{0} (x)}{θ}, u_{t} (x, 0) = \frac{u_{1} (x)}{θ} in Q, \end{matrix} \end{matrix}$ where $A_{0}$ is a constant elliptic matrix defined by $\begin{matrix} A_{0} λ = M (A \nabla ω_{λ}), \end{matrix}$ and $ω_{λ} \in H_{per} (Y)$ is the unique solution of the following boundary value problem: $\begin{array}{l} (4) & \{\begin{matrix} - div (A (y) \nabla ω_{λ}) = 0 in Y^{*}, \\ \frac{\partial ω_{λ}}{\partial ν_{A}} = 0 on \partial S, \\ ω_{λ} - λ \cdot y Y -periodic, \end{matrix} \end{array}$ for any $λ \in R^{n}$ . Next, we show that the associated energy to the problem (1) uniformly converges to the associated energy to the problem (3).

Fig. 2.

Wave generated in a membrane with a random distribution of inclusions.

Problems of type (1) arise in several physical processes in the presence of random fluctuations, for instance, in the modeling of waves generated in a vibrating string, an elastic membrane, see Fig. 2, and a rubbery solid in dimensions 1, 2 and 3 respectively. To illustrate that; for example let us consider the disturbance generated in bridge cables. These cables are made up of composite materials and vibrate continuously with high irregularity as a response to wind blow. In this case the external force is given by $f^{ϵ} (t, x) d t + g^{ϵ} (t, x) d W$ . It is also possible that the disturbance arises via other sources such as birds landing on or taking off from the cable. In this case, the intensity of the disturbance on the cable is moderate. Thus, the force has a more regular behaviour and therefore, the stochastic term may be neglected. In this case, the force is represented by $f^{ϵ} (t, x)$ . In fact problem (1) can also be understood according to the well known Walsh interpretation [61]; since the strings of a guitar have the structure of a composite material, when bombarded by particles of dust, the motion of the strings is subjected to random vibrations. Such a process can be modeled by problem (1). Wave equations in heterogeneous media have applications in several other branches of science such as geoscience, physics and engineering [10,27].

This paper is organized as follows. In Section 2, we introduce some preliminaries about the extension operators. In Section 3, we establish important a priori estimates that will be used in subsequent sections. Section 4 is devoted to the proof of the tightness of probability measures generated by the extension of $u^{ϵ}$ to Q and the Wiener process $W (t)$ ; this will enable us to use Prokhorov’s and Skorokhod’s processes for the construction of a sequence of random variables $(W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}})$ defined on new probability spaces; $(W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}})$ satisfies the original problem (1) and strongly converges in a suitable space to a triple $(W, u, u_{t})$ that solve the homogenized problem (3). Section 5 is devoted to the construction of problem (3) by adapting Tartar’s method of oscillating test functions. In Section 6 we establish the asymptotic relation of the energies, corresponding to the problems (1) and (3).

2. Preliminaries

The following lemma collected from [19] will be needed in the homogenization result.

Lemma 1.
Let $1 ⩽ p < \infty$ and u be a Y-periodic function in $L^{p} (Y)$ . Set $\begin{matrix} u^{ϵ} (x) = u (\frac{x}{ϵ}) a . e on R^{n} . \end{matrix}$ Then, as $ϵ \to 0$ $\begin{matrix} u^{ϵ} (x) ⇀ M_{Y} (u) weakly in L^{p} (K), \end{matrix}$ for any bounded open subset K of $R^{n}$ , where $M_{Y} (u)$ the mean value of u over Y.

The following extension result is proved in [21].
Lemma 2.
There exists a linear continuous extension operator $\begin{matrix} (5) & \tilde{M} \in L (H^{1} (Y^{}), H^{1} (Y)) \end{matrix}$ such that for some constant* $C > 0$ $\begin{matrix} ‖ \nabla \tilde{M} v ‖_{{[L^{2} (Y)]}^{n}} ⩽ C ‖ \nabla v ‖_{{[L^{2} (Y^{})]}^{n}} \end{matrix}$ for any* $v \in H^{1} (Y^{})$ .

We collect the following lemma from [18]. Lemma 3.
There exists a linear continuous extension operator* $\begin{matrix} (6) & P_{ϵ} \in L (L^{\infty} (0, T; H^{k} (Q^{ϵ})), L^{\infty} (0, T; H^{k} (Q))), k = 0, 1 \end{matrix}$ such that for some constant C independent of ϵ and any $v \in L^{\infty} (0, T; H^{1} (Q^{ϵ}))$ with $v^{'} \in L^{\infty} (0, T; L^{2} (Q^{ϵ}))$
$P_{ϵ} v = v$ on $Q^{ϵ} \times (0, T)$ ,

$P_{ϵ} v^{'} = {(P_{ϵ} v)}^{'}$ on $Q \times (0, T)$ ,

$‖ P_{ϵ} v ‖_{L^{\infty} (0, T; L^{2} (Q))} ⩽ C ‖ v ‖_{L^{\infty} (0, T; L^{2} (Q^{ϵ}))}$ ,

$‖ P_{ϵ} v^{'} ‖_{L^{\infty} (0, T; L^{2} (Q))} ⩽ C ‖ v^{'} ‖_{L^{\infty} (0, T; L^{2} (Q^{ϵ}))}$ ,

$‖ \nabla (P_{ϵ} v) ‖_{L^{\infty} (0, T; {[L^{2} (Q)]}^{n})} ⩽ C ‖ \nabla v ‖_{L^{\infty} (0, T; {[L^{2} (Q^{ϵ})]}^{n})}$ .

Lemma 4.
There exists a linear continuous extension operator $\begin{matrix} P_{ϵ} \in L (L^{2} (0, T; H^{- 1} (Q^{ϵ})), L^{2} (0, T; H^{- 1} (Q))) \end{matrix}$ such that for some constant C independent of ϵ
$P_{ϵ} q_{ϵ} = q_{ϵ}$ on $Q^{ϵ} \times (0, T)$ ,

