Omar Anza Hafsa

AND Jean-Philippe Mandallena
email: jean-philippe.mandallena@unimes.fr

G Érard Michaille

Gérard Michaille
email: gerard.michaille@gmail.com

CONTINUITY THEOREM FOR NON-LOCAL FUNCTIONALS INDEXED BY YOUNG MEASURES AND STOCHASTIC HOMOGENIZATION

We establish a continuity theorem for non-local functionals indexed by Young measures that we use to deal with homogenization of stochastic non-diffusive reaction differential equations. Non-local effects induced by homogenization of such stochastic differential equations are studied.

Introduction

Let T ą 0, let N ě 1 be an integer and let Ω Ă R N be a bounded domain. Let X be a polish subspace of a suitable metric space (see §2.1 for more details) and let YpΩ; X q be the class of Young measures on Ω ˆX . Let us set H 1 0 :" ! u P H 1 ps0, T r; L 2 pΩ; Rqq : up0, ¨q " 0

)
and for each µ P YpΩ; X q and each u P H 1 0 , let us consider the following two classes of functions:

H 1 0,µ :" where tµ x u xPΩ denotes the desintegration of µ on Ω ˆX . In this paper we are concerned with non-local functionals Φ µ : H 1 0 R defined by Φ µ puq :" inf ! φ µ pU q : U P S µ puq ) with µ P YpΩ; X q, where φ µ : S µ puq R is given by φ µ pU q :" ż s0,T rˆΩˆX ξ `t, x, U pt, x, ξq, 9 U pt, x, ξq ˘dt b dµpx, ξq `żΩˆX θpU pT, x, ξqqdµpx, ξq with θ : R r0, 8r a convex function of 2-polynomial growth (see (C 1 ), (C 2 ), (C 3 ) and (C 4 ) in §2.1 for more details). Let us set F :" tΦ µ : µ P YpΩ; X qu. Our main result is to prove that the map µ Φ µ is continuous from YpΩ; X q endowed with the narrow convergence to F endowed with the Γ-convergence (see Theorem 2.7 and also Corollaries 2.8 and 2.9), i.e.,

! U P H 1 `s0,
µ ε nar -µ ñ Γ-lim ε 0 Φ µε " Φ µ .
The interest of this result comes from the convergence, when ε goes to zero, of non-diffusive reaction differential equations of the type

pR ε q $ ' & ' % ´Bu ε
Bt pt, xq " Bψ ε Bs pt, x, u ε ptqq for dt b dx-a.a. pt, xq Ps0, T rˆΩ u ε p0, ¨q " 0 with ψ ε : r0, T s ˆRN ˆR R and ε ą 0. In fact, under suitable assumptions (see §2.3 for more details) every (R ε ) admits a unique solution u ε P H 1 0 which is also a minimizer of Φ µε with: µ ε " dx b δ fεpxq ; f ε pxqpt, y, s, 9

sq " ψ ε pt, x, sq `ψε pt, x, ´9 sq;

θpsq "

1 2 |s| 2 ,
where ψ ε denotes the Legendre-Fenchel conjugate of ψ ε with respect to the third variable (see Proposition 2.11). When there exists µ P YpΩ; X q such that dx b δ fεpxq nar µ, Theorem 2.7 implies that any cluster point of tu ε u ε with respect to the weak convergence in H 1 0 is a minimizer of Φ µ with the same θ : R r0, 8r given above (see Corollary 2.12).

This latter result point out the appearance of non-local effects induced by homogenization of non-diffusive reaction equations of type (R ε ) and, in some way, can be seen as a generalization of previous non-local results obtained by Mascarenhas (see [START_REF] Luisa | Memory effect phenomena and Γ-convergence[END_REF]) and Toader (see [START_REF] Toader | Memory effects and Γ-convergence: a time dependent case[END_REF]).

The fact that non-local effects appear when dealing with homogenization of equations of type (R ε ) is illustrated by studying stochastic homogenization of such equations in the framework of a Poisson point process (see Theorem 4.6 and also Theorems 4.3 and 4.4). In particular, we show that the weak limit of solutions of stochastic non-diffusive reaction differential equations of type (R ε ) is characterized by two integro-differential equations (see §4.3).

The plan of the paper is as follows. In Section 2 we state the main result of the paper (see Theorem 2.7) and direct consequences of this result (see Corollaries 2.8 and 2.9) together with results related to homogenization of non-diffusive reaction equations (see Proposition 2.11, Corollary 2.12 and Proposition 2.13). Propositions 2.11 and 2.13 are proved in Section 2 while the proof of Theorem 2.7 is given in Section 3. Section 4 is devoted to stochastic homogenization. In §4.1, as a consequence of Theorem 2.7, we state and prove a non-local stochastic homogenization theorem (see Theorem 4.2) that we apply in §4.2, in the case of a Poisson point process, to homogenization of stochastic integral functionals as well as to homogenization of stochastic non-diffusive reaction equations (see Theorems 4.3, 4.4 and 4.6). Theorems 4.3 and 4.4 and Theorem 4.6 are respectively proved in §4.2 and §4.3. Finally, in the appendix, we recall some standard results on Young measures (see Appendix A) and inf-convolution and parallel sum (see Appendix B).

Notation. Throughout the paper, for any function w, 9 w denotes the partial derivative of w with respect to t, i.e., 9

w " Bw Bt .

Main results

2.1.

A continuity theorem for a class of non-local functionals indexed by Young measures. Let N ě 1 be an integer and let T ą 0. Let α, β Ps0, 8r be such α ď β and let γ : r0, 8r r0, 8r be such that lim r 0 γprq " 0. In what follows, we denote by M α,β,γ the class of functions ξ : r0, T s ˆRN ˆR ˆR R satisfying the following three conditions: (C 1 ) ξpt, x, ¨, ¨q is convex for all pt, xq P r0, T s ˆRN ; (C 2 ) αp|ps, 9 sq| 2 ´1q ď ξpt, x, s, 9 sq ď βp|ps, 9 sq| 2 `1q for all pt, xq P r0, T s ˆRN and all ps, 9 sq P R ˆR; (C 3 ) |ξpt 1 , x 1 , s, 9 sq´ξpt 2 , x 2 , s, 9 sq| ď γp|t 1 ´t2 |`|x 1 ´x2 |q for all pt 1 , x 1 q, pt 2 , x 2 q P r0, T sˆR N and all ps, 9 sq P R ˆR. From now on, to simplify notation we will sometimes write S to denote the couple ps, 9 sq P R 2 . From (C 1 ), (C 2 ) and (C 3 ) it is easy to see that there exists L ą 0 such that |ξpt 1 , x 1 , S 1 q ´ξpt 2 , x 2 , S 2 q| ď γp|t 1 ´t2 | `|x 1 ´x2 |q `L|S 1 ´S2 |p1 `|S 1 | `|S 2 |q (2.1) for all ξ P M α,β,γ , all pt 2 , x 2 q P r0, T s ˆRN and all pS 1 , S 2 q P R 2 ˆR2 . Proposition 2.1. Under (C 1 ), (C 2 ) and (C 3 ) the space M α,β,γ is a compact metric space with respect to the metric d : M α,β,γ ˆMα,β,γ r0, 8r defined by

dpξ 1 , ξ 2 q " 8 ÿ n"0 1 2 n`1 inf ! 1, }ξ 1 ´ξ2 } 8,n ) , (2.2)
where } ¨}8,n is the uniform norm on r0, T s ˆBn p0q, i.e., for each ξ P M α,β,γ , }ξ} 8,n " sup ! |ξpt, x, Sq| : t P r0, T s and px, Sq P B n p0q

) with B n p0q Ă R N `2 being the pN `2q-dimensional closed ball centered at 0 with radius n.

Moreover, the evaluation map E : M α,β,γ ˆr0, T s ˆRN ˆR2 R defined by Epξ, t, x, Sq " ξpt, x, Sq is continuous.

Proof of Proposition 2.1. For each n P N, we consider the normed space pX n , } ¨}8,n q with X n defined by

X n :" ! ξ |r0,T sˆBnp0q : ξ P M α,β,γ
) ,

where ξ |r0,T sˆBnp0q denotes the restriction of ξ to r0, T s ˆBn p0q. Then pM α,β,γ , dq is homeomorphic to the product ś nPN X n . On the other hand, by using (C 2 ) and (2.1), from Ascoli's compactness theorem we see that pX n , } ¨}8,n q is compact for all n P N. By Tychonov's compactness theorem it follows that pM α,β,γ , dq is compact. Let pξ, t, x, Sq ˆr0, T s ˆRN ˆR2 and let tpξ j , t j , x j , S j qu j Ă M α,β,γ ˆr0, T s ˆRN ˆR2 be such that |pt j , x j , S j q ´pt, x, Sq| 0 and dpξ j , ξq 0. Then, there exists n 0 P N such that pt, x, Sq P r0, T s ˆBn 0 p0q and pt j , x j , S j q P r0, T s ˆBn 0 p0q for all j ě 1. Let j 0 ě 1 be such that dpξ j , ξq ă 8 ÿ n"n 0 1 2 n`1 for all j ě j 0 .

(2.3)

Then, for every j ě j 0 we have inf 1, }ξ j ´ξ} 8,n 0 ( " }ξ j ´ξ} 8,n 0 . Indeed, fix any j ě j 0 . If inf 1, }ξ j ´ξ} 8,n 0 ( " 1 then inf 1, }ξ j ´ξ} 8,n

( " 1 for all n ě n 0 because }ξ j ´ξ} 8,n 0 ď }ξ j ´ξ} 8,n for all n ě n 0 . Hence

dpξ j , ξq ě 8 ÿ n"n 0 1 2 n`1 ,
which contradicts (2.3). Using (2.1) we deduce that ˇˇEpξ j , t j , x j , S j q ´Epξ, t, x, Sq ˇˇď ˇˇξ j pt j , x j , S j q ´ξj pt, x, Sq ˇˇ`ˇˇpξ j ´ξqpt, x, Sq ˇď γp|t j ´t| `|x j ´x|q `L|S j ´S|p1 `|S j | `|S|q `}ξ j ´ξ} 8,n 0 ď γp|t j ´t| `|x j ´x|q `LC n 0 |S j ´S| `2n 0 `1dpξ j , ξq for all j ě j 0 with C n 0 :" supt1 `|S j | `|S| : j ě j 0 u Ps0, 8r. Letting j 8 we deduce that |Epξ j , t j , x j , S j q ´Epξ, t, x, Sq| 0. Hence E is continuous.

Let Ω Ă R N be a bounded domain. In what follows, we set

H 1 0 :" ! u P H 1 ps0, T r; L 2 pΩ; Rqq : up0, ¨q " 0 ) .
Let X be a polish subspace of pM α,β,γ , dq and let YpΩ; X q be the class of Young measures on Ω ˆX . For each µ P YpΩ; X q and each u P H 1 0 we consider the following two classes of functions:

H 1 0,µ :" ! U P H 1 `s0, T r; L 2 µ pΩ ˆX ; Rq ˘: U p0, ¨, ¨q " 0 ) ;
S µ puq :"

" U P H 1 0,µ : ż X U pt, x, ξqdµ x pξq " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ * ,
where tµ x u xPΩ denotes the desintegration of µ on Ω ˆX . Let θ : R r0, 8r be a convex function of 2-polynomial growth, i.e., (C 4 ) θpyq ď cp1 `|y| 2 q for all y P R and some c Ps0, 8r.

Remark 2.2. Under (C 4 ) as θ is convex we can assert that there exists C ą 0 such that

|θpy 1 q ´θpy 2 q| ď C |y 1 ´y2 | p1 `|y 1 | `|y 2 |q (2.4)
for all y 1 , y 2 P R.

