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Research article Special Issues

Long-term economic impacts of coastal floods in Europe: a probabilistic analysis

  • Received: 10 July 2023 Revised: 12 September 2023 Accepted: 25 September 2023 Published: 20 October 2023
  • In this article we quantify the long-term economic impacts of coastal flooding in Europe. In particular, how the direct coastal damages generate long-term economic losses that propagate and compound throughout the century. A set of probabilistic projections of inundation-related direct damages (to residential buildings, firms' physical assets and agriculture production) is used as an exogenous shock to a dynamic stochastic economic model. The article considers explicitly the uncertainty related to the economic agents' behaviour and other relevant macroeconomic assumptions, i.e., how would consumers finance the repairing of their homes, how long does it take for a firm to reconstruct, whether firms decide to build-back-better after the inundation and possibly compensate the losses with a productivity gain. Our findings indicate that the long-term impacts of coastal floods could be larger than the direct damages. Under a high emission scenario (RCP8.5) the EU27 plus UK could lose every year between 0.25% and 0.91% of output by 2100, twice as much as the direct damages. The welfare losses present a strong regional variation, with the South (Bulgaria, Greece, Italy, Malta, Portugal and Spain), and United Kingdom (UK) plus Ireland regions showing the highest damages and a significant part of the population that could suffer significant welfare losses by the end of the century.

    Citation: Ignazio Mongelli, Michalis Vousdoukas, Luc Feyen, Antonio Soria, Juan-Carlos Ciscar. Long-term economic impacts of coastal floods in Europe: a probabilistic analysis[J]. AIMS Environmental Science, 2023, 10(5): 593-608. doi: 10.3934/environsci.2023033

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  • In this article we quantify the long-term economic impacts of coastal flooding in Europe. In particular, how the direct coastal damages generate long-term economic losses that propagate and compound throughout the century. A set of probabilistic projections of inundation-related direct damages (to residential buildings, firms' physical assets and agriculture production) is used as an exogenous shock to a dynamic stochastic economic model. The article considers explicitly the uncertainty related to the economic agents' behaviour and other relevant macroeconomic assumptions, i.e., how would consumers finance the repairing of their homes, how long does it take for a firm to reconstruct, whether firms decide to build-back-better after the inundation and possibly compensate the losses with a productivity gain. Our findings indicate that the long-term impacts of coastal floods could be larger than the direct damages. Under a high emission scenario (RCP8.5) the EU27 plus UK could lose every year between 0.25% and 0.91% of output by 2100, twice as much as the direct damages. The welfare losses present a strong regional variation, with the South (Bulgaria, Greece, Italy, Malta, Portugal and Spain), and United Kingdom (UK) plus Ireland regions showing the highest damages and a significant part of the population that could suffer significant welfare losses by the end of the century.



    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 Γ-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 (Pϵ):

    duϵtdiv(Aϵ(x)uϵ)dt+B(t,uϵt)dt=f(t,x,x/ε,uϵ)dt+g(t,x,x/ε,uϵt)dW in (0,T)×Quϵ=0 on(0,T)×Q,uϵ(0,x)=aϵ(x), uϵt(0,x)=bϵ(x) in Q,

    where uϵt denotes the partial derivative uϵ/t of uϵ with respect to t, ϵ>0 is a sufficiently small parameter which ultimately tends to zero, T>0, Q is an open and bounded (at least Lipschitz) subset of Rn, W=(W(t))(t[0,T]) an m-dimensional standard Wiener process defined on a given filtered complete probability space (Ω,F,P,(Ft)0tT); 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,uϵt) stands for damping effects, the term f(t,x,x/ε,uϵ) is the oscillating regular part of the force acting on the system and depending linearly on uϵ, while the term g(t,x,x/ε,uϵt)dW represents the oscillating random component of the force; it depends on uεt. 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 (Pε) is dominated by nonlinear terms such as the damping B(t,uϵt), leading to Lp(Q)-like norms whose combination with the predominently L2-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 2p, we define the Sobolev space

    W1,p(Q)={ϕ:ϕLp(Q),ϕxjLp(Q),j=1,...,n},

    where the derivatives exist in the weak sense, and Lp(Q) is the usual Lebesgue space. For p=2,W1,2(Q) is denoted by H1(Q). By W1,p0(Q) we denote the space of elements ψW1,p(Q) such that ψ|Q=0 with the W1,p(Q)-norm. By (ϕ,ψ) we denote the inner product in L2(Q) and by .,. we denote the duality pairing between W1,p0(Q) and W1,p(Q) (1p+1p = 1). We also consider the following spaces, H(Q)={uH1(Q)|MQ(u)=0} where MQ(u) is the mean value of u over Q, Cper(Y) the subspace of C(Rn) of Y- periodic functions where Y=(0,l1)×...×(0,ln). Let H1per(Y) be the closure of Cper(Y) in the H1-norm, and Hper(Y) the subspace of H1per(Y) with zero mean on Y.

    For a Banach space X, and 1p, we denote by Lp(0,T;X) the space of measurable functions ϕ:t[0,T]ϕ(t)X and p-integrable with the norm

    ||ϕ||Lp(0,T;X)=(T0||ϕ||pXdt)1p,0p<.

    When p=, L(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

    ϕL(0,T;X)=esssup[0,T]ϕX<.

    For 1q,p<, Lq(Ω,F,P,Lp(0,T;X)) ((Ω,F,P) is a probability space with a filtration {Ft}t[0,T]) consists of all random functions ϕ:(ω,t)Ω×[0,T]ϕ(ω,t,)X such that ϕ(ω,t,x) is progressively measurable with respect to (ω,t). We endow this space with the norm

    ||ϕ||Lq(Ω,F,P;Lp(0,T;X))=(E||ϕ||qLp(0,T;X))1/q.

    When p=, the norm in the space Lq(Ω,F,P,L(0,T;X)) is given by

    ||ϕ||Lq(Ω,F,P;L(0,T;X))=(E||ϕ||qL(0,T;X))1/q.

    It is well known that under the above norms, Lq(Ω,F,P,Lp(0,T;X)) is a Banach space.

    We now impose the following hypotheses on the data.

    (A1) Aϵ(x)=A(xϵ)=(ai,j(xϵ))1i,jn is an n×n symmetric matrix, the components ai,j, are Yperiodic and there exists a constant α>0 such that

    ni,j=1ai,jξiξjαni=1ξ2i for, ξRn,ai,jL(Rn),i,j=1,,n.

    (A2) B(t,):uW1,p0(Q)W1,p(Q) such that

    (ⅰ) B(t,) is a hemicontinuous operator, i.e. λB(t,u+λv),w is a continuous operator for all t(0,T) and all u,v,wW1,p0(Q);

    (ⅱ) There exists a constant γ>0 such that B(t,u),uγupW1,p0(Q) for a.e.t(0,T) and all uW1,p0(Q);

    (ⅲ) There exists a positive constant β such that B(t,u)W1,p(Q)βup1W1,p0(Q) for a.e.t(0,T) and all uW1,p0(Q);

    (ⅳ) B(t,u)B(t,v),uv0, for a.e.t(0,T) and all u,vW1,p0(Q);

    (ⅴ) The map tB(t,u) is Lebesgue measurable in (0,T) with values in W1,p(Q) for all uW1,p0(Q).

    (A3) We assume that f(t,x,y,w) is measurable with respect to (x,y) for any (t,w)(0,T)×Rn, continuous with respect to (t,w) for almost every (x,y)Q×Y, and Y-periodic with respect to y. Also there exists an Rn-valued function F=(Fi(t,x,y))1in such that f(t,x,y,w)=F(t,x,y)w. Furthermore,

    ||f(t,x,xε,w)||L2(Q)C||w||L2(Q), 

    for any (t,w,ε)(0,T)×L2(Q)×(0,), with the constant C independent of ε and t. A sufficient requirement for this condition to hold is that Fi(t,)L(Q×Y) for any t(0,T).

    (A4) aϵ(x)H10(Q), bϵ(x)L2(Q), for any ϵ>0.

    (A5) g(t,x,y,ϕ) is an m-dimensional vector-function whose components gj(t,x, y,ϕ) satisfy the following conditions:

    ● gj(t,x,y,ϕ) is Y-periodic with respect to y, measurable with respect to x and y for almost all t(0,T) and for all ϕL2(Q),

    ●  gj(t,x,y,ϕ) is continuous with respect to ϕ for almost all (t,x,y)(0,T)×Q×Y, and there exists a positive constant C independent of t, x and y, such that

    ||gj(t,x,y,ϕ)||L2(Q)C(1+||ϕ||L2(Q)), (1)

    and

    ● gj(t,x,y,) satisfies Lipschitz's condition

    |gj(t,x,y,s1)gj(t,x,y,s2)|L|s1s2|, (2)

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

    If ||gj(t,x,y,0)||L2(Q×Y)< for any i=1,...,m and any t(0,T), the condition (1) is redundant since it follows from the Lipschitz condition (2).

    From now on we use the following oscillating functions

    fϵ(t,x,w)=f(t,x,xε,w), gεj(t,x,ϕ)=gi(t,x,xε,ϕ).

