Depression-induced changes in directed functional brain networks: A source-space resting-state EEG study

1.   Introduction and setting of the problem

Homogenization theory has become an important tool in the investigation of processes taking place in highly heterogenous media ranging from soil to the most advanced aircraft the construction of which uses composite materials. So far, the problems solved by means of homogenization have mainly involved deterministic partial differential equations (PDEs) and the homogenization of PDEs with randomly oscillating coefficients; the great wealth of results obtained over several decades on problems of diverse classes and methodologies can be found for instance in [9,6,40,41,23,34,22,49,31,17,4,32,36,46,50,33], for the deterministic case and [13,14,18,20,24,37,19,47,48]. for the random case. Fundamental methods were subsequently developed such as the method of asymptotic expansions ([9], [6], [40], [41]), the two scale-convergence ([4], [32]), Tartar method of oscillating test functions and H-convergence ([49]), the asymptotic method for non periodically perforated domains ([23], [46]), G-convergence ([36]) and $\Gamma$-convergence developed by De Giorgi and his students; relevant extensions of some of these methods, including their random counterparts, have also emerged in recent times. One rapidly developping important branch of homogenization is that of numerical homogenization; see [1], [2].

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 [3,8,10,11,21,43,44]. We also note the closely related work [3,25,15,16] dealing with stochastic homogenization for SPDEs with small parameters. The list of references is of course not exhaustive, but a representation of the main trends in the field.

The homogenization of hyperbolic SPDEs was initiated in [27], [28,29], [30] where the authors studied the homogenization of Dirichlet problems for linear hyperbolic equation with rapidly oscillating coefficients using the method of the two-scale convergence pioneered by Nguetseng in [32] and developed by Allaire in [4] and [5]; they also dealt with the linear Neumann problem by means of Tartar's method and obtained the corresponding corrector results within these settings; [30] deals with a semilinear hyperbolic SPDE by Tartar's method.

In the present work, following the two-scale convergence method, we investigate the homogenization of a non-linear hyperbolic equation with nonlinear damping, where the intensity of the noise is also nonlinear and is assumed to satisfy Lipschitz's condition. Our investigation relies on crucial compactness results of analytic (Aubin-Lions-Simon's type) and probabilistic (Prokhorov and Skorokhod fundamental theorems) nature. It should be noted that these methods extend readily to the case when Lipschitz condition on the intensity of the noise is replaced by a mere continuity. In contrast to the linear and the semilinear cases considered in previous papers, the type of nonlinear damping and nonlinear noise in the present paper leads to new challenges in obtaining uniform a priori estimates as well as in the passage to the limit. It should be noted that the process of damping in mechanical systems is a crucial stabilizing factor when the system is subjected to very extreme tasks; mathematically this translates in some regularizing effects on the solution of the governing equations.

We are concerned with the homogenization of the initial boundary value problem with oscillating data, referred to throughout the paper as problem $(P_{\epsilon })$:

where $u_{t}^{\epsilon }$ denotes the partial derivative $\partial u^{\epsilon }/\partial t$ of $u^{\epsilon }$ with respect to $t$, $\epsilon >0$ is a sufficiently small parameter which ultimately tends to zero, $T>0$, $Q$ is an open and bounded (at least Lipschitz) subset of $\mathbb{R}^{n}$, $W = (W(t))\left( t\in \lbrack 0,T]\right)$ an $m$-dimensional standard Wiener process defined on a given filtered complete probability space $(\Omega ,\mathcal{F},\mathbb{P},\left( \mathcal{F}_{t}\right) _{0\leq t\leq T})$; $\mathbb{E}$ denotes the corresponding mathematical expectation. For a physical motivation, we refer to [27,28], where the authors discussed real life processes of vibrational nature subjected to random fluctuations; for instance the nonlinear term $B(t,u_{t}^{\epsilon })$ stands for damping effects, the term $f(t,x,x/\varepsilon ,\nabla u^{\epsilon })$ is the oscillating regular part of the force acting on the system and depending linearly on $\nabla u^{\epsilon }$, while the term $g(t,x,x/\varepsilon ,u_{t}^{\epsilon })dW$ represents the oscillating random component of the force; it depends on $u_{t}^{\varepsilon }$. More precise assumptions on the data will be provided shortly.

Few words about the difference between the current work and previous works by the authors on homogenization of SPDEs. Compared to [27,28,29,30], the structure of problem ($P_{\varepsilon }$) is dominated by nonlinear terms such as the damping $B(t,u_{t}^{\epsilon })$, leading to $L^{p}\left( Q\right)$-like norms whose combination with the predominently $L^{2}$-like norms coming from the other terms requires special care, both in the derivation of the a priori estimates, as well as in the passage to the limit. Though, two-scale convergence method is also used in the paper [27], the model there is essentially linear. The works [43,44] deal with stochastic parabolic equations in domains with fine grained boundaries, where no conditions of periodicity hold and the methodology implemented there is a stochastic counterpart of Kruslov-Marchenko's [23] and Skrypnik's [46] homogenization theories based on potiential theory; for instance the homogenized problems in [43,44] involve an additional term of capacitary type. The investigation of a hyperbolic counterpart of these works has still not been undertaken and is somehow overdue. Finally, compared with the above mentioned works, the current paper involves a simpler proof of the convergence of the stochastic nonlinear term (its integral) thanks to a blending of two-scale convergence with a regularizing argument and a result on convergence of stochastic integrals due to Rozovskii [39,Theorem 4,P 63].