To every u P H 1 0 and every µ P YpΩ; X q we associate the integral functional φ µ : S µ puq R defined by φ µ pU q :" ż s0,T rˆΩˆX ξ `t, x, U pt, x, ξq, 9 U pt, x, ξq ˘dt b dµpx, ξq `żΩˆX θpU pT, x, ξqqdµpx, ξq (2.5) and we consider the functional Φ µ : H 1 0 R given by Φ µ puq :" inf ! φ µ pU q : U P S µ puq ) .

(2.6)

Remark 2.3. By Proposition 2.1, the evaluation map E is continuous, and so the map pt, x, ξq ξpt, x, U pt, x, ξq, 9 U pt, x, ξqq is pBps0, T rq b BpΩq b BpX q, BpRqq-measurable. It follows that the integral functional φ µ is well defined.

Remark 2.4. From (C 2 ) and (C 4 ) it is easy to see that

α ż s0,T rˆΩˆX ´|pU, 9 U q| 2 ´1¯d t b dµpx, ξqďφ µ pU qďβ ż s0,T rˆΩˆX ´|pU, 9 U q| 2 `1¯d t b dµpx, ξq `c ż ΩˆX `1 `|U pT, xq| 2 ˘dµpx, ξq
for all U P S µ puq with u P H 1 0 , where α, β, c Ps0, 8r are given by (C 2 ) and (C 4 ).

Let us set F :" Φ µ : µ P YpΩ; X q ( . Remark 2.5. From Remark 2.4 it is easy to see that every F P F satisfies the following growth conditions:

α ż s0,T rˆΩ `|pu, 9 uq| 2 ´1˘d t b dx ď F puq ď β ż s0,T rˆΩ `|pu, 9 uq| 2 `1˘d t b dx `c ż Ω `1 `|upT, xq| 2 ˘dx
for all u P H 1 0 . Let us recall the definition of Γ-convergence (see [START_REF] Dal | An introduction to Γ-convergence, volume 8 of Progress in Nonlinear Differential Equations and their Applications[END_REF][START_REF] Braides | Chapter 2 a handbook of Γ-convergence[END_REF] for more details).

Definition 2.6 (Γ-convergence). Given tF ε u ε Ă F and F P F we say that tF ε u ε Γ-converges to F at u P H 1 0 as ε 0, and we write ´Γlim ε 0

F ε q ¯puq " F puq, if the following two assertions hold: Γ-lower bound at u: for every every

tu ε u ε Ă H 1 0 , if u ε H 1 Ýá u then lim ε 0 F ε pu ε q ě F puq; Γ-upper bound at u: there exists tv ε u ε Ă H 1 0 such that v ε H 1
Ýá u and lim ε 0

F ε pv ε q ď F puq.
When ´Γlim ε 0

F ε q ¯puq " F puq for all u P H 1 0 we say that tF ε u ε Γ-converges to F as ε 0, and we write Γ-lim

ε 0 F ε " F.
The main result of the paper is the following. (This result is a new and more complete version of [MV02, Theorem 3] in the specific case of the one-dimensional distributional derivative.) Theorem 2.7. Under (C 1 ), (C 2 ), (C 3 ) and (C 4 ) the map µ Φ µ is continuous from YpΩ; X q endowed with the narrow convergence to F endowed with the Γ-convergence, i.e., for every µ P YpΩ; X q and every tµ

ε u ε Ă YpΩ; X q, if µ ε nar -µ then Γ-lim ε 0 Φ µε " Φ µ .

2.2.

L N ´ x P Ω : dpf ε pxq, f pxqq ą η ( ¯" 0 for all η ą 0,
where d is the metric defined by (2.2), then

Γ-lim ε 0 Φ fε " Φ f .
2.3. Non-diffusive reaction differential equations. For each ε ą 0, let ψ ε : r0, T s RN ˆR R be a pBpr0, T sq b BpR N q b BpRq, BpRqq-measurable function such that: (A 1 ) ψ ε pt, x, ¨q P C 1 pRq for dt b dx-a.a. pt, xq Ps0, T rˆΩ; (A 2 ) ψ ε pt, x, ¨q is convex for dt b dx-a.a. pt, xq Ps0, T rˆΩ; (A 3 ) there exist c 1 , C 1 ą 0 (which does not depend on ε) such that c 1 p|s| 2 ´1q ď ψ ε p¨, ¨, sq ď C 1 p|s| 2 `1q for all s P R; (A 4 ) there exists δ : r0, 8r r0, 8r (which does not depend on ε) with lim r 0 δprq " 0 such that |ψ ε pt 1 , ¨, ¨q ´ψε pt 2 , ¨, ¨q| ď δp|t 1 ´t2 |q for all t 1 , t 2 P r0, T s; (A 5 ) there exists L ą 0 (which does not depend on ε) such that ˇˇˇB ψ ε Bs p¨, ¨, s 1 q ´Bψ ε Bs p¨, ¨, s 2 q ˇˇˇď L|s 1 ´s2 | for all s 1 , s 2 P R, and consider the non-diffusive reaction differential equation defined in H 1 ps0, T r; L 2 pΩ; Rqq by Remark 2.10. Under (A 1 ), (A 2 ), (A 3 ), (A 4 ) and (A 5 ), it is well known that every (R ε ) has a unique solution u ε P H 0 1 (for a proof, see for instance [AHM18, Theorem 2.3]).

pR ε q $ ' & ' %
The following proposition makes clear the link between non-diffusive reaction differential equations and integral functionals of type (2.7).

Proposition 2.11. If (A 1 ), (A 2 ), (A 3 ), (A 4 ) and (A 5 ) hold then, for every ε ą 0, the nondiffusive reaction differential equation (R ε ) admits a unique solution u ε P H 1 0 . Moreover, there exist α, β ą 0 and γ : r0, 8r r0, 8r with lim r 0 γprq " 0 such that every u ε minimizes the functional Φ fε : H 1 0 R given by (Note that Ψ ε pt, ¨q is convex by (A 2 ).) Then (R ε ) is equivalent to the following differential inclusion defined in H 1 0 by pD ε q ´9 u ε pt, ¨q P BΨ ε pt, u ε pt, ¨qq for dt-a.a. t Ps0, T r, where BΨ ε pt, ¨q denotes the subdifferential of Ψ ε pt, ¨q. According to Fenchel's extremality condition (see [ABM14, Proposition 9.5.1]), we see that pD ε q is equivalent to Ψ ε pt, u ε pt, ¨qq `Ψε pt, ´9 u ε pt, ¨qq `xu ε pt, ¨q, 9 u ε pt, ¨qy " 0 for dt-a.a. it follows that pD ε q is equivalent to F ε pu ε q " 0. (More precisely, we see that pD ε q is equivalent to F ε pu ε q ď 0.) Thus, u ε solves (R ε ) if and only if u ε minimizes F ε .

Φ fε puq " ż s0,T rˆΩ f ε pxq `t,
On the other hand, as ψ ε satisfies (A 3 ) and (A 4 ), so is its Legendre-Fenchel conjugate ψ ε with other constants ĉ1 , Ĉ1 ą 0, instead of c 1 , C 1 ą 0, but with the same function δ. Thus f ε pxq P M α,β,γ for all x P Ω (with suitable constants α, β ą 0 and a suitable function γ) where, for each pt, y, s, 9 sq P r0, T s ˆRN ˆR ˆR, f ε pxqpt, y, s, 9 sq is given by (2.8). Finally, for each u P H 1 0 , as up0, ¨q " 0, by using Legendre-Fenchel's calculus, it is easy to see that

F ε puq " ż s0,T rˆΩ f ε pxqpt, x, upx, tq, 9 upt, xqqdx b dt `żΩ θpupT, xqqdx,
where θ : R r0, 8r is given by (2.9), which means that F ε " Φ fε .

As a direct consequence of Proposition 2.11 and Corollary 2.8 we obtain the following result.

Corollary 2.12. Assume that the assumptions of Proposition 2.11 are satisfied and, for each ε ą 0, let u ε P H 1 0 be the solution of (R ε ). If there exists µ P YpΩ; X q such that dx b δ fε pxq nar µ, where f ε : Ω M α,β,γ is given by (2.8), then any cluster point of tu ε u ε with respect to the weak convergence in H 1 0 is a minimizer of Φ µ : H 1 0 R defined by (2.6) with θ : R r0, 8r given by (2.9).

Thus, the results obtained in [START_REF] Luisa | Memory effect phenomena and Γ-convergence[END_REF][START_REF] Toader | Memory effects and Γ-convergence: a time dependent case[END_REF] concerning non-local effects induced by homogenization can be seen as a particular case of Corollary 2.12 (see §5.2 for more details). Note that if we further assume that (A 6 ) for every ε ą 0, ψ ε does not depend on t, and there exists ψ : R N ˆR R such that for dx-a.e. x P R N , ψpx, ¨q P C 1 pRq, ψpx, ¨q is convex and ψpx, sq " lim ε 0 ψ ε px, sq for all s P R, then the non-local effects disappear. More precisely, we have the following proposition which states a stability result for sequences of non-diffusive reaction differential equations of type (R ε ).

Proposition 2.13. Under the assumptions of Proposition 2.11, if moreover (A 6 ) holds then (up to a subsequence) the sequence tu ε u ε of solutions of (R ε ) weakly converges in H 1 0 to the solution u of the following non-diffusive reaction differential equation: Proof of Proposition 2.13. For each ε ą 0, let Ψ ε , Ψ : L 2 pΩ; Rq R be defined by

$ ' ' ' & ' ' ' % Ψ ε pvq :" ż Ω ψ ε px, vpxqqdx Ψpvq :" ż Ω ψpx, vpxqqdx.
(Note that, by (A 6 ), ψ ε does not depend on t.) We first prove that M-lim

ε 0 Ψ ε " Ψ, (2.10)
where the symbol "M-lim " denotes the Mosco-limit 1 . For any ε ą 0 and any λ ą 0, let Ψ λ ε , Ψ λ : L 2 pΩ; Rq R be the λ-Moreau-Yosida approximation of Ψ ε and Ψ. By using an

1 Let X be a Hilbert space, let h : X R and, for each ε ą 0, let h ε : X R. We say that th ε u ε Mosco-converges to h, and we write M-lim ε 0 h ε " h, if the following two assertions hold: M-lower bound: for every v P X and every tv ε u ε Ă X, if v ε Ýá v then lim ε 0 h ε pv ε q ě hpvq; M-upper bound: for every v P X there exists tw ε u ε Ă X such that w ε v and lim ε 0 h ε pw ε q ď hpvq.

interchange argument of infimum and integral (see [START_REF] Anza | Interchange of infimum and integral[END_REF]) it easily seen that

$ ' ' ' & ' ' ' % Ψ λ ε pvq " ż Ω ψ λ ε px, vpxqqdx Ψ λ pvq " ż Ω ψ λ px, vpxqqdx, (2.11)
where ψ λ ε and ψ λ denotes the λ-Moreau-Yosida approximation of ψ ε and ψ with respect to the second variable. From (A 1 ), (A 2 ) and (A 6 ) we deduce that Ψ and Ψ ε are closed, convex and proper. Consequently, to prove (2.10) it is equivalent to show that lim ε 0 Ψ λ ε pvq " Ψ λ pvq for all λ ą 0 and all v P L 2 pΩ; Rq (2.12)

(see [Att84, Theorem 3.26]). Taking (2.11) and (A 3 ) into account, by Lebesgue's dominated convergence theorem, we see that for establishing (2.10) it suffices to prove that lim ε 0 ψ λ ε px, sq " ψ λ px, sq for dx-a.e. x P R N , all λ ą 0 and all s P R.

(2.13) But (A 1 ), (A 2 ) and (A 6 ) implies that for dx-a.e. x P R N , ψ ε px, ¨q and ψ are closed, convex and proper, and so (2.13) holds if and only if M-lim ε 0 ψ ε px, ¨q " ψpx, ¨q for dx-a.e. x P R N .