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

    Definition 1.1. We define the strong probabilistic solution of the problem (Pϵ) on the prescribed filtered probability space (Ω,F,P,{Ft}t[0,T]) as a process

    uϵ:Ω×[0,T]H10(Q),

    satisfying the following conditions:

    (1) uϵ,uϵt are Ftmeasurable,

    (2)

    uϵL2(Ω,F,P;C(0,T;H10(Q)))uϵtL2(Ω,F,P;C(0,T;L2(Q)))Lp(Ω,F,P;Lp(0,T;W1,p0(Q))),

    (3) t[0,T], uϵ(t,.) satisfies the identity

    (uϵt(t,.),ϕ)(uϵt(0,.),ϕ)+t0(Aϵuϵ(s,.),ϕ)ds+t0Bϵ(s,uϵt),ϕds=t0(fϵ(s,.,uϵ),ϕ)ds+(t0gϵ(s,.,uϵt)dW(s),ϕ),ϕCc(Q).

    The problem of existence and uniqueness of a strong probabilistic solution of (Pϵ) was dealt with in [38]. The relevant result is

    Theorem 1.2. Suppose that the assumptions (A1)(A5) hold and let p2. Then for fixed ϵ>0, the problem (Pϵ) has a unique strong probabilistic solution uϵ in the sense of Definition 1.1.

    Our goal is to show that as ϵ tends to zero the sequence of solutions (uϵ) converge in a suitable sense to a solution u of the following SPDE

    (P){dutdivA0udt+B(t,ut)dt=˜f(t,x,u)dt+˜g(t,x,ut)d˜W in Q×(0,T),u=0 onQ×(0,T),u(0,x)=a(x)H10(Q),ut(0,x)=b(x)L2(Q),

    where A0 is a constant elliptic matrix defined by

    A0=1|Y|Y(A(y)A(y)χ(y))dy,

    χ(y)Hper(Y) is the unique solution of the following boundary value problem:

    {divy(A(y)yχ(y))=yA(y)inYχisYperiodic,

    for any λRn and Y=(0,l1)×...×(0,ln),

    ˜f(t,x,u)=1|Y|YF(t,x,y)[xu(t,x)+yu1(t,x,y)]dy,˜g(t,x,ut)=1|Y|Yg(t,x,y,ut)dy,

    a and b are suitable limits of the oscillating initial conditions aϵ and bϵ, respectively, ˜W is an m-dimensional Wiener process

    Here and in the sequel, C will denote a constant independent of ϵ. In the following lemma, we obtain the energy estimates associated to problem (Pϵ).

    Lemma 2.1. Under the assumptions (A1)-(A5), the solution uϵ of (Pϵ) satisfies the following estimates:

    Esup0tTuϵ(t)2H10(Q)C,Esup0tTuϵt(t)2L2(Q)C, (3)

    and

    ET0uϵt(t)pW1,p0(Q)C. (4)

    Proof. The following arguments are used modulo appropriate stopping times. Itˆo's formula and the symmetry of A give

    d[uϵt2L2(Q)+(Aϵuϵ,uϵ)]+2B(t,uϵt),uϵt)dt=2(fϵ(t,x,uϵ)),uϵt)dt+2(gϵ(t,x,uϵt),uϵt)dW+mj=0gϵj(t,x,uϵt)2L2(Q)dt.

    Integrating over (0,t),tT, we get

    uϵt(t)2L2(Q)+(Aϵuϵ(t),uϵ(t))+2t0B(s,uϵt(s)),uϵt(s))ds=uϵ12L2(Q)+(Aϵuϵ0,uϵ0)+2t0(fϵ(s,x,uϵ),uϵt)ds+2t0(gϵ(s,x,uϵt),uϵt)dW+mj=0t0gϵj(s,x,uϵt)2L2(Q)ds.

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

    E[sup0tTuϵt(t)2L2(Q)+sup0tTuϵ(t)2H10(Q)+2γt0uϵt(s)pW1,p0(Q)ds]C[C1+t0uϵt(t)2L2(Q)dt+2t0|(fϵ(s,x,uϵ),uϵt)|ds+2sup0st|s0(gϵ(σ,x,uϵt),uϵt)dW|], (5)

    where

    C1=C(T)+uϵ12L2(Q)+uϵ02H10(Q).

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

    ET0|(fϵ(s,x,uϵ),uϵt)|dtET0uϵL2(Q)uϵtL2(Q)dtEsup0tTuϵt(t)L2(Q)T0uϵL2(Q)dtϱEsup0tTuϵt(t)2L2(Q)+C(ϱ)T(T0uϵ2L2(Q)dt), (6)

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

    Esup0st|s0(gϵ(σ,x,uϵt(σ)),uϵt(σ))dW(σ)|CE(t0(gϵ(σ,x,uϵt(σ)),uϵt(σ))2dσ)12CE(sup0stuϵt(s)L2(Q)t0gϵ(σ,x,uϵt(σ))2L2(Q)dσ)12.

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

    2Esup0st|s0(gϵ(σ,x,uϵt(σ)),uϵt(σ))dW|ϱEsup0stuϵt(s)2L2(Q)+C(ϱ)T0gϵ(σ,uϵt(σ))2L2(Q)dσϱEsup0stuϵt(s)2L2(Q)+C(ϱ)(T)+C(ϱ)T0uϵt(σ)2L2(Q)dσ, (7)

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

    Esup0tTuϵt(t)2L2(Q)+Esup0tTuϵ(t)2H10(Q)+CEt0uϵt(s)pW1,p0(Q)dsC(T,C1,C2)+CEt0[uϵt(s)2L2(Q)+uϵ(s)2H10(Q)]dt, (8)

    Using Gronwall's inequality, we have

    E[sup0tTuϵt(t)2L2(Q)+sup0tTuϵ(t)2H10(Q)]C,

    and subsequently

    Et0uϵt(s)pW1,p0(Q)dsC.

    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 (Pϵ) and its time derivative.

    Lemma 2.2. Let the conditions of Lemma 2.1 be satisfied and let p2. Then there exists a constant C>0 such that

    Esup|θ|δT0uϵt(t+θ)uϵt(t)pW1,p(Q)dtCδp/p,

    for any ϵ>0 and 0<δ<1.

    Proof..   We consider that div(Aϵϕ) has been restricted to the space W1,p(Q) and that the restriction induces a bounded mapping from W1,p0(Q) to W1,p(Q).

    Assume that uϵt is extended by zero outside the interval [0,T] and that θ>0. We write

    uϵt(t+θ)uϵt(t)=t+θtdiv(Aϵuϵ)dst+θtB(s,uϵt(s))ds+t+θtfϵ(s,x,uϵ)ds+t+θtgϵ(s,uϵt(s))dW(s).

    Then

    uϵt(t+θ)uϵt(t)W1,p(Q)t+θtdiv(Aϵuϵ)dsW1,p(Q)+t+θtB(s,uϵt(s))dsW1,p(Q)+t+θtfϵ(s,x,uϵ)dsW1,p(Q)+t+θtgϵ(s,uϵt(s))dW(s)W1,p(Q). (9)

    Firstly, thanks to assumption (A1), we have

    t+θtdiv(Aϵuϵ)dsW1,p(Q)supϕW1,p0(Q):ϕ=1|t+θtdiv(Aϵuϵ)ds,ϕW1,p(Q),W1,p0(Q)|=supϕW1,p0(Q):ϕ=1Qt+θtAϵuϵϕdxdsCsupϕW1,p0(Q):ϕ=1t+θtuϵLp(Q)ϕLp(Q)dsCt+θtuϵL2(Q)dsCθ1/2(t+θtuϵ2L2(Q)ds)1/2, (10)

    where we have used the fact that p2.

    Secondly, we use assumption (A2)(iii), estimate 4 and H¨older's inequality to get

    t+θtB(s,uϵt(s))dsW1,p(Q)supϕW1,p0(Q):ϕ=1|t+θtB(s,uϵt(s))ds,ϕW1,p(Q),W1,p0(Q)|supϕW1,p0(Q):ϕ=1t+θtB(s,uϵt(s))W1,p(Q)ϕW1,p0(Q)dsCθ1/p(t+θtuϵtpW1,p0(Q)ds)1/p. (11)

    Thirdly,

    t+θtfϵ(s,x,uϵ)dsW1,p(Q)t+θtfϵ(s,x,uϵ)dsL2(Q)Ct+θtuϵL2(Q)θ1/2(t+θtuϵ2L2(Q)ds)1/2, (12)

    where we have used assumption (A3).

    Using 10, 11 and 12 in 9 raised to the power p, for fixed δ>0, we get

    Esup0<θδT0uϵt(t+θ)uϵt(t)pW1,p(Q)dtCEsup0<θδθp/2T0(t+θtuϵ2L2(Q)ds)p/2dt+CEsup0<θδθp/pT0t+θtuϵtpW1,p0(Q)dsdt+Esup0<θδT0t+θtgϵ(s,uϵt(s)dW(s)pW1,p(Q)dt. (13)

    We now estimate the term involving the stochastic integral.