We now introduce some functions spaces needed in the sequel.

For $2\leq p\leq \infty ,$ we define the Sobolev space

where the derivatives exist in the weak sense, and $L^{p}(Q)$ is the usual Lebesgue space. For $p = 2,\,\,W^{1,2}(Q)$ is denoted by $H^{1}(Q)$. By $W_{0}^{1,p}(Q)$ we denote the space of elements $\psi \in W^{1,p}(Q)$ such that $\psi |_{\partial Q} = 0$ with the $W^{1,p}(Q)$-norm. By $(\phi ,\psi )$ we denote the inner product in $L^{2}(Q)$ and by $\langle .,.\rangle$ we denote the duality pairing between $W_{0}^{1,p}(Q)$ and $W^{-1,p^{\prime }}(Q)$ ($\frac{1}{p}+\frac{1}{p^{\prime }}$ = 1). We also consider the following spaces$,\ H(Q) = \{u\in H^{1}(Q)|\mathcal{M}_{Q}(u) = 0\}$ where $\mathcal{M}_{Q}(u)$ is the mean value of $u$ over $Q$, $C_{{\text{per}}}^{\infty }(Y)$ the subspace of $C^{\infty }(\mathbb{R}^{n})$ of $Y$- periodic functions where $Y = (0,l_{1})\times ...\times (0,l_{n})$. Let $H_{{\text{per}}}^{1}(Y)$ be the closure of $C_{{\text{per}}}^{\infty }(Y)$ in the $H^{1}$-norm, and $H_{{\text{per}}}(Y)$ the subspace of $H_{{\text{per}}}^{1}(Y)$ with zero mean on $Y$.

For a Banach space $X$, and $1\leq p\leq \infty$, we denote by $L^{p}(0,T;X)$ the space of measurable functions $\phi :t\in \lbrack 0,T]\longrightarrow \phi (t)\in X$ and $p$-integrable with the norm

When $p = \infty$, $L^{\infty }(0,T;X)$ is the space of all essentially bounded functions on the closed interval $[0,T]$ with values in $X$ equipped with the norm

For $1\leq q,p<\infty$, $L^{q}\left(\Omega ,\mathcal{F},\mathbb{P},L^{p}(0,T;X)\right)$ ($(\Omega ,\mathcal{F},\mathbb{P})$ is a probability space with a filtration $\left\{ \mathcal{F}_{t}\right\} _{t\in \left[ 0,T\right] }$) consists of all random functions $\phi :(\omega ,t)\in \Omega \times \lbrack 0,T]\longrightarrow \phi (\omega ,t,\cdot)\in X$ such that $\phi (\omega ,t,x)$ is progressively measurable with respect to $(\omega ,t)$. We endow this space with the norm

When $p = \infty$, the norm in the space $L^{q}(\Omega ,\mathcal{F},\mathbb{P},L^{\infty }(0,T;X))$ is given by

It is well known that under the above norms, $L^{q}(\Omega ,\mathcal{F}, \mathbb{P},L^{p}(0,T;X))$ is a Banach space.

We now impose the following hypotheses on the data.

$({\rm{A1}})$ $A_{\epsilon}(x) = A(\frac{x}{\epsilon}) = (a_{i,j}(\frac{x}{\epsilon }))_{1\leq i,j\leq n}$ is an $n\times n$ symmetric matrix, the components $a_{i,j}$, are $Y-$periodic and there exists a constant $\alpha >0$ such that

$({\rm{A2}})$ $B(t,\cdot): u \in W_{0}^{1,p}(Q)\longrightarrow W^{-1,p^{\prime }}(Q)$ such that

(ⅰ) $B(t,\cdot)$ is a hemicontinuous operator, i.e. $\lambda \longrightarrow \langle B(t,u+\lambda v), w\rangle$ is a continuous operator for all $t \in (0,T)$ and all $u,v,w \in W_{0}^{1,p}(Q)$;

(ⅱ) There exists a constant $\gamma >0$ such that $\langle B(t,u),u\rangle \geq \gamma \|u\|^{p}_{W_{0}^{1,p}(Q)}$ for $a.e. \,\,t \in (0,T)$ and all $u \in W_{0}^{1,p}(Q);$

(ⅲ) There exists a positive constant $\beta$ such that $\| B(t,u)\|_{W^{-1,p^{\prime }}(Q)} \leq \beta \|u\|^{p-1}_{W_{0}^{1,p}(Q)}$ for $a.e. \,\,t \in (0,T)$ and all $u \in W_{0}^{1,p}(Q);$

(ⅳ) $\langle B(t,u)-B(t,v),u-v\rangle \geq 0,$ for $a.e. \,\, t \in (0,T)$ and all $u,v \in W_{0}^{1,p}(Q);$

(ⅴ) The map $t\longrightarrow B(t,u)$ is Lebesgue measurable in $(0,T)$ with values in $W^{-1,p^{\prime }}(Q)$ for all $u\in W_{0}^{1,p}(Q)$.