(2.14) Fix x P R N and fix any ε ą 0. From (A 1 ) and (A 3 ) we deduce that there exists C ą 0 (which does not depend on ε) such that

|ψ ε px, s 1 q ´ψε px, s 2 q| ď C|s 1 ´s2 |p1 `|s 1 | `|s 2 |q (2.15)
for all s 1 , s 2 P R. Let s P R and let ts ε u ε Ă R. On one hand, by (A 6 ) we have lim

ε 0 ψ ε px, sq " ψpx, sq, (2.16)
which gives the M-upper bound of (2.14). On the other hand, from (2.15) it follows that

ψ ε px, s ε q ď ψ ε px, sq ´C|s ε ´s|p1 `|s ε | `|s|q
for all ε ą 0, and so, by using (2.16), if s ε s in R then lim ε 0 ψ ε px, s ε q ě ψpx, sq, which gives the M-lower bound of (2.14). Thus (2.10) is proved.

For each ε ą 0, let G ε , G : L 2 ps0, T r; L 2 pΩ; Rqq R be defined by $ ' ' ' ' & ' ' ' ' % G ε pvq :" ż T 0 Ψ ε pvpt, ¨qqdt Gpvq :" ż T 0 Ψpvpt, ¨qqdt.
From (2.10) we deduce that M-lim

ε 0 G ε " G (2.17) (see [ABM14, Lemma 17.4.8]). Hence (see [ABM14, Theorem 17.4.3]) M-lim ε 0 G ε " G ˚, (2.18)
with G ε , G ˚: L 2 ps0, T r; L 2 pΩ; Rqq R denoting the Legendre-Fenchel conjugate of G ε and G, where, by using Legendre-Fenchel's calculus, one has

$ ' ' ' ' & ' ' ' ' % G ε pvq " ż T 0 Ψ ε pvpt, ¨qqdt G ˚pvq " ż T 0 Ψ ˚pvpt, ¨qqdt with Ψ ε , Ψ ˚: L 2 pΩ; Rq R denoting the Legendre-Fenchel conjugate of Ψ ε and Ψ. For each ε ą 0, let F ε , F : H 1 0 R be defined by $ ' ' & ' ' % F ε pvq :" G ε pvq `Gε p´9 vq `1 2 }vpT, ¨q} 2 L 2 pΩ;Rq F pvq :" Gpvq `G˚p ´9 vq `1 2 }vpT, ¨q} 2 L 2 pΩ;Rq .
Arguing as in the proof of Proposition 2.11 we can assert that:

for each ε ą 0, v solves (R ε ) if and only if F ε pvq ď 0; (2.19) v solves (R) if and only if F pvq ď 0;
(2.20)

Let tu ε u ε Ă H 1 0 be a sequence of solutions of (R ε ). Taking Remark 2.5 into account, we see that (up to a subsequence) there exists u P H 1 0 such that u ε

H 1 Ýá u, i.e., u ε L 2
Ýá u and

9 u ε L 2
Ýá 9 u. Then, by (2.19), one has F ε pu ε q ď 0 for all ε ą 0. Letting ε 0 we obtain lim ε 0

F ε pu ε q ď 0. (2.21)
On the other hand, from (2.17) and (2.18) we deduce that:

Gpuq ď lim ε 0 G ε pu ε q; (2.22) G ˚p´9 uq ď lim ε 0 G ε p´9 u ε q. (2.23)
As, for any ε ą 0, u ε P H 1 0 and u P H 1 0 we have u ε p0, ¨q " 0 and up0, ¨q " 0, and so u ε pT, ¨q " ş T 0 9 u ε pt, ¨qdt and upT, ¨q "

ş T 0 9 upt, ¨qdt. Since 9 u ε L 2 Ýá 9 u it follows that u ε pT, ¨q L 2
Ýá upT, ¨q, and consequently }upT, ¨q} L 2 pΩ;Rq ď lim ε 0 }u ε pT, ¨q} L 2 pΩ;Rq .

(2.24) From (2.22), (2.23) and (2.24) we deduce that F puq ď lim ε 0 F ε pu ε q. Hence F puq ď 0 by (2.21). From (2.20) we conclude that u solves (R).

Proof of the continuity theorem

In this section we prove Theorem 2.7.

Proof of Theorem 2.7. Let µ P YpΩ; X q and let tµ ε u ε Ă YpΩ; X q be such that µ ε nar µ. According to the left inequality in Remark 2.5, to prove that Γ-lim

ε 0 Φ µε " Φ µ (3.1)
it is equivalent to show that every subsequence of tΦ µε u ε contains a further subsequence which Γ-converges to Φ µ (see [DM93, Proposition 8.17]). Let us consider a subsequence that we still denote by tΦ µε u ε . Taking the left inequality in Remark 2.5 into account, by compactness, we can assert that, up to subsequence, tΦ µε u ε Γ-converges (see [DM93, Corollary 8.12]), i.e., Γ-lim ε 0 Φ µε " Ψ. So, to establish (3.1) it is sufficient to prove that Ψpuq " Φ µ puq for all u P H 1 0 .

(3.2)

For this, we only need to show that for each u P H 1 0 the following two assertions hold: (G 1 ) there exists a subsequence tΦ µ σpεq u ε and tv

ε u ε Ă H 1 0 such that v ε H 1 Ýá u and lim ε 0 Φ µ σpεq pv ε q ď Φ µ puq; (G 2 ) for every tu ε u ε Ă H 1 0 , if u ε H 1 Ýá u then lim ε 0 Φ µ σpεq pu ε q ě Φ µ puq,
where tΦ µ σpεq u ε is given by (G 1 ).

Indeed, let u P H 1 0 . According to Definition 2.6, from (G 1 ) and (G 2 ) we see that tΦ µ σpεq u ε Γ-converges to Φ µ at u as ε 0, i.e., pΓ-lim ε 0 Φ µ σpεq qpuq " Φ µ puq. On the other hand, as tΦ µ σpεq u ε is a subsequence of tΦ µεq u ε we have Γ-lim ε 0 Φ µ σpεq " Γ-lim ε 0 Φ µε , and so Γ-lim ε 0 Φ µ σpεq " Ψ. Hence, in particular, pΓ-lim ε 0 Φ µ σpεq qpuq " Ψpuq. It follows that Ψpuq " Φ µ puq, which proves (3.2).

Let us fix u P H 1 0 . Proof of (G 1 ). By using the direct method of the calculus of variations, we can assert that there exists U P S µ puq such that U is a minimizer of Φ µ puq, i.e., Φ µ puq " φ µ pU q. We then have:

upt, xq " ż X U pt, x, ξqdµ x pξq; (3.3) 9 upt, xq " ż X 9 U pt, x, ξqdµ x pξq, (3.4)
where tµ x u xPΩ denotes the desintegration of µ on Ω ˆX . As C 1 c ps0, T r; C c pΩ ˆX qq is strongly dense in H 1 0,µ , for each δ Ps0, 1s there exists

U δ P C 1 c ps0, T r; C c pΩ ˆX qq such that: ż s0,T rˆΩˆX |U δ pt, x, ξq ´U pt, x, ξq| 2 dt b dµpx, ξq ă δ; (3.5) ż s0,T rˆΩˆX ˇˇ9 U δ pt, x, ξq ´9 U pt, x, ξq ˇˇ2 dt b dµpx, ξq ă δ. (3.6)
In particular, one has:

U δ L 2 U ; (3.7) 9 U δ L 2 9 U . (3.8)
For each δ Ps0, 1s and each ε ą 0, let u δ,ε P H 1 0 be given by

u δ,ε pt, xq :" ż X U δ pt, x, ξqdµ ε x pξq, (3.9)
where tµ ε x u xPΩ denotes the desintegration of µ ε on Ω ˆX . Then

9 u δ,ε pt, xq :" ż X 9 U δ pt, x,
|u δ,ε pt, xq| 2 dt b dx " ż s0,T rˆΩ ˇˇˇż X U δ pt, x, ξqdµ ε x pξq ˇˇˇ2 dt b dx ď ż s0,T rˆΩˆX |U δ pt, x, ξq| 2 dt b dµ ε px, ξq ď 1 `żs0,TrˆΩˆX |U δ pt, x, ξq| 2 dt b dµpx, ξq.
But, taking (3.5) and the left inequality in Remark 2.4 into account and recalling that U P S µ puq, we have

ż s0,T rˆΩˆX |U δ pt, x, ξq| 2 dt b dµpx, ξq ď 2δ `2 ż s0,T rˆΩˆX |U pt, x, ξq| 2 dt b dµpx, ξq ď 2 ˆ1 `1 α φ µ pU q `T |Ω| ˙,
and so, setting R :" 3 `1 α φ µ pU q `T |Ω|, it follows that For each δ Ps0, 1s, consider the mapping ε λ δ pεq given by λ δ pεq :"

" η δ if ε ą η δ ε if ε ď η δ .
(

u δ,λ δ pεq H 1 Ýá ε 0 ż X U δ p¨, ¨, ξqdµ x pξq H 1 Ýá δ 0 u. (3.20)
According to (3.15) and the fact that in B R p0q the weak convergence is metrizable, we can rewrite (3.20) as follows:

d w -lim δ 0 d w -lim ε 0 u δ,λ δ pεq " u, (3.21)
where d w denotes the metric associated with the weak convergence in B R p0q.

On the other hand, by using similar arguments together with (2.1) and (2.4), we can assert that lim δ 0 lim ε 0 φ µ λ δ pεq pU δ q " φ µ pU q.

(3.22)

According to (3.21) and (3.22), by diagonalization there exists a mapping ε δ ε with δ 0 as ε 0 such that:

d w -lim ε 0
u δε,λ δε pεq " u, i.e., u δε,λ δε pεq

H 1 Ýá u; (3.23) lim ε 0
φ µ λ δε pεq pU δε q " φ µ pU q.

(3.24) Since φ µ pU q " Φ µ puq and, for each ε ą 0, φ µ λ δε pεq pU δε q ě Φ µ λ δε pεq pu δε,λ δε pεq q because U δε P C 1 c ps0, T r; C c pΩ ˆX qq Ă H 1 0,µ λ δε pεq and ş X U δε p¨, ¨, ξqdµ λ δεpεq x pξq " u δε,λ δεpεq , from (3.24) we deduce that lim ε 0 Φ µ λ δε pεq pu δε,λ δεpεq q ď Φ µ puq.

(3.25)

Taking (3.23) and (3.25) into account and setting σpεq :" λ δε pεq and v ε :" u δε,λ δεpεq we have σpεq 0 as ε 0 and:

v ε H 1 Ýá u; lim ε 0 Φ µ σpεq pv ε q ď Φ µ puq,
and the proof of (G 1 ) is complete.

Proof of (G 2 ). Let tu ε u ε Ă H 1 0 be such that u ε H 1 Ýá u.
Without loss of generality we can assume that lim ε 0 Φ µ σpεq pu ε q " lim ε 0 Φ µ σpεq pu ε q ă 8, and so sup εą0 Φ µ σpεq pu ε q ă 8.

(3.26)

Fix any ε ą 0. By using the direct method of the calculus of variations, we can assert that there exists U ε P S µ σpεq pu ε q such that U ε is a minimizer of Φ µ σpεq pu ε q, i.e., Φ µ σpεq pu ε q " φ µ σpεq pU ε q (3.27) with φ µ σpεq : S µ σpεq pu ε q R given by (2.5). From (3.26) and the left inequality in (C 2 ) we deduce that

sup εą0 › › ›pU ε , 9 U ε q › › › L 2
dtbµ σpεq ps0,T rˆΩˆX ;R 2 q ă 8.