    We use the embedding

    W1,p0(Q)L2(Q)W1,p(Q)

    to get the estimate

    Esup0<θδT0||t+θtgϵ(s,uϵt(s)dW(s)||pW1,pdtEsup0<θδT0||t+θtgϵ(s,uϵt(s)dW(s)||pL2(Q)dt. (14)

    Thanks to Fubini's theorem and H¨older's inequality, we have

    ET0sup0<θδ||t+θtgϵ(s,uϵt(s)dW(s)||pL2(Q)dtT0(QEsup0<θδ(t+θtgϵ(s,uϵt(s))dW(s))2dx)p/2dtT0(Et+δt||gϵ(s,uϵt(s)||2L2(Q)ds)p/2dt, (15)

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

    Esup0<θδT0||t+θtgϵ(s,uϵt(s)dW(s)||pW1,pdtT0[Et+δt(1+||uϵt(s)||2L2(Q))ds]p/2dtCTδp/2. (16)

    For the first term in the right-hand side of 13, we use Fubini's theorem, H¨older's inequality and estimate 3 to get

    Esup0<θδθp/2T0(t+θtuϵ2L2(Q)ds)p/2δp/2T0(Et+δtuϵ2L2(Q)ds)p/2CTδp. (17)

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

    Esup0<θδθp/pT0t+θtuϵtpW1,p0(Q)dsdtδp/pT0ET0uϵtpW1,p0(Q)dsdtCδp/p. (18)

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

    Esup|θ|δT0uϵt(t+θ)uϵt(t)pW1,p(Q)dtCδp/p.

    This completes the proof.

    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,uϵ,uϵt).

    We have from [45]

    Lemma 3.1. Let B0,  B and B1 be some Banach spaces such that B0B B1 and the injection B0B is compact. For any 1p,q, and 0<s1 let E be a set bounded in Lq(0,T;B0)Ns,p(0,T;B1), where

    Ns,p(0,T;B1)={vLp(0,T;B1):suph>0hsv(t+h)v(t)Lp(0,Tθ,B1)<}.

    Then E is relatively compact in Lp(0,T;B)

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

    Lemma 3.2. (Prokhorov) A sequence of probability measures (Πn)nN on (S,B(S)) is tight if and only if it is relatively compact.

    Lemma 3.3. (Skorokhod) Suppose that the probability measures (μn)nN on (S,B(S)) weakly converge to a probability measure μ. Then there exist random variables ξ,ξ1,ξn,, defined on a common probability space (Ω,F,P), such that L(ξn)=μn and L(ξ)=μ and

    limnξn=ξ,Pa.s.;

    the symbol L() stands for the law of .

    Let us introduce the space Z=Z1×Z2, where

    Z1={ϕ:sup0tTϕ(t)2H10(Q)C1,sup0tTϕ(t)2L2(Q)C1},

    and

    Z2={ψ:sup0tTψ(t)2L2(Q)C3,T0ψ(t)pW1,p0(Q)dtC4,T0ψ(t+θ)ψ(t)pW1,p(Q)C5θ1/p}.

    We endow Z with the norm

    (ϕ,ψ)Z=ϕZ1+ψZ2=sup0tTϕ(t)L2(Q)+sup0tTϕH10(Q)+sup0tTψ(t)2L2(Q)+(T0ψ(t)pW1,p0(Q)dt)1p+(supθ>01θ1/pT0ψ(t+θ)ψ(t)pW1,p(Q))1p.

    Lemma 3.4. The above constructed space Z is a compact subset of L2(0,T;L2(Q))×L2(0,T;L2(Q)).

    Proof. Lemma 3.1 together with suitable arguments due to Bensoussan [7] give the compactness of Z1 and Z2 in L2(0,T;L2(Q)).

    We now consider the space X=C(0,T;Rm)×L2(0,T;L2(Q))×L2(0,T;L2(Q)) and B(X) the σalgebra of its Borel sets. Let Ψϵ be the (X,B(X))-valued measurable map defined on (Ω,F,P) by

    Ψϵ:ω(W(ω),uϵ(ω),uϵt(ω)).

    Define on (X,B(X)) the family of probability measures (Πϵ) by

    Πϵ(A)=P(Ψ1ϵ(A))for allAB(X).

    Lemma 3.5. The family of probability measures {Πϵ:ϵ>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

    \begin{equation*} W_{\delta }\subset C(0,T;\mathbb{R}^{m}),\,\,\,D_{\delta }\subset L^{2}(0,T;L^{2}(Q)),\,\,\,E_{\delta }\subset L^{2}(0,T;L^{2}(Q)) \end{equation*}

    such that

    \begin{equation*} \Pi _{\epsilon }\big\{(W,u^{\epsilon },u_{t}^{\epsilon })\in W_{\delta }\times D_{\delta }\times E_{\delta }\big\}\geq 1-\delta . \end{equation*}

    This is equivalent to

    \begin{equation*} \mathbb{P}\big\{\omega :W(\cdot,\omega )\in W_{\delta },u^{\epsilon }(\cdot,\omega )\in D_{\delta },u_{t}^{\epsilon })(\cdot,\omega )\in E_{\delta }\big\}\geq 1-\delta , \end{equation*}

    which can be proved if we can show that

    \begin{equation*} \mathbb{P}\big\{\omega :W(\cdot,\omega )\notin W_{\delta }\}\leq \delta ,\,\,\,\mathbb{P}\{u^{\epsilon }(\cdot,\omega )\notin D_{\delta }\}\leq \delta ,\,\,\,\mathbb{P}\{u_{t}^{\epsilon })(\cdot,\omega ).\notin E_{\delta }\}\leq \delta . \end{equation*}

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

    \begin{equation*} W_{\delta } = \{W(\cdot)\in C(0,T;\mathbb{R}^{m}):\sup\limits_{t,s\in \lbrack 0,T]}n|W(s)-W(t)|\leq L_{\delta }:|s-t|\leq Tn^{-1}\}. \end{equation*}

    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

    \begin{equation} \mathbb{P}(\omega :\eta (\omega )\geq \alpha )\leq \frac{\mathbb{E}|\eta (\omega )|^{k}}{\alpha ^{k}}, \end{equation} (19)

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

    \begin{align*} \mathbb{P}\big\{\omega & :W(\cdot,\omega )\notin W_{\delta }\}\leq \mathbb{P} \bigg[\bigcup\limits_{n = 1}^{\infty }\bigg(\sup\limits_{t,s\in \lbrack 0,T]}|W(s)-W(t)|\geq \frac{L_{\delta }}{n}:|s-t|\leq Tn^{-1}\bigg)\bigg] \\ & \leq \sum\limits_{n = 0}^{\infty }\mathbb{P}\bigg[\bigcup\limits_{j = 1}^{n^{6}}\bigg( \sup\limits_{Tjn^{-6}\leq t\leq T(j+1)n^{-6}}|W(s)-W(t)|\geq \frac{L_{\delta }}{n} \bigg)\bigg]. \end{align*}

    But

    \mathbb{E}\left( W_{i}(t)-W_{i}(s)\right) ^{2k} = (2k-1)!!(t-s)^{k},\,\,\,k = 1,2,3,\dots ,

    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

    \begin{align*} &\mathbb{P}\left\{ \omega :W(.,\omega )\notin W_{\delta }\right\}\\ & \leq \sum\limits_{n = 0}^{\infty }\sum\limits_{j = 1}^{n^{6}}\left( \dfrac{n}{L_{\delta }}\right) ^{4}\mathbb{E}\left( \sup\limits_{Tjn^{-6}\leq t\leq T(j+1)n^{-6}}\left\vert W(t)-W(jTn^{-6})\right\vert ^{4}\right) \\ & \leq C\sum\limits_{n = 0}^{\infty }\sum\limits_{j = 1}^{n^{6}}\left( \dfrac{n}{L_{\delta }} \right) ^{4}\left( Tn^{-6}\right) ^{2} = \dfrac{CT^{2}}{(L_{\delta })^{4}} \sum\limits_{n = 0}^{\infty }n^{-2}. \end{align*}

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

    \begin{equation*} \mathbb{P}\left\{ \omega :W(.,\omega )\notin W_{\delta }\right\} \leq \frac{\delta }{3}. \end{equation*}

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

    \begin{equation*} D_{\delta } = \big\{z:\sup\limits_{0\leq t\leq T}\Vert z(t)\Vert _{H_{0}^{1}(Q)}^{2}\leq K_{\delta },\,\,\,\sup\limits_{0\leq t\leq T}\Vert z^{\prime }(t)\Vert _{L^{2}(Q)}^{2}\leq M_{\delta }\big\}. \end{equation*}

    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

    \begin{equation*} \mathbb{P}\{u^{\epsilon }\notin D_{\delta }\}\leq \mathbb{P}\big\{\sup\limits_{0\leq t\leq T}\Vert u^{\epsilon }(t)\Vert _{H_{0}^{1}(Q)}^{2}\geq K_{\delta }\}+\mathbb{P}\big\{\sup\limits_{0\leq t\leq T}\Vert u_{t}^{\epsilon }(t)\Vert _{L^{2}(Q)}^{2}\geq M_{\delta }\big\}. \end{equation*}

    Markov's inequality 19 gives

    \begin{equation*} \mathbb{P}\{u^{\epsilon }\notin D_{\delta }\}\leq \frac{1}{K_{\delta }} \mathbb{E}\sup\limits_{0\leq t\leq T}\Vert u^{\epsilon }(t)\Vert _{H_{0}^{1}(Q)}^{2}+\frac{1}{M_{\delta }}\mathbb{E}\sup\limits_{0\leq t\leq T}\Vert u_{t}^{\epsilon }(t)\Vert _{L^{2}(Q)}^{2}\leq \frac{C}{K_{\delta }}+\frac{C}{ M_{\delta }} = \frac{\delta }{3}. \end{equation*}