$‖ P_{ϵ} q_{ϵ} ‖_{L^{2} (0, T; H^{- 1} (Q))} ⩽ C ‖ q_{ϵ} ‖_{L^{2} (0, T; H^{- 1} (Q^{ϵ}))}$
for any $q_{ϵ} \in L^{2} (0, T; H^{- 1} (Q^{ϵ}))$ .
Proof.
Let $R_{ϵ} : L^{2} (0, T; H_{0}^{1} (Q)) \to L^{2} (0, T; H_{ϵ} (Q^{ϵ}))$ be the restriction operator such that $\begin{matrix} R_{ϵ} u = u on Q^{ϵ} \times (0, T) . \end{matrix}$ This operator obviously exists and is bounded. We define the functional $\begin{matrix} F : L^{2} (0, T; H_{0}^{1} (Q)) ⟶ R \end{matrix}$ by $\begin{matrix} {⟨ F, u ⟩}_{L^{2} (0, T; H^{- 1} (Q)), L^{2} (0, T; H_{0}^{1} (Q))} = {⟨ q_{ϵ}, R_{ϵ} u ⟩}_{L^{2} (0, T; H^{- 1} (Q^{ϵ})), L^{2} (0, T; H_{ϵ} (Q^{ϵ}))} . \end{matrix}$ It is clear that $F \in L^{2} (0, T; H^{- 1} (Q))$ and $\begin{matrix} ‖ F ‖_{L^{2} (0, T; H^{- 1} (Q))} = sup_{u \in L^{2} (0, T; H_{0}^{1} (Q)) : ‖ u ‖ = 1} {⟨ F, u ⟩}_{L^{2} (0, T; H^{- 1} (Q)), L^{2} (0, T; H_{0}^{1} (Q))} . \end{matrix}$ Readily $\begin{array}{rcl} ‖ F ‖_{L^{2} (0, T; H^{- 1} (Q))} & = & sup_{u \in L^{2} (0, T; H_{0}^{1} (Q)) : ‖ u ‖ = 1} {⟨ q_{ϵ}, R_{ϵ} u ⟩}_{L^{2} (0, T; H^{- 1} (Q^{ϵ})), L^{2} (0, T; H_{ϵ} (Q^{ϵ}))} \\ ⩽ & ‖ R_{ϵ} u ‖ ‖ q_{ϵ} ‖ ⩽ C ‖ q_{ϵ} ‖ . \end{array}$ Setting $P_{ϵ} q_{ϵ} = F$ , we get the second statement of the lemma. To prove the first statement, we note that $\begin{matrix} {⟨ P_{ϵ} q_{ϵ}, u ⟩}_{L^{2} (0, T; H^{- 1} (Q)), L^{2} (0, T; H_{0}^{1} (Q))} = {⟨ q_{ϵ}, R_{ϵ} u ⟩}_{L^{2} (0, T; H^{- 1} (Q^{ϵ})), L^{2} (0, T; H_{ϵ} (Q^{ϵ}))} . \end{matrix}$ From the definition of the restriction operator $R_{ϵ}$ , we have $\begin{matrix} {⟨ P_{ϵ} q_{ϵ}, u ⟩}_{L^{2} (0, T; H^{- 1} (Q)), L^{2} (0, T; H_{0}^{1} (Q))} = {⟨ q_{ϵ}, u ⟩}_{L^{2} (0, T; H^{- 1} (Q^{ϵ})), L^{2} (0, T; H_{ϵ} (Q^{ϵ}))} . \end{matrix}$ Therefore $\begin{matrix} {⟨ P_{ϵ} q_{ϵ} - q_{ϵ}, u ⟩}_{L^{2} (0, T; H^{- 1} (Q)), L^{2} (0, T; H_{0}^{1} (Q))} = 0, \forall u \in L^{2} (0, T; H_{0}^{1} (Q)) . \end{matrix}$ This implies that (i) holds. The lemma is therefore proved. □
Proposition 1.
Assume that $\begin{matrix} (7) & ‖ u_{0}^{ϵ} ‖_{H_{ϵ}} ⩽ C and {\tilde{u}}_{0}^{ϵ} ⇀ u_{0} weakly in L^{2} (Q), \end{matrix}$ where $C > 0$ is a constant independent of ϵ and ${\tilde{u}}_{0}^{ϵ}$ the extension by zero of $u_{0}^{ϵ}$ to the whole Q. Furthermore assume that ${\tilde{M}}_{ϵ} \in L (H_{ϵ} (Q^{ϵ}), H_{0}^{1} (Q))$ is an extension operator such that $\begin{matrix} ‖ \nabla {\tilde{M}}_{ϵ} v ‖_{{[L^{2} (Q)]}^{n}} ⩽ C ‖ \nabla v ‖_{{[L^{2} (Q^{ϵ})]}^{n}}, \forall v \in H^{1} (Q^{ϵ}), \end{matrix}$ then $\begin{matrix} {\tilde{M}}_{ϵ} u_{0}^{ϵ} ⇀ \frac{u_{0}}{θ} weakly in H_{0}^{1} (Q) . \end{matrix}$
Proof.
From (7), there exists $U_{0}$ , such that $\begin{matrix} (8) & {\tilde{M}}_{ϵ} u_{0}^{ϵ} ⇀ U_{0} weakly in H_{0}^{1} (Q), \end{matrix}$ so $\begin{matrix} (9) & {\tilde{M}}_{ϵ} u_{0}^{ϵ} \to U_{0} strongly in L^{2} (Q) . \end{matrix}$ But, we also have $\begin{matrix} (10) & χ_{(Q^{ϵ})} ⇀ θ weakly* in L^{\infty} (Q) . \end{matrix}$ On the other hand $\begin{matrix} (11) & {\tilde{M}}_{ϵ} u_{0}^{ϵ} . χ_{(Q^{ϵ})} = {\tilde{u}}_{0}^{ϵ} a.e. in Q . \end{matrix}$ Due to (9) and (10), we can pass to the limit in (11) to obtain $\begin{matrix} U_{0} θ = u_{0} . \end{matrix}$ Therefore, we deduce from (8), that $\begin{matrix} (12) & {\tilde{M}}_{ϵ} u_{0}^{ϵ} ⇀ \frac{u_{0}}{θ} weakly in H_{0}^{1} (Q) . \end{matrix}$ □

3. Uniform estimates of $u^{ϵ}$

Here and in the sequel, C will denote a constant independent of ϵ. In this section we establish the a priori estimates announced earlier. In our first lemma, we prove that, both the solution to the problem (1) and its time derivative are bounded in appropriate probabilistic evolution spaces. Likewise in our second lemma we establish a finite difference estimate of the time derivative of the solution in a space involving $H^{- 1} (Q^{ϵ})$ .

Lemma 5.
Under the assumptions (A1)–(A5), the solution $u^{ϵ}$ of (1) satisfies the following estimate $\begin{matrix} (13) & E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{H_{ϵ} (Q^{ϵ})}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} ⩽ C . \end{matrix}$
Proof.
The following arguments are used modulo appropriate stopping times. Itô formula and the symmetry of A give $\begin{matrix} d [{‖ u_{t}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + (A_{ϵ} \nabla u^{ϵ}, \nabla u^{ϵ})] = 2 (f^{ϵ}, u_{t}^{ϵ}) d t + 2 (g^{ϵ}, u_{t}^{ϵ}) d W + {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t . \end{matrix}$ Integrating over $(0, t)$ , $t ⩽ T$ , we get $\begin{array}{l} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + (A_{ϵ} \nabla u^{ϵ} (t), \nabla u^{ϵ} (t)) \\ = {‖ u_{1}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + (A_{ϵ} \nabla u_{0}^{ϵ}, \nabla u_{0}^{ϵ}) + 2 \int_{0}^{t} (f^{ϵ}, u_{t}^{ϵ}) d s + 2 \int_{0}^{t} (g^{ϵ}, u_{t}^{ϵ}) d W + \int_{0}^{t} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d s . \end{array}$ Using the assumptions on the matrix A and taking the supremum over $t \in [0, T]$ , we have $\begin{array}{l} sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{H_{ϵ} (Q^{ϵ})}^{2} \\ ⩽ C {‖ u_{1}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + C {‖ u_{1}^{ϵ} ‖}_{H_{ϵ} (Q^{ϵ})}^{2} \\ + 2 C \int_{0}^{T} (f^{ϵ}, u_{t}^{ϵ}) d t + 2 C sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} (g^{ϵ}, u_{t}^{ϵ}) d W | + C \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t . \end{array}$ Taking the expectation on both sides, we have $\begin{array}{l} E [sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{H_{ϵ} (Q^{ϵ})}^{2}] \\ ⩽ C E [{‖ u_{1}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + {‖ u_{0}^{ϵ} ‖}_{H_{ϵ} (Q^{ϵ})}^{2} \\ + \int_{0}^{T} (f^{ϵ}, u_{t}^{ϵ}) d t + sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} (g^{ϵ}, u_{t}^{ϵ}) d W | + \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t] \\ (14) & ⩽ C [C_{1} + E \int_{0}^{T} (f^{ϵ}, u_{t}^{ϵ}) d t + E sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} (g^{ϵ}, u_{t}^{ϵ}) d W |], \end{array}$ where $\begin{matrix} C_{1} = {‖ u_{1}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + {‖ u_{0}^{ϵ} ‖}_{H_{ϵ} (Q^{ϵ})}^{2} + \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d s . \end{matrix}$

Using Cauchy–Schwarz’s and Young’s inequalities, we have $\begin{array}{rcl} E \int_{0}^{T} (f^{ϵ}, u_{t}^{ϵ}) d t & ⩽ & E \int_{0}^{T} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})} {‖ u_{t}^{ϵ} ‖}_{L^{2} (Q^{ϵ})} d t ⩽ E sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})} \int_{0}^{T} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})} d t \\ (15) & ⩽ & δ E sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + C (δ) {(\int_{0}^{T} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{2} \end{array}$ for any $δ > 0$ .