(3.28)

Let g ε :s0, T rˆΩ ˆX s0, T rˆΩ ˆX ˆR2 be defined by g ε pt, x, ξq :" pt, x, ξ, U ε pt, x, ξq, 9 U ε pt, x, ξqq and let ν ε :" g 7 ε dt b µ σpεq . It is clear that ν ε P Yps0, T rˆΩ; X ˆR2 q whose projection measure is dt b dx. We claim that tν ε u ε is tight (see Definition A.2). Indeed, given η ą 0, as tµ σpεq u ε is tight, there exists a compact set K Ă X such that and let λ ε :" h 7 ε µ σpεq . It is clear that λ ε P YpΩ; X ˆRq. In the same manner we can establish that tλ ε u ε is tight. From Prokhorov's compactness theorem (see Theorem A.3) we deduce that there exist ν P Yps0, T rˆΩ; X ˆR2 q and λ P YpΩ; X ˆRq such that, up to a subsequence, one has:

ν ε nar -ν;
(3.31)

λ ε nar -λ. (3.32)
Let π s0,T rˆΩˆX :s0, T rˆΩ ˆX ˆR2 s0, T rˆΩ ˆX (resp. π ΩˆX : Ω ˆX ˆR Ω ˆX ) be the canonical projection from s0, T rˆΩ ˆX ˆR2 (resp. Ω ˆX ˆR) to s0, T rˆΩ ˆX (resp. Ω ˆX ). From (3.31) and (3.32) it is easy to see that:

π 7 s0,T rˆΩˆX ν " w-lim ε 0 π 7 s0,T rˆΩˆX ν ε " dt b µ; π 7 ΩˆX λ " w-lim ε 0
π 7 s0,T rˆΩˆX λ ε " µ, where "w-lim" denotes the weak limit associated with the weak σpC 1 b , C b q topology (see [ABM14, Definition 4.2.2]). By using desintegration's theorem (see Theorem A.5) it follows that:

ν " dt b µ b ν t,x,ξ ; (3.33) λ " µ b λ x,ξ , (3.34)
where tν t,x,ξ u pt,x,ξqPs0,T rˆΩˆX and tλ x,ξ u px,ξqPΩˆX are families of probability measures on R 2 and R respectively. We will need the following lemma whose proof is given below. (3.37)

Then V " 9 U , U P S µ puq and W T " U pT, ¨, ¨q.

According to (3.27) and the definitions of tν ε u ε and tλ ε u ε , we see that

lim ε 0 Φ µ σpεq pu ε q " lim ε 0 φ µ σpεq pU ε q ě lim ε 0 ż s0,T rˆΩˆX ξ `t, x, U ε pt, x,
" φ µ pU q ě Φ µ puq,
and (G 2 ) is proved. This finishes the proof of Theorem 2.7 (the proof of Lemma 3.1 is given below).

Proof of Lemma 3.1. The proof is divided into six steps.

Step 1: We prove that: Step 4: We prove that U p0, x, ξq " 0 for µ-a.a. px, ξq P Ω ˆX .

U P L 2 ps0,
(3.52)

Let ϕ P C 1 pr0, T sq be such that ϕpT q " 0 and ϕp0q " 0 and let ψ P C c pΩ ˆX q. Then, we have ż for all ψ P C c pΩ ˆX q, and (3.52) follows because ϕp0q " 0.

Step 5: We prove that W T px, ξq " U pT, x, ξq for µ-a.a. px, ξq P Ω ˆX .

(3.55)

Fix any ψ P C c pΩ ˆX q. According to (3.37) and the definition of tλ ε u ε , by using desintegration's theorem (see Theorem A.5) and Theorem A.4(ii) together with (3.32), we see that ż

ΩˆX W T px, ξqψpx, ξqdµpx, ξq " ż ΩˆX ˆżR s T dλ x,ξ ps T q ˙ψpx, ξqdµpx, ξq " ż ΩˆX ˆR s T ψpx, ξqdλpx, ξ, s T q " lim ε 0 ż ΩˆX ˆR s T ψpx, ξqdλ ε px, ξ, s T q " lim ε 0 ż ΩˆX U ε pT, x, ξqψpx, ξqdµ σpεq px, ξq. (3.56)
But, for any ε ą 0, U ε p0, x, ξq " 0 for µ σpεq -a.a. px, ξq P Ω ˆX because U ε P H 1 0,µ σpεq , and so ż for all ψ P C c pΩ ˆX q, and (3.55) follows.

Step 6: end of the proof. From (3.43), (3.44), (3.51) and (3.52) we deduce that U P H 1 0,µ , and so U P S µ puq by (3.48), which completes the proof of Lemma 3.1.

Stochastic homogenization

4.1. A non-local stochastic homogenization theorem. Let pΣ, A, Pq be a probability space and let pT z q zPZ N be a group of P-preserving transformations on pΣ, Aq, i.e., ' (mesurability) T z is A-measurable for all z P Z N ; ' (group property) T z oT z 1 " T z`z 1 and T ´z " T ´1 z for all z, z 1 P Z N ; ' (mass invariance) PpT z pAqq " PpAq for all z P Z N and all A P A. Let f : Σ ˆRN ˆr0, T s ˆRN ˆR ˆR R be a pA b BpR N ˆr0, T s ˆRN ˆR ˆRq, BpRqqmeasurable function such that f pω, xq :" f pω, x, ¨, ¨, ¨, ¨q P X for all ω P Σ and all x P R N . It is easily seen that the map

f : Σ X :" X R N ω f pω, ¨q
is a pA, Bq-random variable with B :" b R N BpX q. We futhermore assume that the random variable f is covariant with respect to the dynamical system pΣ, A, P, pT z q zPZ N q, i.e., f pT z ω, ¨q " f pω, ¨`zq (4.1) for all z P Z N and for P-a.a. ω P Σ. For each z P Z N , let τ z : X X denote the shift map on X, i.e., τ z pwq :" wp¨`zq (4.2) for all w P X. Then, f 7 P, i.e., the law of f , is invariant under the group pτ z q zPZ N , i.e., τ 7 z f 7 P " f 7 P (4.3) for all z P Z N . We finally assume that the dynamical system pΣ, A, P, pT z q zPZ N q, or equivalently pX, B, f 7 P, pτ z q zPZ N q, is ergodic, i.e., for each A P A, if T z pAq " A for all z P Z N then PpAq " 0 or PpAq " 1, or equivalently, for each B P B, if τ z pBq " B for all z P Z N then f 7 PpBq " 0 or f 7 PpBq " 1. (4.4)

For each ω P Σ and each ε ą 0, we consider F ε pωq : H 1 0 R given by

F ε pωqpuq:"Φ f pω, ε q puq" ż s0,T rˆΩ f ´ω, x ε ¯pt, x, upt, xq, 9 upt, xqqdt b dx `żΩ θpupT, xqqdx. (4.5)
For each ε ą 0, let µ ε : Σ YpΩ; X q be the random Young measure defined by

µ ε pωq :" dx b δ f pω, x ε q . (4.6)
Let Y :"s0, 1r N be the unit cell, let f Y : Σ X Y be defined by f Y pωqpyq " f pω, yq and, for each ω P Σ, let f 7 Y pωqdy be the image by f Y pωq of the Lebesgue measure restricted to Y that we denote by dy. Let µ P YpΩ; X q be the Young measure defined by with µ f P YpΩ; X q given by (4.7).

µ f :" dx b dE f 7 Y dy , (
Proof of Proposition 4.1. It is sufficient to prove that there exists p Σ P A with Pp p Σq " 1 such that for every ω P p Σ, one has lim

ε 0 ż ΩˆX 1 A pxqϕpξqdµ ε pωqpx, ξq " ż ΩˆX 1 A pxqϕpξqdx b dE f 7 Y dy pξq
for all A P BpΩq and all ϕ P D with D a dense subset of C c pX q, see [START_REF] Valadier | Young measures[END_REF][START_REF] Valadier | A course on Young measures[END_REF]. Let A P BpΩq and let ϕ P D. Taking (4.8) into account, from the additive ergodic theorem [ABM14, Theorem 12.4.1] (see also [START_REF] Chabi | Ergodic theory and application to nonconvex homogenization[END_REF]), we can assert that there exists N ϕ P F with PpN ϕ q " 0 such that for every ω P ΣzN ϕ , one has

ϕ ´f ´ω, ε ¯¯L 1 Ýá ż X ϕpξqdE f 7 Y dy pξq (4.9)
On the other hand, by (4.6) we see that for every ω P Σ, one has

ż ΩˆX 1 A pxqϕpξqdµ ε pωqpx, ξq " ż Ω 1 A pxqϕ ´f ´ω, x ε ¯¯dx for all ε ą 0. (4.10)
Set p Σ :" Y ψPD pΣzN ψ q (where N ψ corresponds to N ϕ with ϕ " ψ). Then Pp p Σq " 0 and, by using (4.9), from (4.10) we deduce that for every ω P p Σ, one has lim

ε 0 ż ΩˆX 1 A pxqϕpξqdµ ε pωqpx, ξq " lim ε 0 ż Ω 1 A pxqϕ ´f ´ω, x ε ¯¯dx " ż Ω 1 A pxq ˆżX ϕpξqdE f 7 Y dy pξq ˙dx " ż ΩˆX 1 A pxqϕpξqdx b dE f 7
Y dy pξq, and the proof is complete.

Let F hom : H 1 0 R be defined by

F hom puq :" inf "ż ΣˆY Φ f pω,yq pvp¨, ¨, ω, yqqdPpωq b dy : v P S hom puq * , (4.11)
where S hom puq is given by S hom puq:"

" v P H 1 0,dxbPbdy : ż ΣˆY vpt, x,
F ε pωq " Φ µ f with Φ µ f : H 1 0 R
given by (2.6). (Since µ f does not depend on ω, so is for Φ µ f .) Taking (4.7) and (4.8) into account we see that for any u P H 1 0 , one has

S µ f puq" ! U P H 1 0,µ f : ż X U pt, x, ξqdE f 7 Y dy pξq"upt, xqfor dtbdx-a.a. pt, xq Ps0, T rˆΩ ) " ! U P H 1 0,µ f : ż ΣˆY U pt, x, f pω, yqqdPpωqbdy"upt, xqfor dtbdx-a.a. pt, xq Ps0, T rˆΩ ) and Φ µ f puq " inf ! φ µ f pU q : U P S µ f puq ) with φ µ f pU q " ż s0,T rˆΩˆX ξ `t, x, U pt, x, ξq, 9 U pt, x, ξq ˘dt b dµ f px, ξq `żΩˆX θpU pT, x, ξqqdµ f px, ξq " ż s0,T rˆΩˆX ξ `t, x, U pt, x, ξq, 9 U pt, x, ξq ˘dt b dx b dE f 7 Y dy pξq `żΩˆX θpU pT, x, ξqqdx b dE f 7 Y dy pξq " ż s0,T rˆΩˆΣˆY f pω, yq ´t, x, U pt, x, f pω, yqq, 9 U pt, x, f pω, yqq ¯dt b dx b dPpωq b dy `żΩˆΣˆY θ `U pT, x, f pω, yqq ˘dx b dPpωq b dy. (4.13)
So, it remains to prove that Φ µ f " F hom , i.e., for every u P H 1 0 , one has: Φ µ f puq ě F hom puq;

(4.14) Φ µ f puq ď F hom puq.

(4.15)

Let us fix u P H 1 0 . Proof of (4.14). Let U P S µ f puq. Then: and so, from (4.16) we deduce that v P L 2 ps0, T r; L 2 dxbPbdy pΩ ˆΣ ˆY ; Rqq. In the same way, by using (4.17) instead of (4.16) we obtain 9 v P L 2 ps0, T r; L 2 dxbPbdy pΩ ˆΣ ˆY ; Rqq. It follows that v P S µ f puq.

U P L 2 ps0, T r; L 2 µ f pΩ ˆΣ ˆY ; Rqq; (4.16
We have thus proved that for each U P S µ f puq one has v P S hom puq. According to (4.11)-(4.12) and (4.13) we conclude that φ µ f pU q " ż ΣˆY Φ f pω,yq pvp¨, ¨, ω, yqqdPpωq b dy ě F hom puq for all U P S µ f puq, and (4.14) follows.

Proof of (4.15). Consider g : Σ ˆY X ˆΣ ˆY defined by gpω, yq :" pf pω, yq, ω, yq and set λ :" g 7 pP b dyq.