(A3) We assume that $f(t,x,y,w)$ is measurable with respect to$\ \left( x,y\right)$ for any $\left( t,w\right) \in (0,T)\times \mathbb{R}^{n}$, continuous with respect to $\left( t,w\right)$ for almost every $\left( x,y\right) \in Q\times Y$, and $Y$-periodic with respect to $y$. Also there exists an $\mathbb{R}^{n}$-valued function $F = \left( F_{i}\left( t,x,y\right) \right) _{1\leq i\leq n}$ such that $f(t,x,y,w) = F(t,x,y)\cdot w$. Furthermore,

for any $\left( t,w,\varepsilon \right) \in (0,T)\times L^{2}\left( Q\right) \times \left( 0,\infty \right)$, with the constant $C$ independent of $\varepsilon$ and $t$. A sufficient requirement for this condition to hold is that $F_{i}\left( t,\cdot \right) \in L^{\infty }\left( Q\times Y\right)$ for any $t\in (0,T)$.

(A4) $a^{\epsilon }(x)\in H_{0}^{1}\left( Q\right)$, $b^{\epsilon }(x)\in L^{2}\left( Q\right)$, for any $\epsilon >0$.

(A5) $g\left( t,x,y,\phi \right)$ is an $m$-dimensional vector-function whose components $g_{j}\left( t,x,\right.$ $\left.y,\phi \right)$ satisfy the following conditions:

● $g_{j}\left( t,x,y,\phi \right)$ is $Y$-periodic with respect to $y$, measurable with respect to $x$ and $y$ for almost all $t\in \left( 0,T\right)$ and for all $\phi \in L^{2}\left( Q\right)$,

●  $g_{j}\left( t,x,y,\phi \right)$ is continuous with respect to $\phi$ for almost all $\left( t,x,y\right) \in \left( 0,T\right) \times Q\times Y$, and there exists a positive constant $C$ independent of $t$, $x$ and $y$, such that

and

● $g_{j}\left( t,x,y,\cdot \right)$ satisfies Lipschitz's condition

with the constant $L$ independent of $t$, $x$ and $y.$

If $\left\vert \left\vert g_{j}\left( t,x,y,0\right) \right\vert \right\vert _{L^{2}\left( Q\times Y\right) }<\infty$ for any $i = 1,...,m$ and any $t\in \left( 0,T\right)$, the condition (1) is redundant since it follows from the Lipschitz condition (2).

From now on we use the following oscillating functions

We now introduce our notion of solution; namely the strong probabilistic one.

Definition 1.1. We define the strong probabilistic solution of the problem $(P_{\epsilon })$ on the prescribed filtered probability space $\left( \Omega ,\mathcal{F},\mathbb{P},\left\{ \mathcal{F}_{t}\right\} _{t\in \left[ 0,T \right] }\right)$ as a process

satisfying the following conditions:

(1) $u^{\epsilon },u_{t}^{\epsilon }$ are $\mathcal{F}_{t}-\text{measurable}$,

(2)

(3) $\forall t\in \left[ 0,T\right]$, $u^{\epsilon }(t,.)$ satisfies the identity

The problem of existence and uniqueness of a strong probabilistic solution of $(P_{\epsilon})$ was dealt with in [38]. The relevant result is

Theorem 1.2. Suppose that the assumptions $(A1)-(A5)$ hold and let $p\geq 2$. Then for fixed $\epsilon >0$, the problem $\left( P_{\epsilon }\right)$ has a unique strong probabilistic solution $u^{\epsilon }$ in the sense of Definition 1.1.

Our goal is to show that as $\epsilon$ tends to zero the sequence of solutions $\left(u^{\epsilon } \right)$ converge in a suitable sense to a solution $u$ of the following SPDE

where $A_{0}$ is a constant elliptic matrix defined by

$\chi (y)\in H_{\text{per}}(Y)$ is the unique solution of the following boundary value problem:

for any $\lambda \in \mathbb{R}^{n}$ and $Y = (0,l_{1})\times ...\times (0,l_{n})$,

$a$ and $b$ are suitable limits of the oscillating initial conditions $a^{\epsilon }$ and $b^{\epsilon }$, respectively, $\tilde{W}$ is an $m$-dimensional Wiener process

2.   A priori estimates

Here and in the sequel, $C$ will denote a constant independent of $\epsilon$. In the following lemma, we obtain the energy estimates associated to problem $\left( P_{\epsilon}\right)$.