    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

    \begin{align*} B_{\delta }& = \left\{ v:\sup\limits_{0\leq t\leq T}\Vert v(t)\Vert _{L^{2}(Q)}^{2}\leq K_{\delta }^{\prime },\,\,\,\int_{0}^{T}\Vert v(t)\Vert _{W_{0}^{1,p}(Q)}^{p}dt\leq L_{\delta }^{\prime },\right. \\ & \left. \sup\limits_{\theta \leq \mu _{n}}\int_{0}^{T}\Vert v(t+\theta )-v(t)\Vert _{W^{-1,p^{\prime }}(Q)}^{p^{\prime }}dt\leq \nu _{n}M_{\delta }^{\prime }\right\} . \end{align*}

    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

    \begin{align*} \mathbb{P}\{u_{t}^{\epsilon }& \notin B_{\delta }\}\leq \mathbb{P}\left\{ \sup\limits_{0\leq t\leq T}\Vert u_{t}^{\epsilon }(t)\Vert _{L^{2}(Q)}^{2}\geq K_{\delta }^{\prime }\right\} +\mathbb{P}\left\{ \int_{0}^{T}\Vert u_{t}^{\epsilon }(t)\Vert _{W_{0}^{1,p}(Q)}^{p}dt\geq L_{\delta }^{\prime }\right\} \\ & +\mathbb{P}\left\{ \sup\limits_{\theta \leq \mu _{n}}\int_{0}^{T}\Vert u_{t}^{\epsilon }(t+\theta )-u_{t}^{\epsilon }(t)\Vert _{W^{-1,p}(Q)}^{p^{\prime }}dt\geq \nu _{n}M_{\delta }^{\prime }\right\} . \end{align*}

    Again thanks to 19, we obtain

    \begin{eqnarray*} \mathbb{P}\{u_{t}^{\epsilon } \notin B_{\delta }\} \\ &\leq \frac{1}{K_{\delta }^{\prime }}\mathbb{E}\sup\limits_{0\leq t\leq T}\Vert u_{t}^{\epsilon }(t)\Vert _{L^{2}(Q)}^{2}+\frac{1}{L_{\delta }^{\prime }} \mathbb{E}\int_{0}^{T}\Vert u_{t}^{\epsilon }(t)\Vert _{W_{0}^{1,p}(Q)}^{p}dt \\ &+\sum\nolimits_{n = 0}^{\infty }\frac{1}{\nu _{n}M_{\delta }^{\prime }}\mathbb{E} \bigg\{\sup\limits_{\theta \leq \mu _{n}}\int_{0}^{T}\Vert u_{t}^{\epsilon }(t+\theta )-u_{t}^{\epsilon }(t)\Vert _{W^{-1,p}(Q)}^{p^{\prime }}dt\bigg\} \\ &\leq \frac{C}{K_{\delta }^{\prime }}+\frac{C}{L_{\delta }^{\prime }}+\frac{ C}{M_{\delta }^{\prime }}\sum \frac{\mu _{n}^{p^{\prime }/p}}{\nu _{n}} = \frac{\delta }{3}\delta, \end{eqnarray*}

    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

    \begin{equation*} \Pi _{\epsilon _{j}}\rightharpoonup \Pi \end{equation*}

    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

    \begin{equation} (W_{\epsilon _{j}},u^{\epsilon _{j}},u_{t}^{\epsilon _{j}})\rightarrow (\tilde{W},u,u_{t})\,\,\text{in}\,\,\mathcal{X},\,\,\tilde{\mathbb{P}}-a.s.. \end{equation} (20)

    Let us define the filtration

    \begin{equation*} \tilde{\mathcal{F}_{t}} = \sigma \{\tilde{W}(s),u(s),u_{t}(s)\}_{0\leq s\leq t}. \end{equation*}

    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.

    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

    \begin{align} \lim\limits_{\epsilon\rightarrow0}\int_{0}^{T}\int_{Q}v^{\epsilon}\psi^{ \epsilon}dxdt = \dfrac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}v(t,x,y)\psi(t,x,y)dydxdt, \end{align} (21)

    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

    \begin{equation} \bigg\|\psi (\cdot,\cdot,\frac{\cdot}{\epsilon })\bigg\|_{L^{p}(0,T;L^{p}(Q))}\leq \Vert \psi (\cdot,\cdot,\cdot)\Vert _{L^{p}((0,T)\times Q;C_{per}(Y))} \end{equation} (22)

    and

    \begin{equation*} \psi (\cdot,\cdot,\frac{\cdot}{\epsilon })\rightharpoonup \dfrac{1}{|Y|}\int_{Y}\psi (\cdot,\cdot,y)dy\,\,\,weakly\ in\,\,L^{p}(0,T;L^{p}(Q)). \end{equation*}

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

    \begin{equation} \lim\limits_{\epsilon \rightarrow 0}\int_{0}^{T}\int_{Q}\left[ \psi (t,x,\frac{x}{ \epsilon })\right] ^{2}dxdt = \dfrac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}\left[ \psi (t,x,y)\right] ^{2}dtdxdy. \end{equation} (23)

    (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

    \begin{equation*} \frac{1}{r}+\frac{1}{s} = \frac{1}{p}, \end{equation*}

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

    \begin{equation*} \psi (\cdot,\cdot,\frac{\cdot}{\epsilon })\rightharpoonup \dfrac{\psi _{1}(\cdot,\cdot)}{|Y|} \int_{Y}\psi _{2}(y)dy\,\,\,weakly\ in\,\,L^{p}(0,T;L^{p}(Q)). \end{equation*}

    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

    \begin{align} \|u^{\epsilon}\|_{L^{2}\left(0,T;L^{2}(Q)\right)} < \infty. \end{align} (24)

    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

    \begin{equation*} \Vert u^{\epsilon }\Vert _{L^{2}\left( 0,T;H_{0}^{1}(Q)\right) } < \infty . \end{equation*}

    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

    \begin{align} u^{\epsilon }& \rightarrow u\,\,\ \mathit{2 }-s \ in\,\,L^{2}(0,T;L^{2}(Q)), \end{align} (25)
    \begin{align} \nabla u^{\epsilon }& \rightarrow \nabla _{x}u+\nabla _{y}u_{1}\,\,\ \ \mathit{2 }-s \ in\,\,L^{2}(0,T;L^{2}(Q)). \end{align} (26)

    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

    \begin{align} g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) & \rightharpoonup \tilde{g}\left( t,x,u_{t}\right) \\ & = :\frac{1}{\left\vert Y\right\vert }\int_{Y}g\left( t,x,y,u_{t}\right) dy\ \ weakly\ in\ L^{2}\left( \left( 0,T\right) \times Q\right) ,\ \ \tilde{\mathbb{P}}-a.s.. \end{align} (27)

    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:

    \begin{eqnarray} \int_{0}^{T}\int_{Q}g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) \psi \left( t,x,\frac{x}{\varepsilon }\right) dxdt \label{WW0} = I_{1}^{\varepsilon }+I_{2}^{\varepsilon }, \end{eqnarray}

    where

    \begin{eqnarray*} I_{1}^{\varepsilon } & = &\int_{0}^{T}\int_{Q}\left[ g\left( t,x,\frac{x}{ \varepsilon },u_{t}^{\varepsilon _{j}}\right) -g\left( t,x,\frac{x}{ \varepsilon },u_{t}\right) \right] \psi \left( t,x,\frac{x}{\varepsilon } \right) dxdt, \\ I_{2}^{\varepsilon } & = &\int_{0}^{T}\int_{Q}g\left( t,x,\frac{x}{\varepsilon },u_{t}\right) \psi \left( t,x,\frac{x}{\varepsilon }\right) dxdt. \end{eqnarray*}

    Then

    \begin{eqnarray*} I_{1}^{\varepsilon }&\leq \left\vert \left\vert \psi \left( t,x,\frac{x}{ \varepsilon }\right) \right\vert \right\vert _{L^{2}\left( \left( 0,T\right) \times Q\right) }\left\vert \left\vert g\left( t,x,\frac{x}{\varepsilon } ,u_{t}^{\varepsilon }\right) -g\left( t,x,\frac{x}{\varepsilon } ,u_{t}\right) \right\vert \right\vert _{L^{2}\left( \left( 0,T\right) \times Q\right) } \\ &\leq C\left\vert \left\vert u_{t}^{\varepsilon }-u_{t}\right\vert \right\vert _{L^{2}\left( \left( 0,T\right) \times Q\right) }, \end{eqnarray*}

    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

    \begin{equation*} \lim\limits_{\varepsilon \rightarrow 0}I_{2}^{\varepsilon } = \int_{0}^{T}\int_{Q}\int_{Y}g\left( t,x,y,u_{t}\right) \psi \left( t,x,y\right) dxdt,\quad \qquad \quad \tilde{\mathbb{P}}-a.s. \end{equation*}

    Therefore

    \begin{equation} g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) \overset{2-s}{ \rightarrow }g\left( t,x,y,u_{t}\right) ,\ \quad \qquad \quad \tilde{\mathbb{P}}-a.s. \end{equation} (28)

    and this implies the result.