(4.20)

Then π 7 X λ " dE f 7 Y dy with π X : X ˆΣ ˆY
X denoting the canonical projection from X ˆΣ ˆY to X . So, from desintegration's theorem (see Theorem A.5) we can assert that there exists a familiy pλ ξ q ξPX of probability measures on Σ ˆY such that

λ " dE f 7 Y dy b λ ξ .
(4.21)

Fix any v P S hom puq and set U pt, x, ξq :"

ż ΣˆY vpt, x, ω, yqdλ ξ pω, yq. (4.22)
(In particular, U pT, x, ξq :" ş ΣˆY vpT, x, ω, yqdλ ξ pω, yq.) Then, it is easy to see that 9 U pt, x, ξq :"

ż ΣˆY 9 vpt, x, ω, yqdλ ξ pω, yq. (4.23)
We claim that U P S µ f puq. Indeed, as v P S hom puq we have: and so, from (4.24) we deduce that U P L 2 ps0, T r; L 2 µ f pΩ ˆX ; Rqq. In the same way, by using (4.25) instead of (4.24) we obtain 9 U P L 2 ps0, T r; L 2 µ f pΩ ˆX ; Rqq, and the claim is proved. On the other hand, taking (4.22) and (4.23) into account and using Jensen's inequality, we have We have thus prove that for each v P S hom puq there exists U P S µ f puq such that Let D : Σ - -R N be a pA, BpR N qq-measurable multifunction such that for every ω P Σ, the set Dpωq is countable and without cluster point and let N : Σ ˆBpR N q N Y t8u be defined by N pω, Bq :"

v P L 2 `s0,
φ µ f pU q " ż s0,T rˆΩˆX ξ ˆt, x,
φ µ f pU q ď ż ΣˆY Φ f pω,
ÿ zPDpωq δ z pBq " cardpDpωq X Bq,
where δ z denotes the Dirac measure at the point z P R N . (Note that for each ω P Σ, N pω, ¨q is a counting measure.) From now on, we assume that tN p¨, Bqu BPBpR N q is a Poisson point process with intensity λ ą 0, i.e., ' for every bounded set B P BpR N q and every k P N, one has P `rN p¨, Bq " ks ˘" |B| k λ k e ´λ|B| k! (4.28)

where rN p¨, Bq " ks :" tω P Σ : N pω, Bq " ku and |B| denotes the Lebesgue measure of B; ' for every disjoint and bounded sets A, B P BpR N q, N p¨, Aq and N p¨, Bq are independant. Fix r ą 0 and g, h P X and consider the pA, B :" b R N BpX qq-measurable function f p : Σ X :" X R N given by f p pω, xq :"

# g if x P Y zPDpωq B r pzq h otherwise " h `pg ´hq min ! 1, N pω, B r pxqq ) " " g if N pω, B r pxqq ě 1 h if N pω, B r pxqq " 0. (4.29)
Then, f p is covariant with respect to the dynamical system pΣ, A, P, pT z q zPZ N q, i.e., (4.1) hols with f " f p , and the law of f p is invariant under the group pτ z q zPZ N , i.e., (4.3) holds with f " f p and pτ z q zPZ N given by (4.2), and moreover the dynamical system pX, B, f 7 p P, pτ z q zPZ N q is ergodic, i.e., (4.4) holds with f " f p (see [START_REF] Messaoudi | Stochastic homogenization of nonconvex integral functionals[END_REF]§5]). (The ergodic dynamical system pX, B, f 7 p P, pτ z q zPZ N q models environments whose heterogeneities are independently distributed with a frequency λ.) For each ω P Σ and each ε ą 0, we consider F p ε pωq : H 1 0 R given by (4.5) with f " f p , i.e., Let G H : H 1 0 R be the inf-convolution of G and H, i.e., G Hpuq :" inf ! Gpu 1 q `Hpu 2 q : u 1 , u 2 P H 1 0 and u 1 `u2 " u

F
) .

The following result is a consequence of Theorem 4.2. Then, as in §2.3, for P-a.e. ω P Σ and every ε ą 0, (R ω ε ) has a unique solution u ε pω, ¨, ¨q P H 1 0 . Moreover, arguing as in the proof of Proposition 2.11, we can assert that there exist α, β ą 0 and γ : r0, 8r r0, 8r with lim r 0 γprq " 0 such that for each ω P Σ, u ε pω, ¨, ¨q minimizes the functional Φ f ψ pω, ε q : H 1 0 R given by and θ : R r0, 8r given by (2.9), where ψ ˚denotes the Legendre-Fenchel conjugate of ψ with respect to the fourth variable. From (H 0 ) we deduce that f ψ is covariant with respect to the dynamical system pΣ, A, P, pT z q zPZ N q, i.e., (4.1) hols with f " f ψ , and the law of f ψ is invariant under the group pτ z q zPZ N , i.e., (4.3) holds with f " f ψ and pτ z q zPZ N given by (4.2), and moreover the dynamical system pX, B, f 7 ψ P, pτ z q zPZ N q is ergodic, i.e., (4.4) holds with f " f ψ . On the other hand, from (H 2 ) we see that f ψ is strictly convex with respect to ps, 9 sq. Hence F hom defined by (4.11)-(4.12), with f " f ψ and θ given by (2.9), has a unique minimizer u P H 1 0 . So, as a direct consequence of Theorem 4.2 we obtain the following result. Corollary 4.5. Assume that (H 0 ), (H 1 ), (H 2 ), (H 3 ), (H 4 ) and (H 5 ) are satisfied. Then, for P-a.e. ω P Σ, one has u ε pω, ¨, ¨q H 1 Ýá u.

Φ f ψ pω, ε q puq " ż s0,T rˆΩ f ψ ´ω, x ε ¯`t,
In for all s P R; (E 4 ) there exists δ : r0, 8r r0, 8r with lim r 0 δprq " 0 such that for each i P t1, 2u, one has |ψ i pt 1 , ¨, ¨q ´ψi pt 2 , ¨, ¨q| ď δp|t 1 ´t2 |q for all t 1 , t 2 P r0, T s; (E 5 ) there exists L ą 0 such that for each i P t1, 2u, one has ˇˇˇB ψ i Bs p¨, ¨, s 1 q ´Bψ i Bs p¨, ¨, s 2 q ˇˇˇď L|s 1 ´s2 | for all s 1 , s 2 P R, and let ψ p : Σ ˆr0, T s ˆRN ˆR R be defined by ψ p pω, t, x, sq :"

# ψ 1 pt, x, sq if x P Y zPDpωq B r pzq ψ 2 pt, x, sq otherwise " ψ 2 pt, x, sq `pψ 1 pt, x, sq ´ψ2 pt, x, sqq min ! 1, N pω, B r pxqq ) " " ψ 1 pt, x, sq if N pω, B r pxqq ě 1 ψ 2 pt, x, sq if N pω, B r pxqq " 0.
Then, (H 0 ), (H 1 ), (H 2 ), (H 3 ), (H 4 ) and (H 5 ) are satisfied with ψ " ψ p .

For each ω P Σ and each ε ą 0, let F ψp ε pωq : H 1 0 R given by (4.5) with f " f ψp and θ given by (2.9), i.e., F ψp ε pωqpuq :" Φ f ψp pω, ε q puq" ż s0,T rˆΩ

Let Ψ 1 Ψ 2 : H 1 0 R be the inf-convolution of Ψ 1 and Ψ 2 , i.e., Ψ 1 Ψ 2 puq :" inf ! Ψ 1 pu 1 q `Ψ2 pu 2 q : u 1 , u 2 P H 1 0 and u 1 `u2 " u ) .
The following result is a consequence of Theorems 4.3 and 4.4 and Corollary 4.5.

Theorem 4.6. Assume that (E 1 ), (E 2 ), (E 3 ), (E 4 ) and (E 5 ) are satisfied. Then, the following four assertions hold.

(i) For P-a.e. ω P Σ one has

Γ-lim ε 0 F ψp ε pωq " Ψ 1 Ψ 2 .
(ii) For P-a.e. ω P Σ and every ε ą 0, let upω, ¨, ¨q P H 1 0 be the unique solution of (R ω ε ). Then u ε pω, ¨, ¨q H 1 Ýá u, where u P H 1 0 is the unique minimizer of Ψ 1 Ψ 2 . (iii) The function u minimizes Ψ 1 Ψ 2 if and only if there exist v, w P H 1 0 such that u " v `w with v and w satisfying the following differential system: Proof of Theorem 4.6. We only need to prove (iv). For this, it suffices to show that pv, wq P H 1 0 ˆH1 0 satisfies the differential system pS ψp q-pBS ψp q if and only pu 1 , u 2 q " p v σ , w 1´σ q P H 1 0 ˆH1 0 satisfies the integro-differential equations above. First of all, by integrating over st, T r the first equation in pS ψp q, we see that for dt b dx-a.e. pt, xq Ps0, T rˆΩ, and setting u 1 " v σ we obtain the first integro-differential equation. In the same manner, by using the second equations in pS ψp q and pBS ψp q and by setting u 1 " w 1´σ , we obtain the second integro-differential equation. Conversely, setting v " σu 1 and w " p1 ´σqu 2 , by derivating each integro-differential equation with respect to the time variable (resp. by letting t " T in each integro-differential equation), we obtain pS ψp q (resp. pBS ψp q). Appendix A. Young measures Let pV, dq be a polish space, i.e., pV, dq is a separable and complete metric space, let k ě 1 be an integer and let O Ă R k be a bounded domain. Let YpO; Vq be the class of Young measures on O ˆV and let Cth b pO; Vq be the space of all bounded Carathéodory integrands on O ˆV. Definition A.2 (tightness). We say that tµ ε u ε Ă YpO; Vq is tight if for every η ą 0, there exists a compact set K Ă V such that sup εą0 µ ε `O ˆpVzKq ˘ă η.

pS ψp q $ ' ' & ' '
A proof of the following compactness result can be found in [START_REF] Valadier | Young measures[END_REF]Theorem 11] (see also [START_REF] Valadier | A course on Young measures[END_REF]Theorem 7 and Comments 1), 2) and 3)]).

Theorem A.3 (Prokhorov's compactness theorem). If tµ ε u Ă YpO; Vq is tight then there exists µ P YpO; Vq such that, up to a subsequence, µ ε nar µ.

For a proof of the following theorem we also refer to [START_REF] Valadier | Young measures[END_REF][START_REF] Valadier | A course on Young measures[END_REF].

Theorem A.4 (lower semicontinuity, continuity). Let ψ : O ˆV r0, 8s be a BpOqˆBpVqmeasurable function such that ψp¨, yq is lower semicontinuous for all y P O and let Ψ : YpO; Vq r0, 8s be defined by Ψpµ ε q " Ψpµq.

Let pW, δq be a polish space. A proof of the following theorem can be found in [START_REF] Tortrat | Désintégration d'une probabilité, statistiques exhaustives[END_REF] (see also [ABM14, Theorem 4.2.4]).

Theorem A.5 (desintegration). Let µ be a Borel measure on V ˆW and let ν :" π 7 V µ where π V : V ˆW V denotes the canonical projection from V ˆW to V. Then, there exists a unique (up to the equality a.e.) family tµ ξ u ξPV of probability measures on W such that for every ψ P L 1 µ pV ˆWq, one has: The family tµ ξ u ξPV of probability measures on W is called the desintegration of µ on the product V ˆW. To summarize it, we write µ " ν b µ ξ .

U

  T r; L 2 µ pΩ ˆX ; Rq ˘: U p0, ¨, ¨q " pt, x, ξqdµ x pξq " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ * ,

  Bs pt, x, u ε pt, xqq for dt b dx-a.a. pt, xq Ps0, T rˆΩ u ε p0, ¨q " 0.

  , xqq for dt b dx-a.a. pt, xq Ps0, T rˆΩ up0, ¨q " 0.