Lemma 2.1. Under the assumptions $(A1)$-$(A5)$, the solution $u^{\epsilon }$ of $(P_{\epsilon })$ satisfies the following estimates:

and

Proof. The following arguments are used modulo appropriate stopping times. It$\hat{o}$'s formula and the symmetry of $A$ give

Integrating over $(0,t),\,\,t\leq T$, we get

Using the assumptions $(A1)$, $(A2)(ii)$, $(A5)$ and taking the supremum over $t\in \lbrack 0,T]$ and the expectation on both sides of the resulting relation, we have

where

Using assumptions (A3), thanks to Cauchy-Schwarz's and Young's inequalities, we have

where $\varrho >0$. Thanks to Burkholder-Davis-Gundy's inequality, followed by Cauchy-Schwarz's inequality, the last term in 5 can be estimated as

Again using Young's inequality and the assumptions $(A5)$, we get

for $\varrho >0$. Combining the estimates 6, 7, 5 and assumption $(A5)$ and taking $\varrho$ sufficiently small, we infer that

Using Gronwall's inequality, we have

and subsequently

The proof is complete.

The following lemma will be of great importance in proving the tightness of probability measures generated by the solution of problem $(P_{\epsilon})$ and its time derivative.

Lemma 2.2. Let the conditions of Lemma 2.1 be satisfied and let $p\geq 2$. Then there exists a constant $C>0$ such that

for any $\epsilon >0$ and $0<\delta <1$.

Proof.. $\ \ $We consider that $div\left( A_{\epsilon }\nabla \phi \right)$ has been restricted to the space $W^{-1,p^{\prime }}(Q)$ and that the restriction induces a bounded mapping from $W_{0}^{1,p}(Q)$ to $W^{-1,p^{\prime }}(Q)$.

Assume that $u_{t}^{\epsilon }$ is extended by zero outside the interval $[0,T]$ and that $\theta >0$. We write

Then

Firstly, thanks to assumption $(A1)$, we have

where we have used the fact that $p^{\prime }\leq 2$.

Secondly, we use assumption $(A2)(iii),$ estimate 4 and H$\ddot{o}$lder's inequality to get

Thirdly,

where we have used assumption (A3).

Using 10, 11 and 12 in 9 raised to the power $p^{\prime }$, for fixed $\delta >0$, we get

We now estimate the term involving the stochastic integral.

We use the embedding

to get the estimate

Thanks to Fubini's theorem and H$\ddot{o}$lder's inequality, we have

where we have used Burkholder-Davis-Gundy's inequality. We now invoke assumption $(A5)$ and estimate 3 to deduce from 14 and 15 that

For the first term in the right-hand side of 13, we use Fubini's theorem, H$\ddot{o}$lder's inequality and estimate 3 to get

The second term on the right hand side of 13 is estimated using 4 and we get

Combining 13, 16, 17 and 18, and taking into account the fact that the similar estimates hold for $\theta <0$, we conclude that

This completes the proof.

3.   Tightness property of probability measures

The following Lemmas are needed in the proof of the tightness and the study of the properties of the probability measures generated by the sequence $(W,u^{\epsilon },u_{t}^{\epsilon })$.

We have from [45]

Lemma 3.1. 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 $1\leq p,q\leq \infty$, and $0<s\leq 1$ let $E$ be a set bounded in $L^{q}(0,T;B_{0})\cap N^{s,p}(0,T;B_{1})$, where

Then $E$ is relatively compact in $L^{p}(0,T;B)$

The following two lemmas are collected from [12]. Let $\mathcal{S}$ be a separable Banach space and consider its Borel $\sigma$-field to be $\mathcal{B}(\mathcal{S})$. We have

Lemma 3.2. (Prokhorov) A sequence of probability measures $\left( \Pi _{n}\right) _{n\in \mathbb{N}}$ on $(\mathcal{S}, \mathcal{B}(\mathcal{S} ))$ is tight if and only if it is relatively compact.

Lemma 3.3. (Skorokhod) Suppose that the probability measures $\left( \mu _{n}\right) _{n\in \mathbb{N}}$ on $(\mathcal{S}, \mathcal{B}(\mathcal{S}))$ weakly converge to a probability measure $\mu$. Then there exist random variables $\xi ,\xi _{1},\dots \xi _{n},\dots$, defined on a common probability space $(\Omega ,\mathcal{F},P)$, such that $\mathcal{L} (\xi _{n}) = \mu _{n}$ and $\mathcal{L}(\xi ) = \mu$ and

the symbol $\mathcal{L}\left( \cdot \right)$ stands for the law of $\cdot$.