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

    \begin{equation} \lim\limits_{\epsilon \rightarrow 0}\int_{0}^{T}\int_{Q}\left[ g(t,x,\frac{x}{ \epsilon },u_{t}^{\epsilon })\right] ^{2}dxdt = \dfrac{1}{|Y|} \int_{0}^{T}\int_{Q\times Y}\left[ g(t,x,y,u_{t})\right] ^{2}dtdxdy. \end{equation} (29)

    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

    \begin{align} a^{\epsilon _{j}}& \rightharpoonup a,\,\,\,\,weakly\ in \,\,\,H_{0}^{1}(Q), \end{align} (30)
    \begin{align} b^{\epsilon _{j}}& \rightharpoonup b,\,\,\,\,weakly\ in\,\,\,L^{2}(Q). \end{align} (31)

    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

    \begin{align} & u^{\epsilon _{j}}\rightharpoonup u\,\,\,\text{weakly in}\,\,\,L^{\infty }(0,T;H_{0}^{1}(Q))\quad \widehat{\mathbb{P}}-a.s, \end{align} (32)
    \begin{align} & u_{t}^{\epsilon _{j}}\rightharpoonup u_{t}\,\,\,\text{weakly in} \,\,\,L^{\infty }(0,T;L^{2}(Q))\quad \widehat{\mathbb{P}}-a.s, \end{align} (33)
    \begin{align} & u_{t}^{\epsilon _{j}}\rightharpoonup u_{t}\,\,\,\text{weakly in} \,\,\,L^{p}(0,T;W_{0}^{1,p}(Q))\quad \widehat{\mathbb{P}}-a.s, \end{align} (34)
    \begin{align} & B(t,u_{t}^{\epsilon _{j}})\rightharpoonup \chi \,\,\,\text{weakly in} \,\,\,L^{p^{\prime }}(0,T;W^{-1,p^{\prime }}(Q))\quad \widehat{\mathbb{P}} -a.s.. \end{align} (35)

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

    \begin{equation} \int_{0}^{t}\left\langle B(s,u_{t}^{\epsilon _{j}}),u_{t}^{\epsilon _{j}}\right\rangle ds\rightarrow \int_{0}^{t}\langle \chi ,u_{t}\rangle ds,\,\,\text{weakly in}\,\,\,L^{1}(\Omega ),\ \forall t\in \lbrack 0,T]. \end{equation} (36)

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

    \begin{equation} \chi _{\epsilon _{j}} = \widehat{\mathbb{E}}\int_{0}^{T}\left\langle B(t,u_{t}^{\epsilon _{j}})-B(t,v),u_{t}^{\epsilon _{j}}-v\right\rangle dt. \end{equation} (37)

    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

    \begin{equation*} \widehat{\mathbb{E}}\int_{0}^{T}\left\langle \chi -B(t,v),u_{t}-v\right\rangle dt\geq 0. \end{equation*}

    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

    \begin{equation} \widehat{\mathbb{E}}\int_{0}^{T}\left\langle \chi -B(t,u_{t}(t)-\lambda w(t)),w(t)\right\rangle dt\geq 0. \end{equation} (38)

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

    \begin{equation*} \left\langle \chi -B(t,u_{t}(t)-\lambda w(t)),w(t)\right\rangle \longrightarrow \left\langle \chi -B(t,u_{t}(t)),w(t)\right\rangle ,\ \text{ as}\ \lambda \longrightarrow 0,\ \widehat{\mathbb{P}}-a.s.. \end{equation*}

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

    \begin{equation} \widehat{\mathbb{E}}\int_{0}^{T}\left\langle \chi -B(t,u_{t}(t)),w(t)\right\rangle dt\geq 0. \end{equation} (39)

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

    \begin{equation*} \chi = B(t,u_{t}(t),\quad \widehat{\mathbb{P}}-a.s.. \end{equation*}

    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

    \begin{align} & -\int_{0}^{T}\int_{Q} u_{t}^{\epsilon _{j}}\Phi_{t} (t,x)dxdt+\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}\nabla \Phi dxdt+\int_{0}^{T}\int_{Q}\langle B^{\epsilon _{j}}(t,u_{t}^{\epsilon _{j}}),\Phi \rangle dxdt \\ & = \int_{0}^{T}\int_{Q}f^{\epsilon _{j}}(t,x,\nabla u^{\epsilon _{j}})\Phi dxdt+\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,x,u_{t}^{\epsilon _{j}})\Phi dxdW_{\epsilon _{j}}, \end{align} (40)

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

    \begin{equation*} \nabla u^{\epsilon _{j}}\rightarrow \nabla _{x}u+\nabla _{y}u_{1}\,\,\text{ 2-s in},\,\,L^{2}(0,T;L^{2}(Q)). \end{equation*}

    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

    \begin{align} & -\int_{0}^{T}\int_{Q}u_{t}^{\epsilon _{j}}(t,x)\bigg[\phi _{t}(t,x)+\epsilon _{j}\phi _{1t}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}(x)\nabla u^{\epsilon _{j}}(x,t) \bigg [\nabla _{x}\phi (t,x)+\epsilon _{j}\nabla _{x}\phi _{1}(t,x,\frac{x}{ \epsilon _{j}})+\nabla _{y}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}\left\langle B(t,u_{t}^{\epsilon _{j}}),\bigg[\phi _{t}(t,x)+\epsilon _{j}\phi _{1t}(t,x,\frac{x}{\epsilon _{j}})\bigg] \right\rangle dxdt \\ & = \int_{0}^{T}\int_{Q}f^{\epsilon _{j}}(t,x,\nabla u^{\epsilon _{j}})\bigg[ \phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,u_{t}^{\epsilon _{j}})\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg] dxdW_{\epsilon _{j}}. \end{align} (41)

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

    \begin{align*} & \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}u_{t}^{\epsilon _{j}}(t,x)\bigg[\phi _{t}(t,x)+\epsilon _{j}\phi _{1t}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & = \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}u_{t}^{\epsilon _{j}}(t,x)\phi _{t}(t,x)dxdt+\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\int_{0}^{T}\int_{Q}u_{t}^{\epsilon _{j}}(t,x)\phi _{1t}(t,x,\frac{x}{\epsilon _{j}})dxdt \\ & = \int_{0}^{T}\int_{Q}u_{t}(t,x)\phi _{t}(t,x)dxdt,\quad \tilde{\mathbb{P}}-a.s.. \end{align*}

    The second term can be written as follows,

    \begin{align} \lim\limits_{\epsilon _{j}\rightarrow 0}& \int_{0}^{T}\int_{Q}\nabla u^{\epsilon _{j}}(x,t)A_{\epsilon _{j}}\bigg [\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}(x,t)\nabla _{x}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})dxdt. \end{align} (42)

    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

    \begin{align*} & \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}\nabla u^{\epsilon _{j}}(x,t)A_{\epsilon _{j}}\bigg [\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)][\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)]dydxdt. \end{align*}

    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

    \begin{equation*} \lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}(x,t)\nabla _{x}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})dxdt = 0,\quad \tilde{\mathbb{P}}-a.s.. \end{equation*}

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

    \begin{align*} & \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}\left\langle B(t,u_{t}^{\epsilon _{j}}),\bigg[\phi _{t}(t,x)+\epsilon _{j}\phi _{1t}(t,x, \frac{x}{\epsilon _{j}})\bigg]\right\rangle dxdt \\ & = \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}\left\langle B(t,u_{t}^{\epsilon _{j}}),\phi _{t}(t,x)\right\rangle dxdt\\&+\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\int_{0}^{T}\int_{Q}\left\langle B(t,u_{t}^{\epsilon _{j}}),\phi _{1t}(t,x,\frac{x}{\epsilon _{j}} )\right\rangle dxdt \\ & = \int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi _{t}(t,x)\rangle dxdt,\quad \tilde{\mathbb{P}}-a.s.. \end{align*}

    Let us write

    \begin{align} & \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}f^{\epsilon _{j}}(t,x,\nabla u^{\epsilon _{j}})\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & = \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}F^{\epsilon _{j}}(t,x)\cdot \nabla u^{\epsilon _{j}}\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & = \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}F^{\epsilon _{j}}(t,x)\cdot\nabla u^{\epsilon _{j}}\phi (t,x)dxdt\\&\qquad+\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\int_{0}^{T}\int_{Q}F^{\epsilon _{j}}(t,x).\nabla u^{\epsilon _{j}}\phi _{1}(t,x,\frac{x}{\epsilon _{j}} )dxdt, \end{align} (43)

    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

    \begin{align} & \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}F^{\epsilon _{j}}(t,x)\cdot\nabla u^{\epsilon _{j}}\phi (t,x)dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}F(t,x,y)\cdot \left[ \nabla _{x}u+\nabla _{y}u_{1}\right] \phi (t,x)dxdydt,\quad \tilde{\mathbb{P}}-a.s.. \end{align} (44)

    Concerning the stochastic integral, we have

    \begin{align} & \tilde{\mathbb{E}}\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,x,u_{t}^{\epsilon _{j}})\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}}) \bigg]dxdW_{\epsilon _{j}} \\ & = \tilde{\mathbb{E}}\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,x,u_{t}^{\epsilon _{j}})\phi (t,x)dxdW_{\epsilon _{j}}+\tilde{\mathbb{E}}\epsilon _{j}\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,x, u_{t}^{\epsilon _{j}})\phi _{1}(t,x,\frac{x}{\epsilon _{j}})dxdW_{\epsilon _{j}}. \end{align} (45)

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

    \begin{align} \tilde{\mathbb{E}}\int_{0}^{T}&\int_{Q}\phi \left( t,x\right) g\left( t,x, \frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) dW_{t}^{\varepsilon } \\ & = \tilde{\mathbb{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g\left( t,x, \frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) d\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right)\\ &+\tilde{\mathbb{E}}\int_{0}^{T}\int_{Q} \phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) d\tilde{W}_{t}. \end{align} (46)