Thanks to Burkhölder–Davis–Gundy’s inequality, followed by Cauchy–Schwarz’s inequality, the second term in the right-hand side of (14) can be estimated as $\begin{matrix} E sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} (g^{ϵ}, u_{t}^{ϵ}) d W | ⩽ C E {(\int_{0}^{T} {(g^{ϵ}, u_{t}^{ϵ})}^{2} d t)}^{\frac{1}{2}} ⩽ C E {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} {‖ u_{t}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{\frac{1}{2}} . \end{matrix}$ Again using Young’s inequality, we get $\begin{array}{rcl} C E {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} {‖ u_{t}^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{\frac{1}{2}} & ⩽ & C E sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{\frac{1}{2}} \\ (16) & ⩽ & C (δ) E sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + δ C \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t, \end{array}$ for any $δ > 0$ . Using (15) and (16) into (14)with sufficiently small δ and applying assumption (A5), to obtain $\begin{matrix} E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{H_{ϵ} (Q^{ϵ})}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} ⩽ C . \end{matrix}$ The proof is complete. □

Next we have Lemma 6.
Under the assumptions (A1)–(A5) with the replacement of the assumption on $g^{ϵ}$ by the requirement that $g^{ϵ} \in L^{4} ((0, T); H^{- 1} (Q))$ , there exist positive constants C, $δ_{0}$ such that for all $ϵ > 0$ and for all $δ \in (0, δ_{0})$ , $u_{t}^{ϵ}$ satisfies the following estimate: $\begin{matrix} (17) & E sup_{| ρ | ⩽ δ} \int_{0}^{T} {‖ u_{t}^{ϵ} (t + ρ) - u_{t}^{ϵ} (t) ‖}_{H^{- 1} (Q^{ϵ})}^{2} d t < C δ . \end{matrix}$
Proof.
Assume that $u_{t}^{ϵ}$ is extended by zero outside the interval $[0, T]$ . We write $\begin{matrix} u_{t}^{ϵ} (t + ρ) - u_{t}^{ϵ} (t) = \int_{t}^{t + ρ} div (A_{ϵ} \nabla u^{ϵ}) d s + \int_{t}^{t + ρ} f^{ϵ} d s + \int_{t}^{t + ρ} g^{ϵ} d W (s) . \end{matrix}$ Then $\begin{array}{rcl} {‖ u_{t}^{ϵ} (t + ρ) - u_{t}^{ϵ} (t) ‖}_{H^{- 1} (Q^{ϵ})} & ⩽ & {‖ \int_{t}^{t + ρ} div (A_{ϵ} \nabla u^{ϵ}) d s ‖}_{H^{- 1} (Q^{ϵ})} \\ (18) & + {‖ \int_{t}^{t + ρ} f^{ϵ} d s ‖}_{H^{- 1} (Q^{ϵ})} + {‖ \int_{t}^{t + ρ} g^{ϵ} d W (s) ‖}_{H^{- 1} (Q^{ϵ})} . \end{array}$ Using assumption (A2), we have $\begin{array}{rcl} {‖ \int_{t}^{t + ρ} div (A_{ϵ} \nabla u^{ϵ}) d s ‖}_{H^{- 1} (Q^{ϵ})} & ⩽ & sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} | {⟨ \int_{t}^{t + ρ} div (A_{ϵ} \nabla u^{ϵ}) d s, ϕ ⟩}_{H^{- 1} (Q^{ϵ}), H_{ϵ} (Q^{ϵ})} | \\ = & sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} \int_{Q} \int_{t}^{t + ρ} A_{ϵ} \nabla u^{ϵ} \nabla ϕ d x d s \\ (19) & ⩽ & C \int_{t}^{t + ρ} {‖ u^{ϵ} ‖}_{H_{ϵ} (Q^{ϵ})} ‖ ϕ ‖_{H_{ϵ} (Q^{ϵ})} d s ⩽ C ρ^{\frac{1}{2}} . \end{array}$

From assumption (A5), we obtain $\begin{array}{rcl} {‖ \int_{t}^{t + ρ} f^{ϵ} d s ‖}_{H^{- 1} (Q^{ϵ})} & ⩽ & sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} | {⟨ \int_{t}^{t + ρ} f^{ϵ} d s, ϕ ⟩}_{H^{- 1} (Q^{ϵ}), H_{ϵ} (Q^{ϵ})} | \\ = & sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} \int_{Q^{ϵ}} \int_{t}^{t + ρ} f^{ϵ} ϕ d x d s \\ (20) & ⩽ & C \int_{t}^{t + ρ} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})} ‖ ϕ ‖_{L^{2} (Q^{ϵ})} d s ⩽ C ρ^{\frac{1}{2}} . \end{array}$ Since $\begin{matrix} {‖ \int_{t}^{t + ρ} g^{ϵ} d W (s) ‖}_{H^{- 1} (Q^{ϵ})}^{2} ⩽ sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} {| {⟨ \int_{t}^{t + ρ} g^{ϵ} d W (s), ϕ ⟩}_{H^{- 1} (Q^{ϵ}), H_{ϵ} (Q^{ϵ})} |}^{2}, \end{matrix}$ then Fubini’s theorem gives $\begin{array}{l} E sup_{| ρ | ⩽ δ} \int_{0}^{T} {‖ \int_{t}^{t + ρ} g^{ϵ} d W (s) ‖}_{H^{- 1} (Q^{ϵ})}^{2} d t \\ ⩽ sup_{| ρ | ⩽ δ} \int_{0}^{T} sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} E {(\int_{t}^{t + ρ} {⟨ g^{ϵ}, ϕ ⟩}_{H^{- 1} (Q^{ϵ}), H_{ϵ} (Q^{ϵ})} d W (s))}^{2} d t . \end{array}$ Thanks to Burkhölder–Davis–Gundy’s inequality, we get $\begin{array}{l} sup_{| ρ | ⩽ δ} \int_{0}^{T} sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} E {(\int_{t}^{t + ρ} {⟨ g^{ϵ}, ϕ ⟩}_{H^{- 1} (Q^{ϵ}), H_{ϵ} (Q^{ϵ})} d W (s))}^{2} d t \\ ⩽ sup_{| ρ | ⩽ δ} \int_{0}^{T} sup_{ϕ \in H_{ϵ} (Q^{ϵ}) : ‖ ϕ ‖ = 1} \int_{t}^{t + ρ} {⟨ g^{ϵ}, ϕ ⟩}_{H^{- 1} (Q^{ϵ}), H_{ϵ} (Q^{ϵ})}^{2} d s d t ⩽ sup_{| ρ | ⩽ δ} \int_{0}^{T} \int_{t}^{t + ρ} {‖ g^{ϵ} ‖}_{H^{- 1} (Q^{ϵ})}^{2} d s d t . \end{array}$ But Cauchy–Schwarz’s inequality gives $\begin{array}{rcl} sup_{| ρ | ⩽ δ} \int_{0}^{T} \int_{t}^{t + ρ} {‖ g^{ϵ} ‖}_{H^{- 1} (Q^{ϵ})}^{2} d s d t & ⩽ & sup_{| ρ | ⩽ δ} \int_{0}^{T} {(\int_{t}^{t + ρ} d s)}^{\frac{1}{2}} {(\int_{t}^{t + ρ} {‖ g^{ϵ} ‖}_{H^{- 1} (Q^{ϵ})}^{4} d s)}^{\frac{1}{2}} d t \\ ⩽ & δ^{\frac{1}{2}} \int_{0}^{T} {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{H^{- 1} (Q^{ϵ})}^{4} d t)}^{\frac{1}{2}} d t . \end{array}$ Now using the assumption made on $g^{ϵ}$ , we have $\begin{matrix} (21) & E sup_{0 ⩽ ρ ⩽ δ} \int_{0}^{T} {‖ \int_{t}^{t + ρ} g^{ϵ} d W (s) ‖}_{H^{- 1} (Q^{ϵ})}^{2} d t ⩽ C (T) δ^{\frac{1}{2}} . \end{matrix}$ From (19), (20) and (21), we arrive at $\begin{matrix} E sup_{| ρ | ⩽ δ} \int_{0}^{T} {‖ u_{t}^{ϵ} (t + ρ) - u_{t}^{ϵ} (t) ‖}_{H^{- 1} (Q^{ϵ})}^{2} d t ⩽ C δ . \end{matrix}$ □
Remark 1.
From the extensions Lemmas 3, 4, estimates (13) and (17) we have that an extension $P_{ϵ} u^{ϵ}$ of $u^{ϵ}$ to $Ω \times Q \times (0, T)$ satisfying $\begin{array}{l} (22) & E sup_{0 ⩽ t ⩽ T} {‖ P_{ϵ} u^{ϵ} (t) ‖}_{H_{0}^{1} (Q)}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ P_{ϵ} u_{t}^{ϵ} (t) ‖}_{L^{2} (Q)}^{2} ⩽ C, \\ (23) & E sup_{| ρ | ⩽ δ} \int_{0}^{T} {‖ P_{ϵ} u_{t}^{ϵ} (t + ρ) - P_{ϵ} u_{t}^{ϵ} (t) ‖}_{H^{- 1} (Q)}^{2} d t < C δ, \end{array}$ where the data is extended by zero outside $Q^{ϵ} \times (0, T)$ .