Let us introduce the space $Z = Z_{1}\times Z_{2}$, where

and

We endow $Z$ with the norm

Lemma 3.4. The above constructed space $Z$ is a compact subset of $L^{2}(0,T;L^{2}(Q))\times L^{2}(0,T;L^{2}(Q)).$

Proof. Lemma 3.1 together with suitable arguments due to Bensoussan [7] give the compactness of $Z_{1}$ and $Z_{2}$ in $L^{2}(0,T;L^{2}(Q))$.

We now consider the space $\mathcal{X} = C(0,T;\mathbb{R}^{m})\times L^{2}(0,T;L^2(Q)) \times L^{2}(0,T;L^{2}(Q))$ and $\mathcal{B}(\mathcal{X})$ the $\sigma -$algebra of its Borel sets. Let $\Psi _{\epsilon }$ be the $(\mathcal{X},\mathcal{B}(\mathcal{X}))$-valued measurable map defined on $(\Omega ,\mathcal{F},\mathbb{P})$ by

Define on $(\mathcal{X},\mathcal{B}(\mathcal{X}))$ the family of probability measures $\left(\Pi _{\epsilon } \right)$ by

Lemma 3.5. The family of probability measures $\{\Pi _{\epsilon }:\epsilon >0\}$ is tight in $(\mathcal{X},\mathcal{B}(\mathcal{X})).$

Proof. We carry out the proof following a long the lines of the proof of [27,lemma 7]. For $\delta >0$, we look for compact subsets

such that

This is equivalent to

which can be proved if we can show that

Let $L_{\delta }$ be a positive constant and $n\in \mathbb{N}$. Then we deal with the set

Using Arzela's theorem and the fact that $W_{\delta }$ is closed in $C(0,T;\mathbb{R}^{m})$, we ensure the compactness of $W_{\delta }$ in $C(0,T;\mathbb{R}^{m})$. From Markov's inequality

where $\eta$ is a nonnegative random variable and $k$ a positive real number, we have

But

where $(2k-1)!! = 1\cdot 3\cdot \cdot \cdot (2k-1)$ and $W_{i}$ denotes the i-th component of $W$.

For $k = 4$, we have

Choosing $(L_{\delta })^{4} = \dfrac{(\sum n^{-2})^{-1}}{3CT^{2}\delta }$, we have

Now, let $K_{\delta },\,\,M_{\delta }$ be positive constants. We define

Lemma 3.4 shows that $D_{\delta }$ is compact subset of $L^{2}(0,T;L^{2}(Q)),$ for any $\delta >0$. It is therefore easy to see that

Markov's inequality 19 gives

for $K_{\delta } = M_{\delta } = \frac{6C}{\delta }.$

Similarly, we let $\mu _{n},\,\,\nu _{m}$ be sequences of positive real numbers such that $\mu _{n},\,\,\nu _{n}\rightarrow 0$ as $n\rightarrow \infty$, $\sum_{n}\frac{\mu _{n}^{p^{\prime }/p}}{\nu _{n}}<\infty$ (for the series to converge we can choose $\nu _{n} = 1/n^{2}$, $\mu _{n} = 1/n^{\alpha }$, with $\alpha p^{\prime }/p>4$) and define

Owing to Proposition 3.1 in [7], $B_{\delta }$ is a compact subset of $L^{2}(0,T;L^{2}(Q))$ for any $\delta >0$. We have

Again thanks to 19, we obtain

for $K_{\delta }^{\prime } = \frac{9C}{\delta }$, $L_{\delta }^{\prime } = \frac{ 9C}{\delta }$ and $M_{\delta }^{\prime } = \dfrac{9C\sum \frac{\mu _{n}^{p^{\prime }/p}}{\nu _{n}}}{\delta }$. This completes the proof.

From Lemmas 3.2 and 3.5, there exist a subsequence $\{\Pi _{\epsilon _{j}}\}$ and a measure $\Pi$ such that

weakly. From lemma 3.3, there exist a probability space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ and $\mathcal{X}$-valued random variables $(W_{\epsilon _{j}},u^{\epsilon _{j}},u_{t}^{\epsilon _{j}}),\,\,(\tilde{W},u,u_{t})$ such that the probability law of $(W_{\epsilon _{j}},u^{\epsilon _{j}}, u_{t}^{\epsilon _{j}})$ is $\Pi _{\epsilon _{j}}$ and that of $(\tilde{W},u,u_{t})$ is $\Pi$. Furthermore, we have

Let us define the filtration

We show that $\tilde{W}(t)$ is an $\tilde{\mathcal{F}_{t}}$-wiener process following [7] and [42]. Arguing as in [42], we get that $(W_{\epsilon _{j}},u^{\epsilon _{j}},u_{t}^{\epsilon _{j}})$ satisfies $\tilde{\mathbb{P}}-a.s.$ the problem $\left( P_{\epsilon _{j}}\right)$ in the sense of distributions.

4.   Two-scale convergence

In this section, we state some key facts about the powerful two-scale convergence invented by Nguetseng [32].