    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

    \begin{equation} g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) = \frac{1}{\lambda }\int_{0}^{T}\rho \left( -\frac{t-s}{\lambda }\right) g\left( s,x,\frac{x}{\varepsilon },u_{s}^{\varepsilon }\left( s\right) \right) ds\ \text{for}\ \lambda > 0, \end{equation} (47)

    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

    \begin{equation} \mathbb{\tilde{E}}\int_{0}^{T}\left\vert \left\vert g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) \right\vert \right\vert _{L_{2}\left( Q\right) }^{2}dt\leq \mathbb{\tilde{E}} \int_{0}^{T}\left\vert \left\vert g\left( t,x,\frac{x}{\varepsilon } ,u_{t}^{\varepsilon }\left( t\right) \right) \right\vert \right\vert _{L_{2}\left( Q\right) }^{2}dt,\ \text{for any }\lambda > 0, \end{equation} (48)

    and for any \varepsilon >0

    \begin{equation} g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) \rightarrow g^{\varepsilon }\left( t,x,u_{t}^{\varepsilon }\left( t\right) \right) \ \text{strongly in}\ L^{2}\left( \tilde{\Omega},\mathcal{\tilde{F}}, \mathbb{\tilde{P}},L_{2}\left( \left( 0,T\right) \times Q\right) \right) \text{ as }\lambda \rightarrow 0. \end{equation} (49)

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

    \begin{eqnarray} &&\mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g^{\varepsilon }\left( t,x,u_{t}^{\varepsilon }\left( t\right) \right) dxd\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right) \\ && = \mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) dxd\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right) \\ &&+\mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) \left[ g^{\varepsilon }\left( t,x,u_{t}^{\varepsilon }\left( t\right) \right) -g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) \right] dxd\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right) . \end{eqnarray} (50)

    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

    \begin{eqnarray} &&\mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) d\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right) \\ && = \mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right) \frac{\partial }{\partial t}\left[ \phi \left( t,x\right) g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( t\right) \right] dt \\ &&+\mathbb{\tilde{E}}\int_{Q}\phi \left( T,x\right) g_{\lambda }^{\varepsilon }\left( u^{\varepsilon }\right) \left( T\right) \left( W_{T}^{\varepsilon }-\tilde{W}_{T}\right) . \end{eqnarray} (51)

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

    \begin{equation} W_{t}^{\varepsilon }\rightarrow \tilde{W}_{t}\ \text{uniformly in }C\left( \left[ 0,T \right] \right) ,\ \tilde{\mathbb{P}}-\text{a.s.,} \end{equation} (52)

    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

    \begin{equation} \left\vert \mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g^{\varepsilon }\left( t,x,u_{t}^{\varepsilon }\left( t\right) \right) dxd\left( W_{t}^{\varepsilon }-\tilde{W}_{t}\right) \right\vert \leq \sigma _{1}\left( \lambda \right) +\sigma _{2}\left( \lambda \right) \eta _{1}\left( \varepsilon \right) . \end{equation} (53)

    Thus, we infer from 46 that

    \begin{align} \left\vert \mathbb{\tilde{E}}\int_{0}^{T}\int_{Q} \right.&\phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) dxdW_{t}^{\varepsilon } &\\&- \left.\mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) d\tilde{W}_{t}\right\vert \\ &\leq \sigma _{1}\left( \lambda \right) +\sigma _{2}\left( \lambda \right) \eta _{1}\left( \varepsilon \right) \end{align} (54)

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

    \begin{align*} \lim\limits_{\varepsilon \rightarrow 0} \left\vert \mathbb{\tilde{E}} \int_{0}^{T}\int_{Q}\right.&\phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon } ,u_{t}^{\varepsilon }\right) dxdW_{t}^{\varepsilon }\\&-\left.\mathbb{\tilde{E}} \int_{0}^{T}\int_{Q}\phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon } ,u_{t}^{\varepsilon }\right) d\tilde{W}_{t}\right\vert \leq \sigma _{1}\left( \lambda \right) ; \end{align*}

    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

    \begin{align} \lim\limits_{\varepsilon \rightarrow 0}\mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}&\phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) dxdW_{t}^{\varepsilon }\\ & = \lim\limits_{\varepsilon \rightarrow 0}\mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g\left( t,x,\frac{x}{ \varepsilon },u_{t}^{\varepsilon }\right) d\tilde{W}_{t}. \end{align} (55)

    Owing to 27; that is

    \begin{equation*} g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) \rightharpoonup \tilde{g}\left( t,x,u_{t}\right) \text{ weakly in } L^{2}\left( \left( 0,T\right) \times Q\right) ,\ \mathbb{\tilde{P}-} \text{a.s.,} \end{equation*}

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

    \begin{equation*} \mathbb{\tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) g\left( t,x, \frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) dW_{t}\rightarrow \mathbb{ \tilde{E}}\int_{0}^{T}\int_{Q}\phi \left( t,x\right) \tilde{g}\left( t,x,u_{t}\right) d\tilde{W}_{t}. \end{equation*}

    Hence, we deduce from 55 that,

    \begin{equation} \int_{0}^{T}\int_{Q}\phi \left( t,x\right) g\left( t,x,\frac{x}{\varepsilon } ,u_{t}^{\varepsilon }\right) dW_{t}^{\varepsilon }\rightarrow \int_{0}^{T}\int_{Q}\phi \left( t,x\right) \tilde{g}\left( t,x,u_{t}\right) d\tilde{W}_{t},\ \mathbb{\tilde{P}-}\text{a.s.}. \end{equation} (56)

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

    \begin{align*} & \lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\tilde{\mathbb{E}}\sup\limits_{t\in \lbrack 0,T]}\bigg|\int_{0}^{t}\int_{Q}\phi _{1}\left( t,x,\frac{x}{ \varepsilon }\right) g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) dxdW_{t}^{\epsilon _{j}}\bigg| \\ & \leq C\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\tilde{\mathbb{E}} \bigg(\int_{0}^{T}\bigg(\int_{Q}\phi _{1}\left( t,x,\frac{x}{\varepsilon } \right) g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) dx \bigg)^{2}dt\bigg)^{\frac{1}{2}} \\ & \leq C\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\tilde{\mathbb{E}} \bigg(\int_{0}^{T}\Vert g\left( t,x,\frac{x}{\varepsilon } ,u_{t}^{\varepsilon }\right) \Vert _{L^{2}(Q)}\Vert \phi _{1}(t,x,\frac{x}{ \epsilon _{j}})\Vert _{L^{2}(Q)}dt\bigg)^{\frac{1}{2}} \\ & \leq C\lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\bigg( \int_{0}^{T}\Vert g\left( t,x,\frac{x}{\varepsilon },u_{t}^{\varepsilon }\right) \Vert _{L^{2}(Q)}dt\bigg)^{\frac{1}{2}}\rightarrow 0,\quad \tilde{ \mathbb{P}}-a.s. \end{align*}

    Combining the above convergences, we obtain

    \begin{align} -\int_{0}^{T}&\int_{Q}u_{t}(t,x)\phi _{t}(t,x)dxdt \\ & +\frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\\ &\cdot[\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)]dydxdt \\ & +\int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi (t,x)\rangle dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}F(t,x,y).[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\phi (t,x)dxdydt \\ & +\int_{0}^{T}\int_{Q}\tilde{g}\left( t,x,u_{t}\right) \phi (t,x)\tilde{W} dx. \end{align} (57)

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

    \begin{equation} \int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)][\nabla _{y}\phi _{1}(t,x,y)]dydxdt = 0, \end{equation} (58)

    and

    \begin{align} & -\int_{0}^{T}\int_{Q}u_{t}(t,x)\phi _{t}(t,x)dxdt \\ & +\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)][\nabla _{x}\phi (t,x)]dydxdt \\ & +\int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi (t,x)\rangle dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}F(t,x,y).[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\phi (t,x)dxdydt \\ & +\int_{0}^{T}\int_{Q}\tilde{g}\left( t,x,u_{t}\right) \phi (t,x)d\tilde{W} dx. \end{align} (59)

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

    \begin{equation} u_{1}(t,x,y) = -\chi (y)\cdot \nabla _{x}u(t,x)+\tilde{u_{1}}(t,x), \end{equation} (60)

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

    \begin{equation} \left\{ \begin{array}{c} \text{div}_{y}(A(y)\nabla _{y}\chi (y)) = \nabla _{y}\cdot A(y),\,\,\text{in} \,\,Y, \\ \chi \,\,\,\text{is}\,\,Y\,\,\text{periodic}. \end{array} \right. \end{equation} (61)

    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

    \begin{equation} du_{t}-A_{0}\Delta udt+B(t,u_{t})dt = \tilde{f}(t,x,\nabla u)dt+\tilde{g}(t,x,u_{t})d\tilde{W}, \end{equation} (62)

    where

    \begin{array}{*{20}{c}} {A_{0} = \frac{1}{|Y|}\int_{Y}(A(y)-A(y)\nabla _{y}\chi (y))dy,}\\ {\tilde{f}(t,x,\nabla u) = \frac{1}{|Y|}\int_{Y}F(t,x,y)\cdot[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]dy, } \end{array} (63)

    and

    \begin{equation*} \tilde{g}\left( t,x,u_{t}\right) = \frac{1}{\left\vert Y\right\vert }\int_{Y}g\left( t,x,y,u_{t}\right) dy. \end{equation*}