4. Tightness property of probability measures

In this section, we establish the tightness of probability measures generated by the extension of $u^{ϵ}$ to Q (denoted by $u^{ϵ}$ for simplicity) and the Wiener process $W (t)$ . We start by introducing analytic and probabilistic compactness results. We have from [56]

Lemma 7.
Let $B_{0}$ , B and $B_{1}$ be some Banach spaces such that $B_{0} \subset B \subset B_{1}$ and the injection $B_{0} \subset B$ is compact. For any $1 ⩽ p, q ⩽ \infty$ , and $0 < s ⩽ 1$ let E be a set bounded in $L^{q} (0, T; B_{0}) \cap N^{s, p} (0, T; B_{1})$ , where $\begin{matrix} N^{s, p} (0, T; B_{1}) = {v \in L^{p} (0, T; B_{1}) : sup_{h > 0} h^{- s} {‖ v (t + ρ) - v (t) ‖}_{L^{p} (0, T - ρ, B_{1})} < \infty} . \end{matrix}$ Then E is relatively compact in $L^{p} (0, T; B)$ .

The following two lemmas are collected from [9]. Let $S$ be a separable Banach space and consider its Borel σ-field to be $B (S)$ . We have
Lemma 8 (Prokhorov).

A sequence of probability measures ${(Π_{n})}_{n \in N}$ on $(S, B (S))$ is tight if and only if it is relatively compact.

Lemma 9 (Skorokhod).

Suppose that the probability measures ${(μ_{n})}_{n \in N}$ on $(S, B (S))$ weakly converge to a probability measure μ. Then there exist random variables $ξ, ξ_{1}, \dots, ξ_{n}, \dots$ , defined on a common probability space $(Ω, F, P)$ , such that $L (ξ_{n}) = μ_{n}$ and $L (ξ) = μ$ and $\begin{matrix} lim_{n \to \infty} ξ_{n} = ξ, P -a . s .; \end{matrix}$ the symbol $L (\cdot)$ stands for the law of · .

Let us introduce the space $Z = Z_{1} \times Z_{2}$ where $\begin{matrix} Z_{1} = {ϕ : sup_{0 ⩽ t ⩽ T} {‖ ϕ (t) ‖}_{H_{0}^{1} (Q)}^{2} ⩽ C_{1}, sup_{0 ⩽ t ⩽ T} {‖ ϕ^{'} (t) ‖}_{L^{2} (Q)}^{2} ⩽ C_{1}} \end{matrix}$ and $\begin{matrix} Z_{2} = {ψ : sup_{0 ⩽ t ⩽ T} {‖ ψ (t) ‖}_{L^{2} (Q)}^{2} ⩽ C_{3} and sup_{n} \frac{1}{ν_{n}} sup_{ρ ⩽ μ_{n}} {(\int_{0}^{T} {‖ ψ (t + ρ) - ψ (t) ‖}_{H^{- 1} (Q)}^{2})}^{\frac{1}{2}} < \infty} . \end{matrix}$ We endow Z with the norm $\begin{array}{rcl} {‖ (ϕ, ψ) ‖}_{Z} & = & ‖ ϕ ‖_{Z_{1}} + ‖ ψ ‖_{Z_{2}} \\ = & sup_{0 ⩽ t ⩽ T} {‖ ϕ^{'} (t) ‖}_{L^{2} (Q)} + sup_{0 ⩽ t ⩽ T} ‖ ϕ ‖_{H_{0}^{1} (Q)} + sup_{0 ⩽ t ⩽ T} {‖ ψ (t) ‖}_{L^{2} (Q)} \\ + sup_{n} \frac{1}{ν_{n}} sup_{ρ ⩽ μ_{n}} {(\int_{0}^{T} {‖ ψ (t + ρ) - ψ (t) ‖}_{H^{- 1} (Q)}^{2})}^{\frac{1}{2}} . \end{array}$

Lemma 10.
The above constructed space Z is a compact subset of $L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; H^{- 1} (Q))$ .
Proof.
Lemma 7 together with a suitable argument due to Bensoussan [7] give the compactness of $Z_{1}$ and $Z_{2}$ in $L^{2} (0, T; L^{2} (Q))$ and $L^{2} (0, T; H^{- 1} (Q))$ respectively. □

Now consider the space $X = C (0, T; R^{m}) \times L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; H^{- 1} (Q))$ and $B (X)$ the σ-algebra of the Borel sets of $X$ . Let $Ψ_{ϵ}$ be the $(X, B (X))$ -valued measurable map defined on $(Ω, F, P)$ by $\begin{matrix} Ψ_{ϵ} : ω \mapsto (W (ω), u^{ϵ} (ω), u_{t}^{ϵ} (ω)) . \end{matrix}$ Define on $(X, B (X))$ the probability measures $Π_{ϵ}$ by $\begin{matrix} Π_{ϵ} (A) = P (Ψ_{ϵ}^{- 1} (A)) for all A \in B (X) . \end{matrix}$
Lemma 11.
The family of probability measures ${Π_{ϵ} : ϵ > 0}$ is tight in $(X, B (X))$ .
Proof.
The proof follow along the lines of the proof of [35, Lemma 7], see also [7,22,45,46] and [51]. □

From Lemmas 8, 11, the estimates (22) and (23), there exist a subsequence ${Π_{ϵ_{j}}}$ and a measure Π such that $\begin{matrix} Π_{ϵ_{j}} ⇀ Π \end{matrix}$ weakly. From Lemma 9 there exist a probability space $(\hat{Ω}, \hat{F}, \hat{P})$ and $X$ -valued random variables $(W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}})$ , $(\hat{W}, u, u_{t})$ such that the probability law of $(W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}})$ is $Π_{ϵ_{j}}$ and that of $(\hat{W} u, u_{t})$ is Π. Furthermore, we have $\begin{matrix} (24) & (W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}}) \to (\hat{W} u, u_{t}) in X, \tilde{P} -a.s. \end{matrix}$ Let us define the filtration $\begin{matrix} \tilde{F_{t}} = σ {\hat{W} (s), u (s), u_{t} (s)}_{0 ⩽ s ⩽ t} . \end{matrix}$ We show that $\hat{W} (t)$ is an $\hat{F_{t}}$ -Wiener process following [7] and [51]. Arguing as in [51] we get that $(W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}})$ satisfies $\hat{P}$ -a.s. the problem (1) on the fixed domain $Ω \times Q \times (0, T)$ in the sense of distributions.
5. The homogenization result

In this section we formulate our first main result which is:

Theorem 2.
Suppose that the assumptions (A1)–(A5) are satisfied with the replacement of the assumption on $g^{ϵ}$ by $g^{ϵ} \in L^{4} ((0, T); H^{- 1} (Q^{ϵ}))$ . Furthermore let $g_{t}^{ϵ_{j}} \in L^{2} (Q^{ϵ} \times (0, T))$ and $\begin{array}{l} (25) & {\tilde{f}}^{ϵ_{j}} ⇀ f weakly in L^{2} (Q \times (0, T)), \\ (26) & u_{0}^{ϵ_{j}} ⇀ \frac{u_{0}}{θ} weakly in H_{0}^{1} (Q), \\ (27) & {\tilde{u}}_{1}^{ϵ_{j}} ⇀ u_{1} weakly in L^{2} (Q), \\ (28) & {\tilde{g}}^{ϵ_{j}} ⇀ g weakly in L^{2} (Q \times (0, T)), \\ (29) & {\tilde{g}}_{t}^{ϵ_{j}} ⇀ g_{t} weakly in L^{2} (Q \times (0, T)) . \end{array}$ Then there exist a probability space $(\tilde{Ω}, \tilde{F}, \tilde{P}, {({\tilde{F}}_{t})}_{0 ⩽ t ⩽ T})$ and random variables $(W_{ϵ_{j}}, u^{ϵ_{j}}, u_{t}^{ϵ_{j}})$ and $(\hat{W}, u, u_{t})$ such that the convergence (24) holds and $\begin{matrix} (30) & A_{ϵ_{j}} \tilde{\nabla u^{ϵ_{j}}} ⇀ A_{0} \nabla u weakly in {[L^{2} (0, T; L^{2} (Q))]}^{n}, \tilde{P} -a . s ., \end{matrix}$ and $(\hat{W}, u, u_{t})$ satisfies the homogenized problem (3 ) .
Proof.
We shall omit j from the index of the sequence $ϵ_{j}$ . Let us define the vector functions $ξ^{ϵ}$ by $\begin{matrix} ξ^{ϵ} (x, t) = A_{ϵ} (x) \nabla u^{ϵ} (x, t) . \end{matrix}$ Then $ξ^{ϵ}$ satisfies $\begin{array}{l} (31) & - div ξ^{ϵ} d t = f^{ϵ} d t + g^{ϵ} d W_{ϵ} - d u_{t}^{ϵ} in Q^{ϵ} \times (0, T), \\ (32) & ξ^{ϵ} \cdot n = 0 on \partial S^{ϵ} \times (0, T) . \end{array}$