Definition 4.1. A sequence $\{v^{\epsilon}\}$ in $L^{p}(0,T;L^p(Q))(1<p<\infty)$ is said to be two-scale converge to $v = v(t,x,y),\,\ v\in L^{p}(0,T;L^p(Q\times Y)),$ as $\epsilon \rightarrow 0$ if for any $\psi = \psi(t,x,y)\in L^p((0,T)\times Q ;C_{\text{per}}^{\infty }(Y))$, one has

where $\psi^{\epsilon}(t,x) = \psi(t,x, \frac{x}{\epsilon})$. We denote this by $\{v^{\epsilon}\}\rightarrow v \,\,\,\text{2-s in}\,\,\, L^{p}(0,T;L^p(Q))$.

The following result deals with some of the properties of the test functions which we are considering; it is a modification of Lemma 9.1 from [17,p.174].

Lemma 4.2. (i) Let $\psi \in L^{p}((0,T)\times Q;C_{per}(Y)),\,\,1<p<\infty$. Then $\psi (\cdot,\cdot,\frac{\cdot}{\epsilon })\in L^{p}(0,T;L^{p}(Q))$ with

and

Furthermore if $\psi \in L^{2}((0,T)\times Q;C_{per}(Y))$, then

(ii) If $\psi (t,x,y) = \psi _{1}(t,x)\psi _{2}(y),\,\ \psi _{1}\in L^{p}(0,T;L^{s}(Q)),\,\psi _{2}\in L_{per}^{r}(Y),\,\,1\leq s,r<\infty$ are such that

then $\psi (\cdot,\cdot,\frac{\cdot}{\epsilon })\in L^{p}(0,T;L^{p}(Q))$ and

The following theorems are of great importance in obtaining the homogenization result; for their proofs, we refer to [4], [17] and [26].

Theorem 4.3. Let $\{u^{\epsilon}\}$ be a sequence of functions in $L^{2}\left(0,T;L^{2}(Q)\right)$ such that

Then up to a subsequence $u^{\epsilon}$ is two-scale convergent in $L^{2}\left(0,T;L^{2}(Q)\right)$.

Theorem 4.4. Let $\{u^{\epsilon }\}$ be a sequence satisfying the assumptions of Theorem 4.3. Furthermore, let $\{u^{\epsilon }\}\subset L^{2}\left( 0,T;H_{0}^{1}(Q)\right)$ be such that

Then, up to a subsequence, there exists a couple of functions $(u,u_{1})$ with $u\in L^{2}(0,T;H_{0}^{1}(Q))$ and $u_{1}\in L^{2}((0,T)\times Q;H_{{\text{per}}}(Y))$ such that

The following lemma is crucial in obtaining the convergence of the stochastic integral in the next section

Lemma 4.5. The oscillating data given in (A5) satisfies the following convergence

Proof. Test with $\psi \left( t,x,\frac{x}{\varepsilon }\right) ,$ where $\psi \left( t,x,y\right) \in L^{2}\left( \left( 0,T\right) \times Q,C_{per}^{\infty }\left( Y\right) \right) ,$ as follows:

where

Then

thanks to the Lipschitz condition on $g\left( t,x,\cdot \right)$. Now due to the strong convergence 20 of $u_{t}^{\varepsilon }-u_{t}$ to zero in $L^{2}\left( \left( 0,T\right) \times Q\right)$, $\mathbb{\tilde{P}}$-a.s., we get that $I_{1}^{\varepsilon }\rightarrow 0$, $\tilde{\mathbb{P}}-a.s.$

Now we can apply 2-scale convergence for the limit of $I_{2}^{\varepsilon }$ and indeed

Therefore

and this implies the result.

Remark 1. From the assumption (A5), 28 and 23, we have the following strong convergence

5.   The homogenization result

We will now study the asymptotic behaviour of the problem $(P_{\epsilon_j})$, when $\epsilon_j \rightarrow 0$.

Theorem 5.1. Suppose that the assumptions on the data are satisfied. Let

Then there exist a probability space $\left( \tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}},\left( \tilde{\mathcal{F}}_{t}\right) _{0\leq t\leq T}\right)$ and random variables $(u^{\epsilon _{j}},u_{t}^{\epsilon _{j}},W_{\epsilon _{j}})$ and $(u,u_{t},\tilde{W})$ such that the convergences 20 and 26 hold. Furthermore $(u,u_{t},\tilde{W})$ satisfies the homogenized problem $(P)$.