    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

    \begin{align*} & -\int_{0}^{T}\int_{Q}u_{t}^{\epsilon _{j}}(t,x)\bigg[\phi _{t}(t,x)+\epsilon _{j}\phi _{1t}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}(x)\nabla u^{\epsilon _{j}}(x,t)\cdot \bigg [\nabla _{x}\phi (t,x)+\epsilon _{j}\nabla _{x}\phi _{1}(t,x,\frac{x}{ \epsilon _{j}})+\nabla _{y}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}\left \langle B(t,u_{t}^{\epsilon }),\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]\right \rangle dxdt \\ & = \int_{0}^{T}\int_{Q}f^{\epsilon _{j}}(t,x,\nabla u^{\epsilon _{j}})\bigg[ \phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,x,u_{t}^{\epsilon })\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg] dxdW_{\epsilon _{j}}+\int_{Q}u_{t}^{\epsilon _{j}}(x,0)v(x)dx, \end{align*}

    where we pass to the limit, to get

    \begin{align*} & -\int_{0}^{T}\int_{Q}u_{t}(t,x)\phi _{t}(t,x)dxdt \\ & +\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\cdot [\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)]dydxdt \\ & +\int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi (t,x)\rangle dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}F(t,x,y)\cdot [\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\phi (t,x)dxdydt \\ & +\int_{0}^{T}\int_{Q}\tilde{g}\left( t,x,u_{t}\right) \phi (t,x)\tilde{W} dxdt+\int_{Q}b(x)v(x)dx. \end{align*}

    The integration by parts, in the first term gives

    \begin{align*} & \int_{0}^{T}\int_{Q}du_{t}(t,x)\phi (t,x)dx+\int_{Q}u_{t}(x,0)v(x)dx \\ & +\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\cdot [\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)]dydxdt \\ & +\int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi (t,x)\rangle dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}F(t,x,y)\cdot [\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\phi (t,x)dxdydt \\ & +\int_{0}^{T}\int_{Q}\tilde{g}(t,x,u_{t})\phi (t,x)\tilde{W} dxdt+\int_{Q}b(x)v(x)dx. \end{align*}

    In view of equation 57, we deduce that

    \begin{equation*} \int_{Q}u_{t}(x,0)v(x)dx = \int_{Q}b(x)v(x)dx, \end{equation*}

    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

    \begin{align*} & \int_{0}^{T}\int_{Q}u^{\epsilon _{j}}(t,x)\bigg[\phi _{tt}(t,x)+\epsilon _{j}\phi _{1tt}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}(x)\nabla u^{\epsilon _{j}}(x,t)\cdot \bigg [\nabla _{x}\phi (t,x)+\epsilon _{j}\nabla _{x}\phi _{1}(t,x,\frac{x}{ \epsilon _{j}})+\nabla _{y}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}\left \langle B(t,u_{t}^{\epsilon }),\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]\right \rangle dxdt \\ & = \int_{0}^{T}\int_{Q}f^{\epsilon _{j}}(t,x,\nabla u^{\epsilon _{j}})\bigg[ \phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg]dxdt \\ & +\int_{0}^{T}\int_{Q}g^{\epsilon _{j}}(t,x,u_{t}^{\epsilon })\bigg[\phi (t,x)+\epsilon _{j}\phi _{1}(t,x,\frac{x}{\epsilon _{j}})\bigg] dxdW_{\epsilon _{j}}-\int_{Q}u^{\epsilon _{j}}(x,0)v(x)dx. \end{align*}

    Passing to the limit in this equation, we obtain

    \begin{align*} & \int_{0}^{T}\int_{Q}u(t,x)\phi _{tt}(t,x)dxdt \\ & +\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\cdot[\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)]dydxdt \\ & +\int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi (t,x)\rangle dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times ,Y}F(t,x,y)\cdot [\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\phi (t,x)dxdydt \\ & +\int_{0}^{T}\int_{Q}\tilde{g}(t,x,u_{t})\phi (t,x)\tilde{W} dxdt-\int_{Q}a(x)v(x)dx. \end{align*}

    We integrate by parts again to obtain

    \begin{align*} & -\int_{0}^{T}\int_{Q}u_{t}(t,x)\phi _{t}(t,x)dxdt-\int_{Q}u(x,0)v(x)dx \\ & +\int_{0}^{T}\int_{Q\times Y}A(y)[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\cdot [\nabla _{x}\phi (t,x)+\nabla _{y}\phi _{1}(t,x,y)]dydxdt \\ & +\int_{0}^{T}\int_{Q}\langle B(t,u_{t}),\phi (t,x)\rangle dxdt \\ & = \frac{1}{|Y|}\int_{0}^{T}\int_{Q\times Y}F(t,x,y)\cdot [\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\phi (t,x)dxdydt \\ & +\int_{0}^{T}\int_{Q}\tilde{g}(t,x,u_{t})\phi (t,x)\tilde{W} dxdt-\int_{Q}a(x)v(x)dx. \end{align*}

    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.

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

    \begin{align*} \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)& = \frac{1}{2}\tilde{ \mathbb{E}}\Vert u_{t}^{\epsilon _{j}}(t)\Vert _{L^{2}(Q)}^{2}+\frac{1}{2} \tilde{\mathbb{E}}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}(x,t)\cdot\nabla u^{\epsilon _{j}}(x,t)dx \\ & +\tilde{\mathbb{E}}\int_{0}^{t}\langle B(s,u_{t}^{\epsilon _{j}}),u_{t}^{\epsilon _{j}}\rangle ds \\ \mathcal{E}(u)(t)& = \frac{1}{2}\tilde{\mathbb{E}}\Vert u_{t}(t)\Vert _{L^{2}(Q)}^{2}+\frac{1}{2}\tilde{\mathbb{E}}\int_{Q}A_{0}\nabla u(x,t)\cdot\nabla u(x,t)dx \\ & +\tilde{\mathbb{E}}\int_{0}^{t}\langle B(s,u_{t}),u_{t}\rangle ds. \end{align*}

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

    \begin{align*} & \frac{1}{2}\tilde{\mathbb{E}}\Vert u_{t}^{\epsilon _{j}}(t)\Vert _{L^{2}(Q)}^{2}+\frac{1}{2}\tilde{\mathbb{E}}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}(t)\cdot\nabla u^{\epsilon _{j}}(t)dx+\tilde{\mathbb{E}} \int_{0}^{t}\langle B(s,u_{t}^{\epsilon _{j}}),u_{t}^{\epsilon _{j}}\rangle ds \\ & = \tilde{\mathbb{E}}\bigg[\frac{1}{2}\Vert u_{1}^{\epsilon _{j}}\Vert _{L^{2}(Q)}^{2}+\frac{1}{2}\int_{Q}A_{\epsilon _{j}}\nabla u_{0}^{\epsilon _{j}}\cdot \nabla u_{0}^{\epsilon _{j}}dx+\int_{0}^{t}(f^{\epsilon _{j}}(s,x,\nabla u^{\epsilon _{j}}),u_{t}^{\epsilon _{j}})ds \\ & +\frac{1}{2}\int_{0}^{t}\Vert g^{\epsilon _{j}}(s,u_{t}^{\epsilon _{j}})\Vert _{L^{2}(Q)}^{2}ds+\int_{0}^{t}(g^{\epsilon _{j}}(s,u_{t}^{\epsilon _{j}}),u_{t}^{\epsilon _{j}})dW_{\epsilon _{j}}\bigg] . \end{align*}

    Thus

    \begin{align} \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)& = \frac{1}{2}\tilde{ \mathbb{E}}\Vert u_{1}^{\epsilon _{j}}\Vert _{L^{2}(Q)}^{2}+\frac{1}{2} \tilde{\mathbb{E}}\int_{Q}A_{\epsilon _{j}}\nabla u_{0}^{\epsilon _{j}}\cdot\nabla u_{0}^{\epsilon _{j}}dx \\ & +\tilde{\mathbb{E}}\int_{0}^{t}(f^{\epsilon _{j}}(s,x,\nabla u^{\epsilon _{j}}),u_{t}^{\epsilon _{j}})ds+\frac{1}{2}\tilde{\mathbb{E}} \int_{0}^{t}\Vert g^{\epsilon _{j}}(s,u_{t}^{\epsilon _{j}})\Vert _{L^{2}(Q)}^{2}ds, \end{align} (64)
    \begin{align} \mathcal{E}(u)(t)& = \frac{1}{2}\tilde{\mathbb{E}}\Vert u_{1}\Vert _{L^{2}(Q)}^{2}+\frac{1}{2}\tilde{\mathbb{E}}\int_{Q}A_{0}\nabla u_{0}\cdot \nabla u_{0}dx \\ & +\tilde{\mathbb{E}}\int_{0}^{t}(\tilde{f}(s,x,\nabla u),u_{t})ds+\frac{1}{2 }\tilde{\mathbb{E}}\int_{0}^{t}\Vert \tilde{g}\left( s,x,u_{t}\right) \Vert _{L^{2}(Q)}^{2}ds. \end{align} (65)

    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

    \begin{align} & -div(A_{\epsilon _{j}}\nabla a^{\epsilon _{j}})\rightarrow -div(A_{0}\nabla a),\,\,\,\, strongly\ in\,\,\,H^{-1}(Q), \end{align} (66)
    \begin{align} & b^{\epsilon _{j}}\rightarrow b,\,\,\,\,\ strongly\ in\,\,\,L^{2}(Q). \end{align} (67)

    Then

    \begin{equation*} \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)\rightarrow \mathcal{E} (u)(t)\,\,\,in\,\,C([0,T]), \end{equation*}

    where u is the solution of the homogenized problem.