ż

  s0,T rˆΩ |u δ,ε pt, xq| 2 dt b dx ď R for all δ Ps0, 1s and all ε Ps0, η δ s.(3.13)In the same manner, by considering (3.10), (3.12) and (3.6) instead of (3.9), (3.11) and (3.5), we obtain ż s0,T rˆΩ | 9 u δ,ε pt, xq| 2 dt b dx ď R for all δ Ps0, 1s and all ε Ps0, η δ s.(3.14)

  28) and Markov's inequality we can assert that there exists R ą 0 such that supεą0 dt b µ σpεq pE ε,R q ă η 2 (3.30) with E ε,R :" ! pt, x, ξq Ps0, T rˆΩ ˆX : pU ε pt, x, ξq, 9 U ε pt, x, ξqq P R 2 zB R p0q) ,where B R p0q Ă R 2 denotes the 2-dimensional closed ball centered at the origin with radius R. On the other hand, we have ν ε ps0, T rˆΩ ˆpX ˆR2 qzpK ˆBR p0qqq ď dt b µ σpεq ps0, T rˆΩ ˆpX zKqq `dt b µ σpεq pE ε,R q " T µ σpεq pΩ ˆpX zKqq `dt b µ σpεq pE ε,R q for all ε ą 0, and consequently sup εą0 ν ε ps0, T rˆΩ ˆpX ˆR2 qzpK ˆBR p0qqq ă η, which proves the claim. Let h ε : Ω ˆX Ω ˆX ˆR defined by h ε px, ξq :" px, ξ, U ε pT, x, ξqq

ΩˆXU

  ε pT, x, ξqψpx, ξqdµ σpεq px, ξq " pt, x, ξqdt ˙ψpx, ξqdµ σpεq px, ξq" ż s0,T rˆΩˆX 9 U ε pt, x,ξqψpx, ξqdt b dµ σpεq px, ξq. (3.57) Moreover, according to (3.51) and (3.36) and the definition of tν ε u ε , by using the same arguments as in above with (3.31) instead of (3.32), we have lim pt, x, ξqψpx, ξqdt b dµ σpεq px, ξq " lim ε 0 ż s0,T rˆΩˆX ˆR2 9 sψpx, ξqdν ε pt, x, ξ, s, 9 sq " ż s0,T rˆΩˆX ˆR2 9 sψpx, ξqdνpt, x, ξ, s, 9 sq " ż s0,T rˆΩˆX ˆżR 2 9 sdν t,x,ξ ps, 9 sq ˙ψpx, ξqdt b dµpx, ξq " ż s0,T rˆΩˆX 9 U pt, x, ξqψpx, ξqdt b dµpx, ξq. (3.58) From (3.52) we see that ż s0,T rˆΩˆX 9 U pt, x, ξqψpx, ξqdt b dµpx, ξq " x, ξqdt ˙ψpx, ξqdµpx, ξq " ż ΩˆX U pT, x, ξqψpx, ξqdµpx, ξq, (3.59) and consequently, from (3.56), (3.57), (3.58) and (3.59), we conclude that ż ΩˆX W T px, ξqψpx, ξqdµpx, ξq " ż ΩˆX U pT, x, ξqψpx, ξqdµpx, ξq

) 9 U

 9 P L 2 ps0, T r; L 2 µ f pΩ ˆΣ ˆY ; Rqq; (4.17) U p0, ¨, ¨q " 0; (4.18) ż ΣˆY U pt, x, f pω, yqqdPpωq b dy " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ. (4.19) Set vpt, x, ω, yq :" U pt, x, f pω, yqq. By (4.18) we have vp0, ¨, ¨, ¨q " 0 and from (4.19) it is clear that ż ΣˆY vpt, x, ω, yqdPpωq b dy " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ. On the other hand, taking (4.8) into account, we see that ż s0,T rˆΩˆΣˆY |vpt, x, ω, yq| 2 dt b dx b dPpωq b dy " ż s0,T rˆΩ ˆżΣˆY |U pt, x, f pω, yqq| 2 dPpωq b dy ˙dt b dx " ż s0,T r ˆżΩˆX |U pt, x, ξq| 2 dx b dE f 7 Y dy pξq ˙dt " ż s0,T rˆΩˆX |U pt, x, ξq| 2 dt b dµ f px, ξq,

  T r; L 2 dxbPbdy pΩ ˆΣ ˆY ; Rq ˘; (4.24) 9 v P L 2 `s0, T r; L 2 dxbPbdy pΩ ˆΣ ˆY ; Rq ˘; (4.25) vp0, ¨, ¨, ¨q " 0; (4.26) ż ΣˆY vpt, x, ω, yqdPpωq b dy " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ. (4.27) Firstly, by (4.26) it is clear that U p0, ¨, ¨q " 0. Secondly, by using (4.8), (4.22), (4.21), (4.20) and (4.27) we see thatż ΣˆY U pt, x, f pω, yqqdPpωq b dy " ż X U pt, x, ξqdE f 7 Y dy pξq " ż X ˆżΣˆY vpt, x, ω, yqdλ ξ pω, yq ˙dE f 7 Y dy pξq " ż X ˆΣˆY vpt, x,ω, yqdλpξ, ω, yq " ż ΣˆY vpt, x, ω, yqdPpωq b dy " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ. Thirdly, taking (4.22) into account and using Jensen's inequality, we have ż s0,T rˆΩˆX |U pt, x, ξq| 2 dt b dµ f px, ξq " ż s0,T rˆΩˆX ˇˇˇż ΣˆY vpt, x, ω, yqdλ ξ pω, yq ˇˇˇ2 dt b dµ f px, ξq ď ż s0,T rˆΩˆX ˆΣˆY |vpt, x, ω, yq| 2 dt b dµ f px, ξq b dλ ξ pω, yq. But, by (4.7), (4.21) and (4.20) we have ż s0,T rˆΩˆX ˆΣˆY |vpt, x, ω, yq| 2 dt b dµ f px, ξq b dλ ξ pω, yq " ż s0,T rˆΩˆX ˆΣˆY |vpt, x, ω, yq| 2 dt b dx b dE f 7 Y dy pξq b dλ ξ pω, yq " ż s0,T rˆΩˆX ˆΣˆY |vpt, x, ω, yq| 2 dt b dx b dλpξ, ω, yq " ż s0,T rˆΩˆΣˆY |vpt, x, ω, yq| 2 dt b dx b dPpωq b dy hence ż s0,T rˆΩˆX |U pt, x, ξq| 2 dt b dµ f px, ξq ď ż s0,T rˆΩˆΣˆY |vpt, x, ω, yq| 2 dt b dx b dPpωq b dy,

  ż ΣˆY vpt, x, ω, yqdλ ξ pω, yq, ż ΣˆY 9 vpt, x, ω, yqdλ ξ pω, yq ˙dt b dµ f px, ξq `żΩˆX θ ˆżΣˆY vpT, x, ω, yqdλ ξ pω, yq ˙dµ f px, ξq ď ż s0,T rˆΩˆX ˆΣˆY ξ pt, x, vpt, x, ω, yq, 9 vpt, x, ω, yqq dt b dµ f px, ξq b dλ ξ pω, yq `żΩˆXˆΣˆY θ pvpT, x, ω, yqq dµ f px, ξq b dλ ξ pω, yq, hence, by using (4.7), (4.21), we obtain φ µ f pU q ď ż s0,T rˆΩˆX ˆΣˆY ξ pt, x, vpt, x, ω, yq, 9 vpt, x, ω, yqq dt b dx b dE f 7 Y dy pξq b dλ ξ pω, yq `żΩˆXˆΣˆY θ pvpT, x, ω, yqq dx b dE f 7 Y dy pξq b dλ ξ pω, yq ď ż s0,T rˆΩˆX ˆΣˆY ξ pt, x, vpt, x, ω, yq, 9 vpt, x, ω, yqq dt b dx b dλpξ, ω, yq `żΩˆXˆΣˆY θ pvpT, x, ω, yqq dx b dλpξ, ω, yq, and consequently, from (4.20) we conclude that φ µ f pU q ď ż s0,T rˆΩˆΣˆY f pω, yq pt, x, vpt, x, ω, yq, 9 vpt, x, ω, yqq dt b dx b dPpωq b dy `żΩˆΣˆY θ pvpT, x, ω, yqq dx b dPpωq b dy " ż ΣˆY Φ f pω,yq pvp¨, ¨, ω, yqqdPpωq b dy.

  Definition A.1 (narrow convergence). Let tµ ε u ε Ă YpO; Vq and let µ P YpO; Vq. We say that tµ ε u ε narrow converges to µ as ε 0, and we write µ ε nar ε py, ξq " ż OˆV ψpy, ξqdµpy, ξq for all ψ P Cth b pO; Vq.

µ then lim ε 0

 0 The functional Ψ is lower semicontinuous, i.e., for every tµ ε u ε Ă YpO; Vq and every µ P YpO; Vq, if µ ε nar -Ψpµ ε q ě Ψpµq.(ii) If ψp¨, yq is continuous for all y P O then for every tµ ε u ε Ă YpO; Vq and every µ P YpO; Vq such that µ ε nar ξq|dµ ε py, ξq ˙" 0, one has lim ε 0

(i) the function ξ ż W

 ż ψpξ, ζqdµ ξ pζq is ν-measurable; ξ pζq ˙dνpξq,

  From local to non-local functionals. It is easy to see that if µ " dx b δ f pxq with In what follows, when µ " dx b δ f pxq we use the notation "Φ f " instead of "Φ µ ". As a direct consequence of Theorem 2.7 we have Corollary 2.8. Assume that (C 1 ), (C 2 ), (C 3 ) and (C 4 ) hold and, for each ε ą 0, consider a pBpΩq, BpX qq-measurable function f ε : Ω X . If there exists µ P YpΩ; X q such that dx b δ fε pxq

	1 0 one has θpupT, xqqdx θpupT, xqqdx In corollary 2.8, although the functionals Φ fε are local, i.e., are integral functionals, the φ µ pU q " ż s0,T rˆΩ f pxq `t, x, upt, xq, 9 upt, xq ˘dt b dx `żΩ for all U P S µ puq. Hence Φ µ puq " ż s0,T rˆΩ f pxq `t, x, upt, xq, 9 upt, xq ˘dt b dx `żΩ (2.7) for all u P H 1 -µ then Γ-lim ε 0 Φ fε " Φ µ . Γ-limit Φ µ is in general non-local (here infimum of integral functionals). However, since dx b δ fε pxq nar -dx b δ f pxq if and only if tf ε u ε converges in measure to f (see [ABM14, Proposition 4.3.8] and also [Val90, Proposition 6]), we have Corollary 2.9. Assume that (C 1 ), (C 2 ), (C 3 ) and (C 4 ) hold and, for each ε ą 0, consider a pBpΩq, BpX qq-measurable function f ε : Ω X . If there exists a pBpΩq, BpX qq-measurable function f : Ω X such that tf ε u ε converges in measure to f , i.e., 0 . nar lim ε 0

f : Ω X a pBpΩq, BpX qq-measurable function, then for any u P H

  where ψ ε denotes the Legendre-Fenchel conjugate of ψ ε with respect to the third variable.

		x, upt, xq, 9 upt, xq ˘dt b dx	`żΩ	θpupT, xqqdx
	with f ε : Ω	M α,β,γ (here X " M α,β,γ ) and θ : R	r0, 8r respectively defined by
		f ε pxqpt, y, s, 9 sq :" ψ ε pt, x, sq `ψε pt, x, ´9 sq	(2.8)
	and	θpsq :"	1 2	|s| 2 ,		(2.9)
	Proof of Proposition 2.11. For ε ą 0 and define Ψ ε : r0, T s ˆL2 pΩ; Rq	R by
		ż
		Ψ ε pt, vq :"

Ω ψ ε pt, x, vpxqqdx.