From the assumptions on $A_{ϵ}$ and estimate (13), we obtain $\begin{matrix} \hat{E} {‖ ξ^{ϵ} ‖}_{{[L^{2} (0, T; L^{2} (Q^{ϵ}))]}^{n}} ⩽ C . \end{matrix}$ Then up to a subsequence still denoted by $ξ^{ϵ}$ , the extension ${\tilde{ξ}}^{ϵ}$ of $ξ^{ϵ}$ satisfies $\begin{matrix} (33) & {\tilde{ξ}}^{ϵ} ⇀ ξ weakly in {[L^{2} (\hat{Ω}, \hat{F}, \hat{P}; L^{2} (0, T; L^{2} (Q)))]}^{n} . \end{matrix}$ Using the definition of this vector in problem $(P_{ϵ})$ , we see that $ξ^{ϵ}$ satisfies $\begin{array}{l} \int_{0}^{T} \int_{Q} d P_{ϵ} u_{t}^{ϵ} ϕ (t) v (x) \cdot χ_{(Q^{ϵ})} d x + \int_{0}^{T} \int_{Q} {\tilde{ξ}}^{ϵ} ϕ \nabla v d x d t \\ (34) & = \int_{0}^{T} \int_{Q} {\tilde{f}}^{ϵ} ϕ v d x d t + \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x d W_{ϵ}, \end{array}$ for any $v \in D (Q)$ and $ϕ \in D (0, T)$ . Integrating the first term by parts twice and using property (2) of Lemma 3, we have $\begin{array}{l} \int_{0}^{T} \int_{Q} P_{ϵ} u^{ϵ} ϕ_{t t} (t) v (x) \cdot χ_{(Q^{ϵ})} d x + \int_{0}^{T} \int_{Q} {\tilde{ξ}}^{ϵ} ϕ \nabla v d x d t \\ (35) & = \int_{0}^{T} \int_{Q} {\tilde{f}}^{ϵ} ϕ v d x d t + \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x d W_{ϵ} . \end{array}$ We pass to the limit, in the two terms on the left-hand side and in the first term on the right-hand side of (35), using convergences (10), (24), (33) and (25), then we get $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ_{t t} (t) v (x) d x + \int_{0}^{T} \int_{Q} ξ ϕ \nabla v d x d t \\ (36) & = \int_{0}^{T} \int_{Q} f ϕ v d x d t + lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x d W_{ϵ} . \end{array}$ In the following we show that $\begin{matrix} (37) & lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x d W_{ϵ} = \int_{0}^{T} \int_{Q} g ϕ v d x d \hat{W} . \end{matrix}$ Using integration by parts, we have $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x d W_{ϵ} \\ (38) & = lim_{ϵ \to 0} [W_{ϵ} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x |_{0}^{T} - \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v W_{ϵ} d x d t - \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v W_{ϵ} d x d t] . \end{array}$ In the first term on the right hand side of (38), we pass to the limit using convergences (24) and (28) and obtain $\begin{array}{l} (39) & lim_{ϵ \to 0} W_{ϵ} \int_{Q} {\tilde{g}}^{ϵ} ϕ v d x |_{0}^{T} = \hat{W} \int_{Q} g ϕ v d x |_{0}^{T}, \end{array}$ the second term can be written as $\begin{array}{l} \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v W_{ϵ} d x d t \\ (40) & = lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v (W_{ϵ} - W) d x d t + lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v \hat{W} d x d t . \end{array}$ The first term on the right hand side of (40) will converge to zero using convergence (24) and the assumptions on ${\tilde{g}}_{t}^{ϵ}$ , v and ϕ. Indeed $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v (W_{ϵ} - \hat{W}) d x d t \\ ⩽ lim_{ϵ \to 0} sup_{t \in [0, T]} | W_{ϵ} - \hat{W} | \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v d x d t ⩽ lim_{ϵ \to 0} C sup_{t \in [0, T]} | W_{ϵ} - \hat{W} | = 0 . \end{array}$ Now using convergence (29), we can pass to the limit in the second term on the right hand side of (40) to obtain $\begin{matrix} (41) & lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}_{t}^{ϵ} ϕ v \hat{W} d x d t = \int_{0}^{T} \int_{Q} g_{t} ϕ v \hat{W} d x d t . \end{matrix}$ Similarly, we treat the last term in the right hand side of (38) by writing $\begin{matrix} (42) & \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v W_{ϵ} d x d t = lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v (W_{ϵ} - \hat{W}) d x d t + lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v \hat{W} d x d t . \end{matrix}$ Using the assumptions on ${\tilde{g}}^{ϵ}$ , v and ϕ, we have $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v (W_{ϵ} - \hat{W}) d x d t \\ ⩽ lim_{ϵ \to 0} sup_{t \in [0, T]} | W_{ϵ} - \hat{W} | \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v d x d t ⩽ lim_{ϵ \to 0} C sup_{t \in [0, T]} | W_{ϵ} - \hat{W} | . \end{array}$ Hence the first term on the right hand side of (42) converges to zero by (24). Now using convergence (28), we can pass to the limit in the second term on the right hand side of (42) to obtain $\begin{matrix} (43) & lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ^{'} v \hat{W} d x d t = \int_{0}^{T} \int_{Q} g ϕ^{'} v \hat{W} d x d t . \end{matrix}$ From the convergences (43), (41) and (39), we obtain (37). Therefore we have $\begin{matrix} (44) & θ \int_{0}^{T} \int_{Q} u ϕ_{t t} (t) v (x) d x d t + \int_{0}^{T} \int_{Q} ξ ϕ \nabla v d x d t = \int_{0}^{T} \int_{Q} f ϕ v d x d t + \int_{0}^{T} \int_{Q} g ϕ v d x d \hat{W} \end{matrix}$ for all $v \in D (Q)$ and $ϕ \in D (0, T)$ . Using integration by parts, we obtain $\begin{matrix} (45) & θ d u_{t} (x, t) - div ξ (x, t) = f (x, t) + g (x, t) d \hat{W} on Q \times (0, T) \end{matrix}$ in the sense of distribution. We will have the homogenized SPDE, if we can show that $\begin{matrix} (46) & ξ (x, t) = A_{0} \nabla u . \end{matrix}$ In order to show (46), we use the argument of the elliptic Neumann problem, see [21]. Let $\begin{matrix} ω_{λ}^{ϵ} (x) = ϵ (\tilde{M} ω_{λ}) (\frac{x}{ϵ}), \end{matrix}$ where $\tilde{M}$ is defined in Lemma 2 and $ω_{λ}$ is defined by the cell problem (4). Using Lemma 1 and the definition of $ω_{λ}^{ϵ}$ , it is an easy task to show that $\begin{array}{l} (47) & ω_{λ}^{ϵ} ⇀ λ \cdot x weakly in H^{1} (Q), \\ (48) & ω_{λ}^{ϵ} \to λ \cdot x strongly in L^{2} (Q) . \end{array}$ We also introduce the following vector function $\begin{matrix} η_{λ}^{ϵ} = A_{ϵ} \nabla ω_{λ}^{ϵ} . \end{matrix}$ Since the vector $A_{ϵ} \nabla ω_{λ}^{ϵ}$ is Y-periodic, Lemma 1 gives $\begin{matrix} (49) & {\tilde{η}}_{λ}^{ϵ} ⇀ M (A \nabla ω_{λ}) = A_{0} λ weakly in {[L^{2} (Q)]}^{n} . \end{matrix}$ Also from (4), we have that $\begin{matrix} (50) & - div ({\tilde{η}}_{λ}^{ϵ}) = 0 in Q . \end{matrix}$ We test this equation with $ϕ (t) v (x) P_{ϵ} u^{ϵ} (x)$ for any $v \in D (Q)$ and $ϕ \in D (0, T)$ , integrate in t over $(0, T)$ and in x over Q to get $\begin{matrix} (51) & \int_{0}^{T} \int_{Q^{ϵ}} η_{λ}^{ϵ} \cdot \nabla P_{ϵ} u^{ϵ} ϕ (t) v (x) d x d t + \int_{0}^{T} \int_{Q} {\tilde{η}}_{λ}^{ϵ} \cdot \nabla v (x) ϕ (t) P_{ϵ} u^{ϵ} d x d t = 0 . \end{matrix}$ Using $ϕ (t) v (x) ω_{λ}^{ϵ} (x)$ as a test function in (35), one has $\begin{array}{l} \int_{0}^{T} \int_{Q} P_{ϵ} u^{ϵ} ϕ_{t t} (t) v (x) ω_{λ}^{ϵ} (x) \cdot χ_{(Q^{ϵ})} d x d t + \int_{0}^{T} \int_{Q} {\tilde{ξ}}^{ϵ} ϕ \nabla v ω_{λ}^{ϵ} d x d t + \int_{0}^{T} \int_{Q^{ϵ}} ξ^{ϵ} ϕ \nabla ω_{λ}^{ϵ} v d x d t \\ (52) & = \int_{0}^{T} \int_{Q} {\tilde{f}}^{ϵ} ϕ ω_{λ}^{ϵ} v d x d t + \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ ω_{λ}^{ϵ} v d x d W_{ϵ} . \end{array}$ Now subtracting (51) from (52), we obtain $\begin{array}{l} \int_{0}^{T} \int_{Q} P_{ϵ} u^{ϵ} ϕ_{t t} (t) v (x) \cdot χ_{(Q^{ϵ})} d x d t + \int_{0}^{T} \int_{Q} {\tilde{ξ}}^{ϵ} ϕ \nabla v ω_{λ}^{ϵ} d x d t \\ - \int_{0}^{T} \int_{Q} {\tilde{η}}_{λ}^{ϵ} \cdot \nabla v (x) ϕ (t) P_{ϵ} u^{ϵ} d x d t \\ (53) & = \int_{0}^{T} \int_{Q} {\tilde{f}}^{ϵ} ϕ ω_{λ}^{ϵ} v d x d t + \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ ω_{λ}^{ϵ} v d x d W_{ϵ} . \end{array}$ We pass to the limit in all the terms in (53) excluding the stochastic integral using convergences (10), (24), (25), (33), (48) and (49), to obtain $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ_{t t} (t) v (x) d x + \int_{0}^{T} \int_{Q} ξ ϕ \nabla v (λ \cdot x) d x d t \\ - \int_{0}^{T} \int_{Q} A_{0} λ \cdot \nabla v (x) ϕ (t) u d x d t \\ (54) & = \int_{0}^{T} \int_{Q} f ϕ (λ \cdot x) v d x d t + lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ ω_{λ}^{ϵ} v d x d W_{ϵ} . \end{array}$ Concerning the stochastic integral, we have $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v ω_{λ}^{ϵ} d x d W_{ϵ} \\ = lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v (ω_{λ}^{ϵ} - λ \cdot x) d x d W_{ϵ} + lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v (λ \cdot x) d x d W_{ϵ} . \end{array}$ Thanks to Burkhölder–Davis–Gundy’s inequality, (48) and the assumptions $g^{ϵ}$ we show that the first term in the right hand side, converges to zero $\tilde{P}$ -a.s. Indeed we have $\begin{array}{l} lim_{ϵ \to 0} \hat{E} sup_{0 \in [0, T]} | \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v (ω_{λ}^{ϵ} - λ \cdot x) d x d W_{ϵ} | \\ ⩽ C lim_{ϵ \to 0} \hat{E} {(\int_{0}^{T} {(\int_{Q} {\tilde{g}}^{ϵ} ϕ v (ω_{λ}^{ϵ} - λ \cdot x) d x)}^{2} d t)}^{\frac{1}{2}} \\ ⩽ C_{1} lim_{ϵ \to 0} \hat{E} {(\int_{0}^{T} {‖ {\tilde{g}}^{ϵ} ‖}_{L^{2} (Q)} {‖ ω_{λ}^{ϵ} - λ \cdot x ‖}_{L^{2} (Q)} d t)}^{\frac{1}{2}} \\ ⩽ C_{1} lim_{ϵ \to 0} {‖ ω_{λ}^{ϵ} - λ \cdot x ‖}_{L^{2} (Q)} \hat{E} {(\int_{0}^{T} {‖ {\tilde{g}}^{ϵ} ‖}_{L^{2} (Q)} d t)}^{\frac{1}{2}} . \end{array}$ Since $ω_{λ}^{ϵ}$ converges strongly to $λ \cdot x$ in $L^{2} (Q)$ we get $\begin{matrix} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v (ω_{λ}^{ϵ} - λ \cdot x) d x d W_{ϵ} = 0 \tilde{P} -a.s. \end{matrix}$ By the same steps as in the proof of (37), we show that $\begin{matrix} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ϕ v (λ \cdot x) d x d W_{ϵ} = \int_{0}^{T} \int_{Q} g ϕ v (λ \cdot x) d x d \hat{W} . \end{matrix}$ Therefore we obtain the corresponding limit equation to (53) $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ_{t t} (t) (λ \cdot x) v (x) d x d t + \int_{0}^{T} \int_{Q} ξ ϕ \nabla v (λ \cdot x) d x d t \\ - \int_{0}^{T} \int_{Q} A_{0} λ \cdot \nabla v (x) ϕ (t) u d x d t \\ = \int_{0}^{T} \int_{Q} f ϕ (λ \cdot x) v d x d t + \int_{0}^{T} \int_{Q} g ϕ (λ \cdot x) v d x d \hat{W} . \end{array}$ We rewrite it as $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ_{t t} (t) (λ \cdot x) v (x) d x d t + \int_{0}^{T} \int_{Q} ξ ϕ \nabla (λ \cdot x v) d x d t - \int_{0}^{T} \int_{Q} ξ \cdot λ ϕ v d x d t \\ - \int_{0}^{T} \int_{Q} A_{0} λ \cdot \nabla v (x) ϕ (t) u d x d t \\ (55) & = \int_{0}^{T} \int_{Q} f ϕ (λ \cdot x) v d x d t + \int_{0}^{T} \int_{Q} g ϕ (λ \cdot x) v d x d \hat{W} . \end{array}$ Replacing $ϕ v$ by $ϕ λ \cdot x v$ as test function in (44), we get $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ_{t t} (t) (λ \cdot x) v (x) d x d t + \int_{0}^{T} \int_{Q} ξ ϕ \nabla (λ \cdot x v) d x d t \\ (56) & = \int_{0}^{T} \int_{Q} f ϕ (λ \cdot x) v d x d t + \int_{0}^{T} \int_{Q} g ϕ (λ \cdot x) v d x d \hat{W} . \end{array}$ Comparing equations (55) and (56), we obtain $\begin{array}{rcl} \int_{0}^{T} \int_{Q} ξ \cdot λ ϕ v d x d t & = & - \int_{0}^{T} \int_{Q} A_{0} λ \cdot \nabla v (x) ϕ (t) u d x d t = \int_{0}^{T} \int_{Q} A_{0} λ \cdot \nabla u (x) ϕ (t) v d x d t \\ = & \int_{0}^{T} \int_{Q} A_{0} \nabla u (x) \cdot λ ϕ (t) v d x d t, \end{array}$ for any $v \in D (Q)$ and $ϕ \in D (0, T)$ . This implies that $\begin{matrix} ξ \cdot λ = A_{0} \nabla u (x) \cdot λ . \end{matrix}$ Since λ is an arbitrary in $R^{n}$ , we deduce (46).