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

    \begin{equation*} \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)\rightarrow \mathcal{E} (u)(t),\,\,\,\forall t \in [0,T]. \end{equation*}

    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

    \begin{align*} \left\vert \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)\right\vert & \leq \frac{1}{2}\tilde{\mathbb{E}}\Vert b^{\epsilon _{j}}\Vert _{L^{2}(Q)}^{2}+\frac{\alpha }{2}\tilde{\mathbb{E}}\Vert a^{\epsilon _{j}}\Vert _{H_{0}^{1}}+\tilde{\mathbb{E}}\int_{0}^{t}\left\vert (f^{\epsilon _{j}}(s,x,\nabla u^{\epsilon _{j}}),u_{t}^{\epsilon _{j}})\right\vert ds \\ & +\frac{1}{2}\int_{0}^{t}\Vert g^{\epsilon _{j}}(s,u_{t}^{\epsilon _{j}})\Vert _{L^{2}(Q)}^{2}ds. \end{align*}

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

    \begin{equation*} \left\vert \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)\right\vert \leq C,\quad \forall t\in \lbrack 0,T]. \end{equation*}

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

    \begin{align*} |\mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})&(t+h)- \mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)| \\ & \leq \tilde{\mathbb{E}}\int_{t}^{t+h}|(f^{\epsilon _{j}}(s,x,\nabla u^{\epsilon _{j}}),u_{t}^{\epsilon _{j}})|ds+\frac{1}{2}\tilde{\mathbb{E}} \int_{t}^{t+h}\Vert g^{\epsilon _{j}}(s,u_{t}^{\epsilon _{j}})\Vert _{L^{2}(Q)}^{2}ds. \end{align*}

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

    \begin{equation*} |\mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t+h)-\mathcal{E}^{\epsilon _{j}}(u^{\epsilon _{j}})(t)|\leq C\left( h+h^{\frac{1}{2}}\right) . \end{equation*}

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

    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

    \begin{equation*} \frac{1}{r}+\frac{1}{s} = \frac{1}{2}. \end{equation*}

    Then

    \begin{align} u_{t}^{\epsilon _{j}}-u_{t}-\epsilon _{j}u_{1t}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}} )& \rightarrow 0\,\,\,\ strongly\ in\,\,\,L^{2}(0,T;L^{2}(Q))\quad \tilde{\mathbb{P}}-a.s., \end{align} (68)
    \begin{align} u^{\epsilon _{j}}-u-\epsilon _{j}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})& \rightarrow 0\,\,\,\ strongly\ in\,\,\,L^{2}(0,T;H^{1}(Q))\quad \tilde{ \mathbb{P}}-a.s.. \end{align} (69)

    Proof. It is easy to see that

    \begin{equation*} \lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}u_{1t}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})\rightarrow 0\,\,\,\text{in}\,\,\,L^{2}(0,T;L^{2}(Q))\quad \tilde{ \mathbb{P}}-a.s.. \end{equation*}

    Then convergence 20 gives

    \begin{equation*} u_{t}^{\epsilon _{j}}-u_{t}-\epsilon _{j}u_{1t}(\cdot,\cdot,\frac{.}{\epsilon _{j}} )\rightarrow 0\,\,\,\text{in}\,\,\,L^{2}(0,T;L^{2}(Q))\quad \tilde{\mathbb{P} }-a.s.. \end{equation*}

    Thus 68 holds. Similarly we show that

    \begin{equation*} u^{\epsilon _{j}}-u-\epsilon _{j}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}} )\rightarrow 0\,\,\,\text{strongly in}\,\,\,L^{2}(0,T;L^{2}(Q))\quad \tilde{ \mathbb{P}}-a.s.. \end{equation*}

    It remains to show that

    \begin{equation*} \nabla (u^{\epsilon _{j}}-u-\epsilon _{j}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}} ))\rightarrow 0\,\,\,\text{strongly in}\,\,\,L^{2}(0,T;[L^{2}(Q)]^{n})\quad \tilde{\mathbb{P}}-a.s.. \end{equation*}

    We have

    \begin{equation*} \nabla (u^{\epsilon _{j}}-u-\epsilon _{j}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}} )) = \nabla u^{\epsilon _{j}}-\nabla u-\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}}))-\epsilon _{j}\nabla u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})). \end{equation*}

    Again

    \begin{equation*} \lim\limits_{\epsilon _{j}\rightarrow 0}\epsilon _{j}\nabla u_{1}(\cdot,\cdot,\frac{\cdot}{ \epsilon _{j}})\rightarrow 0\,\,\,\text{in}\,\,\,L^{2}(0,T;[L^{2}(Q)]^{n}), \quad \tilde{\mathbb{P}}-a.s.. \end{equation*}

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

    \begin{align} \alpha \mathbb{E}\int_{0}^{T}&\Vert \nabla u^{\epsilon _{j}}-\nabla u-\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})\Vert _{L^{2}(Q)}^{2}dt \\ & \leq \mathbb{E}\int_{0}^{T}\int_{Q}A\left( \frac{x}{\epsilon _{j}}\right) \left( \nabla u^{\epsilon _{j}}-\nabla u-\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{ \epsilon _{j}})\right) \\ & \cdot \left( \nabla u^{\epsilon _{j}}-\nabla u-\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})\right) dxdt \\ & = \mathbb{E}\int_{0}^{T}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}\cdot\nabla u^{\epsilon _{j}}dxdt \\ &-2\mathbb{E}\int_{0}^{T}\int_{Q}\nabla u^{\epsilon _{j}}A\left( \frac{x}{\epsilon _{j}}\right)\cdot \left( \nabla u+\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})\right) dxdt \\ & +\mathbb{E}\int_{0}^{T}\int_{Q}A\left( \frac{x}{\epsilon _{j}}\right) \left( \nabla u+\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})\right) \\ &\cdot \left( \nabla u+\nabla _{y}u_{1}(\cdot,\cdot,\frac{\cdot}{\epsilon _{j}})\right) dxdt. \end{align} (70)

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

    \begin{equation*} \mathbb{E}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}\cdot \nabla u^{\epsilon _{j}}dx. \end{equation*}

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

    \begin{align} & \lim\limits_{\epsilon _{j}\rightarrow 0}\mathbb{E}\int_{Q}A_{\epsilon _{j}}\nabla u^{\epsilon _{j}}\cdot \nabla u^{\epsilon _{j}}dx \\ & = \mathbb{E}\int_{Q\times Y}A(y)\cdot[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]\cdot[\nabla _{x}u(t,x)+\nabla _{y}u_{1}(t,x,y)]dydx. \end{align} (71)

    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

    \begin{align} \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}&\nabla u^{\epsilon _{j}}(t,x)\cdot A\left( \frac{x}{\epsilon _{j}}\right)\cdot \left( \nabla u+\nabla _{y}u_{1}(t,x,\frac{x}{\epsilon _{j}})\right) dxdt \\ & = \int_{0}^{T}\int_{Q\times Y}\left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right) \\ &\cdot A\left( y\right) \cdot\left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right) dxdydt. \end{align} (72)

    Now, let us write

    \begin{align*} \psi (t,x,y)& = A\left( y\right) \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right) \cdot \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right) \\ & = A\left( y\right) \nabla u(t,x)\cdot\nabla u(t,x)+2A\left( y\right) \nabla u(t,x)\cdot\nabla _{y}u_{1}(t,x,y)\\ &+A\left( y\right) \nabla _{y}u_{1}(t,x,y)\cdot\nabla _{y}u_{1}(t,x,y). \end{align*}

    For u_{1} given by 60, we have

    \begin{align*} \psi (t,x,y) = & A\left( y\right) \nabla u(t,x)\cdot\nabla u(t,x)-2A\left( y\right) \nabla u(t,x)\cdot\nabla _{y}[\chi (y)\cdot \nabla _{x}u(t,x)] \\ & +A\left( y\right) \nabla _{y}[\chi (y)\cdot \nabla _{x}u(t,x)]\nabla _{y}[\chi (y)\cdot \nabla _{x}u(t,x)]. \end{align*}

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

    \begin{align} \lim\limits_{\epsilon _{j}\rightarrow 0}\int_{0}^{T}\int_{Q}&A\left( \frac{x}{ \epsilon _{j}}\right) \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,\frac{x}{ \epsilon _{j}})\right)\\ & \cdot \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,\frac{ y}{\epsilon _{j}})\right) dxdt \\ & = \int_{0}^{T}\int_{Q\times Y}A\left( y\right) \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right)\\ & \cdot \left( \nabla u(t,x)+\nabla _{y}u_{1}(t,x,y)\right) dxdydt. \end{align} (73)

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

    \begin{equation*} \lim\limits_{\epsilon _{j}\rightarrow 0}\mathbb{E}\int_{0}^{T}\Vert \nabla u^{\epsilon _{j}}-\nabla u-\nabla _{y}u_{1}(.,.,\frac{.}{\epsilon _{j}} )\Vert _{L^{2}(Q)}^{2}dt = 0\quad \tilde{\mathbb{P}}-a.s.. \end{equation*}

    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.

    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.



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