Proof. From estimates 3 and 4 and assumption $(A2)(iii)$, we have the following convergences

Now let us identify the limit in 35. By arguing as in [38,Lemma 2.6,p. 51], we get

Having this in hand, let $v\in L^{p}(0,T;W_{0}^{1,p}(Q))$ and define

From the monotonicity assumption $(A2)(iv)$, we have $\chi _{\epsilon _{j}}\geq 0$. Now using 34, 35 and 36 to pass to the limit in 37, we get

For $\lambda >0$ and $w\in L^{p}(0,T;W_{0}^{1,p}(Q))$, we can chose $v(t) = u_{t}(t)-\lambda w(t)$. Hence

Using the hemicontinuty assumption $(A2)(i)$, we have

Now, from assumptions $(A2)(ii)$ and $(A2)(v)$, we use the Lebesgue dominated convergence theorem to pass to the limit in 38. This implies

But the inequality 39 is true for all $w(t)\in L^{p}(0,T;W_{0}^{1,p}(Q)))$. Therefore

Testing problem $(P_{\epsilon _{j}})$ by the function $\Phi \in C^{\infty}_{\text{c}}((0,T)\times Q)$ and integrating the first term in the right-hand side by parts, we have

Using estimate 3, the convergence 20 and Theorems 4.3 and 4.4, we show the two-scale convergence

Let $\Phi ^{\epsilon _{j}}(t,x) = \phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}}),$ where $\phi \in C^{\infty}_{\text{c}}((0,T)\times Q)$ and $\phi _{1}\in C^{\infty}_{\text{c}}((0, T)\times Q;C_{\text{per}}^{\infty }(Y))$. Then we can still consider $\Phi ^{\epsilon _{j}}$ as test function in 40. Thus

Let us deal with these terms one by one, when $\epsilon _{j}\rightarrow 0$. Thanks to estimate 22 and convergence 33, we have

The second term can be written as follows,

Since $A_{\epsilon _{j}}\in L^{\infty }(Y)$ and $\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)\in L_{\text{per}}^{2}(Y;C(Q\times (0,T)))$, we regard $A_{\epsilon _{j}}[\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})]$ as a test function in the two-scale limit of the gradient in the first term in 42. Therefore

Thanks to H$\ddot{o}$lder inequality, 22 and the fact that $A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}$ is bounded in $L^{\infty }(0,T; L^{2}(Q)$, we have

Again, thanks to estimate 22 and convergence 35, we have

Let us write

where we have used the assumption (A3). It is easy to see that the second term in 43, converges to zero. For the first term in the right-hand side of 43, we readily have

Concerning the stochastic integral, we have

We deal with the term involving $\phi \left( t,x\right)$. We have

In view of the unbounded variation of $W_{t}^{\varepsilon }-\tilde{W}_{t}$, the convergence of the first term on the right-hand side of 46 needs appropriate care, in order to take advantage of the $\mathbb{\tilde{P}}-$ a.s. uniform convergence of $W_{t}^{\varepsilon }$ to $\tilde{W}_{t}$ in $C\left( \left[ 0,T\right] \right)$. We adopt the idea of regularization of $g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right)$ with respect to the variable $t$, by means of the following sequence

where $\rho$ is a standard mollifier.

We have that $g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right)$ is a differentiable function of $t$ and satisfies the relations

and for any $\varepsilon >0$

We split the first term in the right-hand side of 46 as

Owing to 49, and Burkholder-Davis-Gundy's inequality, it readily follows that the second term in 50 is bounded by a function $\sigma _{1}\left( \lambda \right)$ which converges to zero as $\lambda \rightarrow 0$. In the first term in the same relation, we take advantage of the differentiability of $g_{\lambda }^{\varepsilon }$ with respect to $t$ in order to integrate by parts. As a result we get

Thanks to the conditions on $\phi$ and $g$ and the uniform convergence obtained from the application of Skorokhod's compactness result, namely

we get that both terms on the right-hand side of 51 are bounded by the product $\sigma _{2}\left( \lambda \right) \eta _{1}\left( \varepsilon \right)$ such that $\sigma _{2}\left( \lambda \right)$ is finite and $\eta _{1}\left( \varepsilon \right)$ vanishes as $\varepsilon$ tends to zero. Summarizing these facts, we deduce from 50 that

Thus, we infer from 46 that

Taking the limit in 54 as $\varepsilon \rightarrow 0$, we get

but the left-hand side of this relation being independent of $\lambda$, we can pass to the limit on both sides as $\lambda \rightarrow 0$, to arrive at the crucial statement

Owing to 27; that is

we can call upon the convergence theorem for stochastic integrals due to Rozovskii [39,Theorem 4,p. 63] to claim that

Hence, we deduce from 55 that,

For the second term in 45, thanks to Burkholder-Davis-Gundy's inequality, the assumptions on $g^{\epsilon _{j}}$ and 22, we have

Combining the above convergences, we obtain

Choosing in the first stage $\phi = 0$ and after $\phi _{1} = 0$, the problem 57 is equivalent to the following system of integral equations

and

By standard arguments (see [17]), equation 58 has a unique solution given by

where $\chi (y),$ known as the first order corrector, is the unique solution to the following equation:

As for the uniqueness of the solution of 59, we prove it as follows. Using 60 in 59, one obtains that 59 is the weak formulation of the equation

where

and

But the initial boundary value problem corresponding to 62 has a unique solution by [38]. It remains to show that $u(x,0) = a(x)$ and $u_{t}(x,0) = b(x)$. Notice that equation 40 is valid for $\Phi ^{\epsilon _{j}}(t,x) = \phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{ \epsilon _{j}})$ where $\phi \in C^{\infty }_{\text{c}}((0,T)\times Q)$ and $\phi _{1}\in C^{\infty }_{\text{c}}((0,T)\times Q;C_{\text{per}}^{\infty }(Y))$, such that $\phi (0,x) = v(x)$ and $\phi (T,x) = 0$. Thus, we have

where we pass to the limit, to get

The integration by parts, in the first term gives

In view of equation 57, we deduce that

for any $v\in C^{\infty }_{\text{c}}(Q)$. This implies that $u_{t}(x,0) = b(x)$. For the other initial condition, we consider $\Phi ^{\epsilon _{j}}(t,x) = \phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})$ as a test function in 40, where $\phi \in C^{\infty }_{\text{c}}((0,T)\times Q)$ and $\phi _{1}\in C^{\infty }_{\text{c}}((0,T)\times Q;C_{\text{per}}^{\infty }(Y))$, such that $\phi (0,x) = 0,\phi _{t}(0,x) = v(x)$ and $\phi (T,x) = 0 = \phi _{t}(T,x)$. Integration by parts in the first term of 40, gives

Passing to the limit in this equation, we obtain

We integrate by parts again to obtain

Using the same argument as before, we show that $u(x,0) = a(x)$. We note the triple $\left( \tilde{W},u,u_{t}\right)$ is a probabilistic weak solution of $(P)$ which is unique. Thus by the infinite dimensional version of Yamada-Watanabe's theorem (see [35]), we get that $\left( W,u,u_{t}\right)$ is the unique strong solution of $(P)$. Thus up to distribution (probability law) the whole sequence of solutions of $(P_{\epsilon })$ converges to the solution of problem $(P)$. Thus the proof of Theorem 5.1 is complete.

6.   Convergence of the energy

Let us introduce the energies associated with the problems ($P_{\epsilon _{j}}$) and ($P$), as follows:

But from It$\hat{o}$'s formula, we have

Thus

The vanishing of the expectation of the stochastic integrals is due to the fact that $(g^{\epsilon }(u_{t}^{\epsilon }),\tilde{u}_{t}^{\epsilon })$ and $(g(u),u_{t})$ are square integrable in time. We want to prove that the energy associated with the problem ($P_{\epsilon _{j}}$), uniformly converges to that of the corresponding homogenized problem ($P$). For this purpose we need to assume some stronger assumptions on the initial data. We have the following result

Theorem 6.1. Assume that the assumptions of Theorem 5.1 are fulfilled and

Then

where $u$ is the solution of the homogenized problem.

Proof. Thanks to the convergences 20, 44, 29, 66 and 67, we show that

Now we need to show that $\left( \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)\right)$, is uniformly bounded and equicontinuous on $[0,T]$ and hence Arzela-Ascoli's theorem concludes the proof. We have

Thanks to the assumptions on the data $(A3),\,(A4)$ and $(A5)$, the a priori estimates 3 and 4, we show that

For any $h>0$ and $t\in \lbrack 0,T]$, we get

Again assumptions (A3), (A5) and Cauchy-Schwarz's inequality, give

This implies the equicontinuity of the sequence $\{\mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)\}_{\epsilon _{j}}$, and therefore the proof is complete.

7.   The corrector result

In this section, we establish a corrector result stated in the following

Theorem 7.1. Let the assumptions of Theorems 5.1 and 6.1 be fulfilled. Assume that $\nabla _{y}\chi (y)\in \lbrack L^{r}(Y)]^{n}$ and $\nabla u\in L^{2}(0,T;[L^{s}(Y)]^{n})$ with $1\leq r,s<\infty$ such that

Then

Proof. It is easy to see that

Then convergence 20 gives

Thus 68 holds. Similarly we show that

It remains to show that

We have

Again

Now from the ellipticity assumption on the matrix $A$, we have

Let us pass to the limit in this inequality. We start with

From the convergence of the energies in Theorem 6.1 and using 63 and 60, we have

Next, using the two-scale convergence of $\nabla u^{\epsilon _{j}}$, with the test function $A\left( y\right) \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right)$, we obtain

Now, let us write

For $u_{1}$ given by 60, we have

Now using $(ii)$ of Lemma 4.2, for $p = 2$, we obtain

Combining 71, 72 and 73 with 70, we deduce that

Thus 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.

Acknowledgments

The authors express their deepest gratitude to the reviewers for their careful reading of the paper and their insightful comments which have improved the paper. Part of this work was conducted when the first author visited the African Institute for Mathematical Sciences (AIMS), South Africa, he is grateful to the generous hospitality of AIMS.