It remains to check that the initial conditions $u (x, 0) = u_{0} / θ$ and $u_{t} (x, 0) = b_{1} / θ$ are satisfied. Notice that equation (34) is valid for $ψ \in C^{\infty} ([0, T])$ , such that $ψ (0) = 1$ , $ψ (T) = 0$ , $ψ_{t} (0) = 0$ and $ψ_{t} (T) = 0$ . Therefore we have $\begin{array}{l} - \int_{0}^{T} \int_{Q} {\tilde{u}}_{t}^{ϵ} ψ_{t} (t) v (x) d x d t + \int_{0}^{T} \int_{Q} {\tilde{ξ}}^{ϵ} ψ \nabla v d x d t \\ = \int_{0}^{T} \int_{Q} {\tilde{f}}^{ϵ} ψ v d x d t + \int_{0}^{T} \int_{Q} {\tilde{g}}^{ϵ} ψ v d x d W_{ϵ} + \int_{Q} {\tilde{u}}_{t}^{ϵ} (x, 0) v d x . \end{array}$ We pass to the limit in this equation and get $\begin{array}{l} - lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{u}}_{t}^{ϵ} ψ^{'} (t) v (x) d x d t + \int_{0}^{T} \int_{Q} ξ ψ \nabla v d x d t \\ = \int_{0}^{T} \int_{Q} f ψ v d x d t + \int_{0}^{T} \int_{Q} g ψ v d x d \hat{W} + \int_{Q} u_{1} v d x . \end{array}$ But $\begin{array}{rcl} - lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} {\tilde{u}}_{t}^{ϵ} ψ^{'} (t) v (x) d x d t & = & - lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} P_{ϵ} u_{t}^{ϵ} ψ^{'} (t) v (x) \cdot χ_{Q^{ϵ}} d x d t \\ = & lim_{ϵ \to 0} \int_{0}^{T} \int_{Q} P_{ϵ} u^{ϵ} ψ_{t t} (t) v (x) \cdot χ_{Q^{ϵ}} d x d t \\ = & θ \int_{0}^{T} \int_{Q} u ψ_{t t} (t) v (x) d x d t \\ = & θ \int_{0}^{T} \int_{Q} d u_{t} ψ (t) v (x) d x + θ \int_{Q} u_{t} (x, 0) v d x . \end{array}$ Thus $\begin{array}{l} θ \int_{0}^{T} \int_{Q} d u_{t} ψ (t) v (x) d x + θ \int_{Q} u_{t} (x, 0) v d x + \int_{0}^{T} \int_{Q} ξ ψ \nabla v d x d t \\ = \int_{0}^{T} \int_{Q} f ψ v d x d t + \int_{0}^{T} \int_{Q} g ψ v d x d \hat{W} + \int_{Q} u_{1} v d x . \end{array}$

Comparing this with (45) tested by $ψ (t) v (x)$ , we deduce that $\begin{matrix} u_{t} (x, 0) = \frac{u_{1}}{θ} . \end{matrix}$ Choosing $ψ \in C^{\infty} ([0, T])$ , such that $ψ (0) = 0$ , $ψ^{'} (0) = 1$ and $ψ (T) = ψ^{'} (T) = 0$ as test function in (34) we obtain $u (x, 0) = u_{0} / θ$ ; using similar arguments.

We note that the triple $(\tilde{W}, u, u_{t})$ is a probabilistic weak solution of (3) which is unique. Thus by the infinite dimensional version of Yamada–Watanabe’s theorem (see [40]), we get that $(W, u, u_{t})$ is unique strong solution of (3). Thus up to distribution (probability law) the whole sequence of solutions of (1) converges to the solution of problem (3). Thus the proof is complete. □

6. Convergence of the energy

This section is devoted to the second main result of the paper, concerning the convergence of the energy.

Let us introduce the energies associated with the problems (1) and (3), as follows $\begin{array}{l} (57) & E^{ϵ} (u^{ϵ}) (t) = \frac{1}{2} \hat{E} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + \frac{1}{2} \hat{E} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} (x, t) \nabla u^{ϵ} (x, t) d x, \\ (58) & E (u) (t) = \frac{θ^{2}}{2} \hat{E} {‖ u_{t} (t) ‖}_{L^{2} (Q)}^{2} + \frac{θ}{2} \hat{E} \int_{Q} A_{0} \nabla u (x, t) \nabla u (x, t) d x . \end{array}$ By Itô’s formula, it follows that $\begin{array}{l} \frac{1}{2} \hat{E} {‖ u_{t}^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + \frac{1}{2} \hat{E} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} (x, t) \nabla u^{ϵ} (x, t) d x \\ = \hat{E} [\frac{1}{2} {‖ {\tilde{u}}_{1}^{ϵ} ‖}_{L^{2} (Q)}^{2} + \frac{1}{2} \int_{Q} χ_{Q^{ϵ}} A_{ϵ} \nabla u_{0}^{ϵ} \nabla u_{0}^{ϵ} d x \\ + \int_{0}^{t} ({\tilde{f}}^{ϵ}, {\tilde{u}}_{t}^{ϵ}) d s + \int_{0}^{t} ({\tilde{g}}^{ϵ}, {\tilde{u}}_{t}^{ϵ}) d W_{ϵ} + \frac{1}{2} \int_{0}^{t} {‖ {\tilde{g}}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s] \end{array}$ and $\begin{array}{l} \frac{θ^{2}}{2} \hat{E} {‖ u_{t} (x, t) ‖}_{L^{2} (Q)}^{2} + \frac{θ}{2} \hat{E} \int_{Q} A_{0} \nabla u (x, t) \nabla u (x, t) d x \\ = \hat{E} [\frac{θ^{2}}{2} {‖ u_{t} (x, 0) ‖}_{L^{2} (Q)}^{2} + \frac{θ}{2} \int_{Q} A_{0} \nabla u (x, 0) \nabla u (x, 0) d x \\ + θ \int_{0}^{t} (f, u_{t}) d s + \frac{1}{2} \int_{0}^{t} ‖ g ‖_{L^{2} (Q)}^{2} d s + θ \int_{0}^{t} (g, u_{t}) d \hat{W}] . \end{array}$ Thus, we can write the energies as follows: $\begin{array}{l} E^{ϵ} (u^{ϵ}) (t) = \frac{1}{2} {‖ {\tilde{u}}_{1}^{ϵ} ‖}_{L^{2} (Q)}^{2} + \frac{1}{2} \int_{Q} χ_{Q^{ϵ}} A_{ϵ} \nabla u_{0}^{ϵ} \nabla u_{0}^{ϵ} d x \\ (59) & + \hat{E} \int_{0}^{t} ({\tilde{f}}^{ϵ}, {\tilde{u}}_{t}^{ϵ}) d s + \frac{1}{2} \int_{0}^{t} {‖ {\tilde{g}}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s, \\ E (u) (t) = \frac{θ^{2}}{2} {‖ u_{t} (x, 0) ‖}_{L^{2} (Q)}^{2} + \frac{θ}{2} \int_{Q} A_{0} \nabla u (x, 0) \nabla u (x, 0) d x \\ (60) & + θ \hat{E} \int_{0}^{t} (f, u_{t}) d s + \frac{1}{2} \int_{0}^{t} ‖ g ‖_{L^{2} (Q)}^{2} d s . \end{array}$ The vanishing of the expectation of the stochastic integrals is due to the fact that $({\tilde{g}}^{ϵ}, {\tilde{u}}_{t}^{ϵ})$ and $(g, u_{t})$ are square integrable in time (see assumption (A5) and estimate (13)).