  Lemma 3.1. For each pt, x, ξq Ps0, T rˆΩ ˆX , let us denote by U pt, x, ξq, V pt, x, ξq and W T px, ξq the 1 th moments of ν t,x,ξ and λ x,ξ , i.e.,

	ż
	U pt, x, ξq :"	sdν t,x,ξ ps, 9 sq;	(3.35)
	R 2
	ż
	V pt, x, ξq :"	9 sdν t,x,ξ ps, 9 sq;	(3.36)
	R 2

W T px, ξq :" ż R s T dλ x,ξ ps T q.

  ξq, 9 U ε pt, x, ξq ˘dt b dµ σpεq px, ξq ΩˆX ˆR θps T qdλ ε px, ξ, s T q.

	where U P S µ puq by Lemma 3.1. Consequently, combining (3.39) with (3.41) and (3.40) with
	(3.42), and using (3.38) we conclude that
		lim ε 0	Φ µ σpεq pu ε q ě	s0,T rˆΩˆX ż	ξ ´t, x, U pt, x, ξq, 9 U pt, x, ξq ¯dt b dµpx, ξq
						ż `żΩˆX	θ pU pT, x, ξqq dµpx, ξq
				`lim		θpU ε pT, x, ξqqdµ σpεq px, ξq
					ε 0	ΩˆX
						ż
			" lim			ξpt, x, Sqdν ε pt, x, ξ, Sq
				ε 0	s0,T rˆΩˆX ˆR2
							ż
				`lim		(3.38)
					ε 0
	But, recalling (3.31) and (3.32), by Theorem A.4(i) have:
		ż						ż
	lim		ξpt, x, Sqdν ε pt, x, ξ, Sq ě	ξpt, x, Sqdνpt, x, ξ, Sq;	(3.39)
	ε 0	s0,T rˆΩˆX ˆR2					s0,T rˆΩˆX ˆR2
	lim
	ε 0
	ż s0,T rˆΩˆX ˆR2 ξpt, x, Sqdνpt, x, ξ, Sq "	ż	s0,T rˆΩˆX ˆżR 2	ξpt, x, Sqdν t,x,ξ pSq ˙dt b dµpx, ξq
			ě	ż			ξ ˆt, x,	ż	Sdν t,x,ξ pSq ˙dt b dµpx, ξq
					s0,T rˆΩˆX	R 2
			"	ż			ξ ˆt, x,	ż	sdν t,x,ξ ps, 9 sq,	ż	9 sdν t,x,ξ ps, 9 sq ˙dt b dµpx, ξq
					s0,T rˆΩˆX	R 2	R 2
			"	ż			ξ ´t, x, U pt, x, ξq, 9 U pt, x, ξq ¯dt b dµpx, ξq;	(3.41)
					s0,T rˆΩˆX
		ż ΩˆX ˆR θps T qdλpx, ξ, s T q "	ż ΩˆX ˆżR	θps T qdλ x,ξ ps T q ˙dµpx, ξq
								ě	ż	θ	ˆżR	s T dλ x,ξ ps T q ˙dµpx, ξq
								ΩˆX
								ż
								"	θ pW T px, ξqq dµpx, ξq
								ΩˆX
								ż
								"	θ pU pT, x, ξqq dµpx, ξq,	(3.42)
								ΩˆX

ż ΩˆX ˆR θps T qdλ ε px, ξ, s T q ě ż ΩˆX ˆR θps T qdλpx, ξ, s T q.

(3.40) Moreover, by using (C 1 ), (3.33) and (3.34), Jensen's inequality and Lemma 3.1, we obtain the following two inequalities:

  , ξqdµ x pξq " upt, xq for dt b dx-a.a. pt, xq Ps0, T rˆΩ, (3.48)where tµ x u xPΩ denotes the desintegration of µ on Ω ˆX . Fix any ϕ P C 8 ϕpt, xqs and recalling that U ε P S µ σpεq for any ε ą 0, we see that According to (3.36) and the definition of tν ε u ε , by using desintegration's theorem (see Theorem A.5) and

	Step 2: We prove that Theorem A.4(ii) together with (3.31), we see that
		ż
	ż s0,T rˆΩˆX ϕptqφpxqV pt, x, ξqdt b dµpx, ξq "	ż	s0,T rˆΩˆX ϕptqφpxq	ˆżR 2	9 sdν t,x,ξ ps, 9 sq ˙dt b dµpx, ξq
	u ε	H 1 Ýá u we have		"	ż	s0,T rˆΩˆX ˆR2	c ps0, T rˆΩq. As sqdνpt, x, ξ, s, 9 ψ 1 pt, x, ξ, s, 9 sq
						ż
		ż s0,T rˆΩ	ϕpt, xqupt, xqdt b dx " lim ε 0 " lim ε 0 s0,T rˆΩˆX ˆR2 ż s0,T rˆΩ ψ 1 pt, x, ξ, s, 9 ϕpt, xqu ε pt, xqdt b dx. sqdν ε pt, x, ξ, s, 9 sq ż	(3.49)
	ż s0,T rˆΩˆX s0,T rˆΩˆX ψ 1 pt, x, ξ, U ε pt, x, ξq, 9 ˇˇˇż R 2 sdν t,x,ξ ps, 9 U ε pt, x, ξqqdt b dµ σpεq px, ξq ż s0,T rˆΩˆX ϕptqφpxq 9 U with sq ˇˇˇ2 dt b dµpx, ξq s0,T rˆΩˆX ˆR2 ϕpt, xqu ε pt, xqdt b dx " Setting ψpt, x, ξ, s, 9 " lim ε 0 sq :" ż s0,T rˆΩ " lim ε 0 s0,T rˆΩ ε 0 ż ż ψpt, x, ξ, s, 9 sqdνpt, x, ξ, s, 9 sq (3.35) instead of (3.36), it follows that
	ď |U pt, x, ξq| 2 dt b dµpx, ξq ď ż s0,T rˆΩˆX ˆżR 2 ż ż ż " s0,T rˆΩˆX ˆR2 ϕpt, xqsdνpt, x, ξ, s, 9 |s| 2 dν t,x,ξ ps, 9 sq ˙dt b dµpx, ξq, |s| 2 dνpt, x, ξ, s, 9 sq (3.46) sq. (3.50) Combining (3.49) with (3.50) and by using desintegration's theorem (see Theorem A.5) and ż and so ż (3.35), we obtain s0,T rˆΩˆX ϕptqφpxqV pt, x, ξqdt b dµpx, ξq " ´lim ε 0 s0,T rˆΩˆX ψ 2 pt, x, ξ, U ε pt, x, ξq, 9 ż U " ´lim s0,T rˆΩˆX ˆR2 ε 0 ψ 2 pt, x, ξ, s, 9 sqdν ε pt, x, ξ, s, 9 sq
	s0,T rˆΩˆX because of (3.33). On the other hand, according to (3.31) and the definition of tν ε u ε , From s0,T rˆΩˆX ˆR2 Theorem A.4(i) we have ż s0,T rˆΩˆX ˆR2 |s| 2 dνpt, x, ξ, s, 9 sq ď lim ε 0 ż s0,T rˆΩˆX ˆR2 |s| 2 dν ε pt, x, ξ, s, 9 sq " lim ε 0 ż s0,T rˆΩˆX ż ż s0,T rˆΩ ϕpt, xqupt, xqdt b dx " ż s0,T rˆΩˆX ϕpt, xq sdν t,x,ξ ps, 9 sq ˙dt b dµpx, ξq " ´żs0,TrˆΩˆXˆR 2 ψ 2 pt, x, ξ, s, 9 sqdνpt, x, ξ, s, 9 sq ˆżR 2 " ż s0,T rˆΩˆX " ϕ 1 ptqφpxqsdνpt, x, ξ, s, 9 sq ´żs0,TrˆΩˆXˆR 2 ϕpt, xqU pt, x, ξqdt b dµpx, ξq " ż ϕpt, xq ˆżX " ´żs0,TrˆΩˆX ϕ 1 ptqφpxq ˆżR 2 sdν t,x,ξ ps, 9 sq ˙dt b dµpx, ξq U pt, x, ξqdµ x pξq ˙dt b dx, which proves (3.48). |U and so s0,T rˆΩ " ´żs0,TrˆΩˆX ϕ 1 ptqφpxqU pt, x, ξqdt b dµpx, ξq,
	Step 3: We prove that which establishes (3.51).	s0,T rˆΩˆX ˆR2	|s| 2 dνpt, x, ξ, s, 9 sq ă 8	(3.47)
							V " 9 U .	(3.51)

T r; L 2 µ pΩ ˆX ; Rqq; (3.43) V P L 2 ps0, T r; L 2 µ pΩ ˆX ; Rqq; (3.44) W T P L 2 µ pΩ ˆX ; Rq. (3.45) Taking (3.35) into account and using Jensen's inequality, we see that ż s0,T rˆΩˆX |U pt, x, ξq| 2 dt b dµpx, ξq " ε pt, x, ξq| 2 dt b dµ σpεq px, ξq, by (3.28). From (3.46) and (3.46) we get (3.43). In the same manner, by considering (3.36) instead of (3.35) we obtain (3.44). By using (3.37), (3.34), (3.32) and the definition of tλ ε u ε instead of (3.35), (3.33), (3.28) and the definition of tν ε u ε , a similar calculation gives (3.45). X U pt, xϕpt, xqu ε pt, xqdt b dx " ż s0,T rˆΩˆX ψpt, x, ξ, U ε pt, x, ξq, 9 U ε pt, x, ξqqdt b dµ σpεq px, ξq, and so, according to the definition of tν ε u ε , one has ż s0,T rˆΩ ϕpt, xqu ε pt, xqdt b dx " ż s0,T rˆΩˆX ˆR2 ψpt, x, ξ, s, 9 sqdν ε pt, x, ξ, s, 9 sq for all ε ą 0. Taking (3.31) into account, from Theorem A.4(ii) we deduce that lim Fix any ϕ P C 1 c ps0, T rq and any φ P C c pΩq. Set ψ 1 pt, x, ξ, s, 9 sq :" ϕptqφpxq 9 s. ε pt, x, ξqdt b dµ σpεq px, ξq.

But, for any ε ą 0, as ϕ P C 1 c ps0, T rq one has

ż s0,T rˆΩˆX ϕptqφpxq 9 U ε pt, x, ξqdt b dµ ε px, ξq " ´żs0,TrˆΩˆX ϕ 1 ptqφpxqU ε pt, x, ξqdt b dµ σpεq px, ξq,

and so, setting ψ 2 pt, x, ξ, s, 9 sq :" ϕ 1 ptqφpxqs and using the same arguments as in above ε pt, x, ξqqdt b dµ σpεq px, ξq

  yq pvp¨, ¨, ω, yqqdPpωq b dy,

	which implies (4.15).
	This finishes the proof of Theorem 4.2.
	4.2. Non-local stochastic homogenization in the setting of a Poisson point pro-
	cess.

  p ε pωqpuq :" Φ fppω, ε q puq "

						ż s0,T rˆΩ	f p ´ω,	x ε	¯pt, x, upt, xq, 9 upt, xqqdt b dx	`żΩ	θpupT, xqqdx.
	Set σ :" PprN p¨, B r p0qq ě 1sq " 1 ´e´λ|Brp0q| and consider the integral functionals G, H :
	H 1 0	R given by:
	Gpuq :" σ	"ż s0,T rˆΩ	g ˆt, x,	upt, xq σ	,	9 upt, xq σ	˙dt b dx	`żΩ	θ	ˆupT, xq σ	˙dx		;	(4.30)
	Hpuq :" p1 ´σq	"ż s0,T rˆΩ	h ˆt, x,	upt, xq 1 ´σ ,	9 upt, xq 1 ´σ ˙dt b dx	`żΩ	θ	ˆupT, xq 1 ´σ ˙dx		. (4.31)