We want to prove that the energy associated with the problem (1), uniformly converges to that of the corresponding homogenized problem (3). For this purpose we need to assume some stronger assumptions on the data. The main result of this section is

Theorem 3.

Let the assumptions of Theorem 2 be fulfilled. Let $\begin{array}{l} (61) & - div (A_{ϵ} \nabla u_{0}^{ϵ}) \to - \frac{1}{θ} div (A_{0} \nabla u_{0}) strongly in H^{- 1} (Q), \\ (62) & u_{0}^{ϵ} \to \frac{u_{0}}{θ} strongly in H_{0}^{1} (Q), \\ (63) & {\tilde{u}}_{1}^{ϵ} \to u_{1} strongly in L^{2} (Q), \\ (64) & {\tilde{f}}^{ϵ} \to f strongly in L^{2} (Q \times (0, T)), \\ (65) & {\tilde{g}}^{ϵ} \to g strongly in L^{2} (Q \times (0, T)) . \end{array}$ Then $\begin{matrix} E^{ϵ} (u^{ϵ}) (t) \to E (u) (t) in C ([0, T]), \end{matrix}$ where u is the solution of the homogenized problem ( 3 ) .

Proof.

Let us first show that $\begin{matrix} (66) & {\tilde{u}}_{t}^{ϵ} ⇀ θ u_{t} weakly in L^{2} (Q \times (0, T)) . \end{matrix}$ From estimate (13) there exists a function $V \in L^{2} (Q \times (0, T))$ such that ${\tilde{u}}_{t}^{ϵ} ⇀ V$ weakly in $L^{2} (Q \times (0, T))$ . Thus for any function $Ψ \in D ((0, T) \times Q)$ , we have $\begin{matrix} (67) & \int_{0}^{T} \int_{Q^{ϵ}} u_{t}^{ϵ} Ψ d x d t = \int_{0}^{T} \int_{Q} {\tilde{u}}_{t}^{ϵ} Ψ d x d t \to \int_{0}^{T} \int_{Q} V Ψ d x d t . \end{matrix}$

Using convergences (24) and (10) and integration by parts, we have $\begin{array}{l} \int_{0}^{T} \int_{Q^{ϵ}} u_{t}^{ϵ} Ψ d x d t = - \int_{0}^{T} \int_{Q} χ_{Q^{ϵ}} P_{ϵ} u^{ϵ} Ψ_{t} d x d t \\ (68) & \to - \int_{0}^{T} \int_{Q} θ u Ψ_{t} d x d t = \int_{0}^{T} \int_{Q} θ u_{t} Ψ d x d t . \end{array}$ Thus (67) and (6) give (66). Now let us show $\begin{matrix} E^{ϵ} (u^{ϵ}) (t) \to E (u) (t), \forall t \in [0, T] . \end{matrix}$ Using convergences (10), (61), (62), (63), (64), (65) and (66) we obtain the pointwise convergence $\begin{matrix} E^{ϵ} (u^{ϵ}) (t) \to E (u) (t) in [0, T], \forall t \in [0, T] . \end{matrix}$ Now we need to show that $(E^{ϵ} (u^{ϵ}) (t))$ , is uniformly bounded and equicontinuous on $[0, T]$ and hence Arzela–Ascoli’s theorem will imply the proof. Thanks to the assumptions on the data, we have $\begin{array}{rcl} | E^{ϵ} (u^{ϵ}) (t) | & ⩽ & \frac{1}{2} {‖ {\tilde{u}}_{1}^{ϵ} ‖}_{L^{2} (Q)}^{2} + \frac{C}{2} {‖ u_{0}^{ϵ} ‖}_{H_{0}^{1} (Q)} + \hat{E} | \int_{0}^{t} ({\tilde{f}}^{ϵ}, {\tilde{u}}_{t}^{ϵ}) d s | + \frac{1}{2} \int_{0}^{t} {‖ {\tilde{g}}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s \\ (69) & ⩽ & C + \hat{E} | \int_{0}^{t} ({\tilde{f}}^{ϵ}, {\tilde{u}}_{t}^{ϵ}) d s | . \end{array}$ The last term on the right-hand side of (69) can be estimated as we did in (15). Thus $(E^{ϵ} (u^{ϵ}) (t))$ , uniformly bounded on $[0, T]$ . For any $h > 0$ and $t \in [0, T]$ , we get $\begin{matrix} | E^{ϵ} (u^{ϵ}) (t + h) - E^{ϵ} (u^{ϵ}) (t) | ⩽ \hat{E} \int_{t}^{t + h} | ({\tilde{f}}^{ϵ}, {\tilde{u}}_{t}^{ϵ}) | d s + \frac{1}{2} \int_{t}^{t + h} {‖ {\tilde{g}}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s . \end{matrix}$ Thanks to Cauchy–Schwarz’s inequality, we have $\begin{array}{rcl} | E^{ϵ} (u^{ϵ}) (t + h) - E^{ϵ} (u^{ϵ}) (t) | & ⩽ & \hat{E} \int_{t}^{t + h} {‖ {\tilde{f}}^{ϵ} ‖}_{L^{2} (Q)} ‖ {\tilde{u}}_{t}^{ϵ}) ‖_{L^{2} (Q)} d s \\ + \frac{1}{2} {(\int_{t}^{t + h} d s)}^{\frac{1}{2}} {(\int_{t}^{t + h} {‖ {\tilde{g}}^{ϵ} (s) ‖}_{L^{2} (Q)}^{4} d s)}^{\frac{1}{2}} . \end{array}$ Applying once more Cauchy–Schwarz’s inequality to the first term in the right hand side of this inequality and using the assumption on $g^{ϵ}$ , we have $\begin{array}{rcl} | E^{ϵ} (u^{ϵ}) (t + h) - E^{ϵ} (u^{ϵ}) (t) | & ⩽ & \tilde{E} {(\int_{t}^{t + h} {‖ {\tilde{f}}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s)}^{\frac{1}{2}} {(\int_{t}^{t + h} {‖ {\tilde{u}}_{t}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s)}^{\frac{1}{2}} + C (T) h^{\frac{1}{2}} \\ ⩽ & h^{\frac{1}{2}} \hat{E} sup {‖ {\tilde{u}}_{t}^{ϵ} (t) ‖}_{L^{2} (Q)} {(\int_{t}^{t + h} {‖ {\tilde{f}}^{ϵ} ‖}_{L^{2} (Q)}^{2} d s)}^{\frac{1}{2}} + C (T) h^{\frac{1}{2}} . \end{array}$ Thanks to assumption (A5) and estimate (13), we obtain $\begin{matrix} | E^{ϵ} (u^{ϵ}) (t + h) - E^{ϵ} (u^{ϵ}) (t) | ⩽ C h^{\frac{1}{2}} . \end{matrix}$ This implies the equicontinuity of the sequence ${E^{ϵ} (u^{ϵ}) (t)}_{ϵ}$ . The proof is complete. □

As a closing remark, we note that our results can readily be extended to the case of infinite dimensional Wiener processes taking values in appropriate Hilbert spaces; for instance cylindrical Wiener processes.

Footnotes

Acknowledgements

The authors thank the anonymous reviewers for their useful comments that improved the paper. They gratefully acknowledge the support of the National Research Foundation of South Africa and the University of Pretoria.

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