  , taking ξpt, xq " aptqbpxq with aptq " p t T q n and b P C 8 c pΩq, integrating by part with respect to the time variable and letting n 8, from (I) we obtain (BS g,h ). 4.3. Non-local stochastic homogenization of non-diffusive reaction equations. Let ψ : Σ ˆr0, T s ˆRN ˆR R be a pA b Bpr0, T sq b BpR N q b BpRq, BpRqq-measurable function such that: (H 0 ) ψpT z ω, ¨, x, ¨q " ψpω, ¨, x `z, ¨q for P-a.a. ω P Σ, all x P R N and all z P Z N ; (H 1 ) ψpω, t, x, ¨q P C 1 pRq for P b dt b dx-a.a. pω, t, xq P Σˆs0, T rˆR N ; (H 2 ) ψpω, t, x, ¨q is strictly convex for P b dt b dx-a.a. pω, t, xq P Σˆs0, T rˆR N ; (H 3 ) there exist c 1 , C 1 ą 0 such that for P-a.e. ω P Σ, one hasc 1 p|s| 2 ´1q ď ψpω, ¨, ¨, sq ď C 1 p|s| 2 `1qfor all s P R; (H 4 ) there exists δ : r0, 8r r0, 8r with lim r 0 δprq " 0 such that for P b dx-a.e. pω, xq P Σ ˆRN , one has |ψpω, t 1 , ¨, ¨q ´ψpω, t 2 , ¨, ¨q| ď δp|t 1 ´t2 |q for all t 1 , t 2 P r0, T s; (H 5 ) there exists L ą 0 such that for P-a.e. ω P Σ, one has ˇˇˇB ψ ¨, s 2 q ˇˇˇď L|s 1 ´s2 | for all s 1 , s 2 P R, and, for each ω P Σ, consider the non-diffusive reaction differential equation defined in H 1 ps0, T r; L 2 pΩ; Rqq by

	Theorem 4.3. For P-a.e. ω P Σ one has Γ-lim % ε 0 F p ' ε pωq " G H. Bs pω, ¨, ¨, s 1 q ´Bψ Bs pω, ¨, pR ω ε q $ ' & ´Bu ε Bt pω, t, xq " Bψ Bs ´ω, t, x ε , u
	Proof of Theorem 4.3. Applying Theorem 4.2 with f " f p we deduce that for P-a.e.
	ω P Σ, one has
	Γ-lim ε 0	F p ε pωq " F hom
	with F hom : H 1 0 F hom " G H, i.e., for every u P H 1 R given by (4.11)-(4.12) with f " f p . So, it remains to prove that 0 , one has:
		F hom puq ě G Hpuq;	(4.32)
		F hom puq ď G Hpuq.	(4.33)
	Let us fix u P H 1 0 and let us set:
			!	)
	" N p¨, B r p¨qq ě 1 ‰	:"	pω, yq P Σ ˆY : N pω, B r pyqq ě 1	;
	" N p¨, B r p¨qq " 0 ‰	:"

!

pω, yq P Σ ˆY : N pω, B r pyqq " 0

) .

handε pω, t, xq ¯for dt b dx-a.a. pt, xq Ps0, T rˆΩ u ε pω, 0, ¨q " 0.

  the setting of a Poisson point process, we can precise the (non-local) equations satisfied by u making clear the non-local effects induced by homogenization of stochastic non-difusive reaction differential equations of type (R ω ε ) (see Theorem 4.6(iv)). 4.3.1. The setting of a Poisson point process. Let ψ 1 , ψ 2 : r0, T s ˆRN ˆR R be two pBpr0, T sq b BpR N q b BpRq, BpRqq-measurable functions such that: (E 1 ) for each i P t1, 2u, ψ i pt, x, ¨q, ψ i pt, x, ¨q P C 1 pRq for dt b dx-a.a. pt, xq P Σˆs0, T rˆR N ; (E 2 ) for each i P t1, 2u, ψ i pt, x, ¨q is strictly convex for dt b dx-a.a. pt, xq P Σˆs0, T rˆR N ; (E 3 ) there exist c 1 , C 1 ą 0 such that for each i P t1, 2u, one has c 1 p|s| 2 ´1q ď ψ i p¨, ¨, sq ď C 1 p|s| 2 `1q

  % dt b dx-a.e. in s0, T rˆΩ subjected to the time-boundary conditions:where ψ 1 (resp. ψ 2 ) denotes the Legendre-Fenchel conjugate of ψ 1 (resp. ψ 2 ) with respect to the third variable.(iv) The function u minimizes Ψ 1 Ψ 2 if and only if there exist u 1 , u 2 P H 1 0 such that u " σu 1 `p1 ´σqu 2 with u 1 and u 2 being the unique solutions of the following integro-differential equations: Bs pτ, x, u 1 pτ, xqq ˙dτ ˙dt b dx-a.e. in s0, T rˆΩ Bs pτ, x, u 2 pτ, xqq ˙dτ ˙dt b dx-a.e. in s0, T rˆΩ u 2 p0, ¨q " 0.

	$ & ´Bu 1 Bt	pt, xq"	Bψ 1 Bs	ˆt, x,	u 1 pT, xq σ	`ż T t Bψ 1 Bs	ˆτ, x,	Bψ 1
	%	u 1 p0, ¨q " 0
	$ & ´Bu 2 Bt	pt, xq"	Bψ 2 Bs	ˆt, x,	u 2 pT, xq 1 ´σ	`ż T t Bψ 2 Bs	ˆτ, x,	Bψ 2
	%
			Bψ 1 Bs	ˆt, x,	vpt, xq σ ˙´B Bt	ˆBψ B 9 1 s	ˆt, x,	´9 vpt, xq σ ˙˙" 0 dt b dx-a.e. in s0, T rˆΩ
			Bψ 2 Bs ´σ ˙˙" 0 pBS ψp q ˆt, x, wpt, xq 1 ´σ ˙´B Bt ˆBψ 2 B 9 s ˆt, x, ´9 wpt, xq 1 $ ' ' ' ' ' vpT, xq σ ´σ Bψ 1 s ˆT, x, ´9 vpT, xq σ ˙" 0 dx-a.e. in Ω B 9 ' & ' wpT, xq 1 ´σ ´p1 ´σq Bψ 2 B 9 s ˆT, x, ´9 wpT, xq 1 ´σ ˙" 0 dx-a.e. in Ω ' ' ' ' vp0, ¨q " 0
					' %	wp0, ¨q " 0,

  Bs˘´1 denotes the inverse of the subdifferential of ψ 1 with respect to s, and so from (4.42) we deduce that Moreover, according to the first equation in pBS ψp q, one has

		Bψ B 9 1 s	ˆt, x,	´9 vpt, xq σ	˙" Bψ B 9 1 s	ˆT, x,	´9 vpT, xq σ	˙`ż T t	Bψ 1 Bs	ˆτ, x,	vpτ, xq σ	˙dτ.	(4.42)
	But	Bψ B 9 1
	´1 σ	Bv Bt	pt, xq "		Bψ 1 Bs	ˆt, x,	Bψ B 9 1 s	ˆT, x,	´9 vpT, xq σ	˙`ż T t	Bψ 1 Bs	ˆt, x	Bψ 1 Bs	ˆτ, x,	σ vpτ, xq	˙˙dτ	˙.
												vpT, xq σ 2 "	Bψ B 9 1 s	ˆT, x,	σ ´9 vpT, xq	˙.
	It follows that
			´1 σ	Bv Bt	pt, xq "	Bψ 1 Bs	ˆt, x,	vpT, xq σ 2	`ż T t	Bψ 1 Bs	ˆt, x	Bψ 1 Bs	ˆτ, x,	σ vpτ, xq	˙˙dτ	˙.

s pt, x, ¨q " `Bψ 1

Bs

˘´1

pt, x, ¨q (see [ABM14, Theorem 9.5.1]), where `Bψ 1

Key words and phrases. Γ-convergence, Stochastic homogenization, Non-local functionals, Non-diffusive reaction differential equations, Non-local effects, Young measures.

Taking (4.29) into account, as rN p¨, B r p¨qq ě 1s and rN p¨, B r p¨qq " 0s are disjoints and Σ ˆY " rN p¨, B r p¨qq ě 1s Y rN p¨, B r p¨qq " 0s, for any v P S hom puq, one has Φ g pvp¨, ¨, ω, yqqdPpωq b dy ě Gpu 1 q; ż rN p¨,Brp¨qq"0s Φ h pvp¨, ¨, ω, yqqdPpωq b dy ě Hpu 2 q.

Consequently, one has

for all v P S hom puq, and (4.32) follows by taking (4.34) into account.

Conversely, for any u 1 , u 2 P H 1 0 such that u 1 `u2 " u, set vp¨, ¨, ω, yq :"

Then, v P S hom puq and, from (4.12), (4.30), (4.31), (4.34) and (4.35), we see that Gpu 1 q `P b dyprN p¨, B r p¨qq " 0sq 1 1 ´σ Hpu 2 q " Gpu 1 q `Hpu 2 q, which gives (4.33), and the proof of Theorem 4.3 is complete. Assume futhermore that:

(P 1 ) for dt b dx-a.e. pt, xq Ps0, T rˆΩ, the functions gpt, x, ¨, ¨q and hpt, x, ¨, ¨q are strictly convex, gpt, x, ¨, ¨q P C 1 pR 2 q and hpt, x, ¨, ¨q P C 1 pR 2 q; (P 2 ) the function θ is strictly convex and θ P C 1 pRq.

The following result is somehow a version of Theorem 4.3 in terms of weak convergence in H 1 and Euler-Lagrange's equations.

Theorem 4.4. Assume that (P 1 ) and (P 2 ) hold and, for each ω P Σ and each ε ą 0, let u ε pωq be the minimizer of F p ε pωq. Then, for P-a.e. ω P Σ one has

where u P H 1 0 is the minimizer of G H. Moreover, u minimizes G H if and only if there exist v, w P H 1 0 such that u " v `w with v and w satisfying the following differential system: In what follows, E is a Banach space and E 1 is its dual.

Definition B.3 (parallel sum). Let Γ, Λ : E - -E 1 be two multifunctions. By the parallel sum of Γ and Λ we mean the multifunction Γ {{ Λ : E - -E 1 defined by Γ {{ Λpuq :" ! v P E 1 : Dpu 1 , u 2 q P E ˆE such that u " u 1 `u2 and v P Γpu 1 q X Λpu 2 q

) .

For a proof of the following theorem, we refer to [Str94, Theorem 3.7] (see also [START_REF] Strömberg | The operation of infimal convolution[END_REF]).

Theorem B.4. Assume that E is reflexive, G : E s ´8, 8s is convex, proper and lower semicontinuous and H : E s ´8, 8r is convex and Gâteaux differentiable. Then G H is Gâteaux differentiable and DpG Hq " BG {{ DH.


Lacerda Ângela

Rogério Nobre

Marc Duarte

Jacquinet

Information Systems Governance and Industry 4.0 -Epistemology of Data and Semiotics Methodologies of IS in Digital Ecosystems

Keywords: Semiotic learning, Social semiotics, Material phenomenology Poetic rationality, Data epistemology

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Information Systems Governance and Industry 4.0 -Epistemology of Data and Semiotics Methodologies of IS in Digital Ecosystems

Ângela Lacerda Nobre 1(&) , Rogério Duarte 2 , and Marc Jacquinet 3 Abstract. Contemporary Information Systems management incorporates the need to make explicit the links between semiotics, meaning-making and the digital age. This focus addresses, at its core, pure rationality, that is, the capacity of human interpretation and of human inscription upon reality. Creating the new real, that is the motto. Humans are intrinsically semiotic creatures. Consequently, semiotics is not a choice or an option but something that works like a second skin, establishing limits and permeable linkages between: (i) human thought and human's infinite world of imagination; and (ii) human action, with its correspondent infinite world of intentionality, of desire and of unexplored possibilities. Two instances are contrasted as two reading lenses of current business reality: IS governance and industry 4.0. These phenomena correspond to the need to take accountability, transparency and responsibility into account, when designing IS and when using such systems through the ecology of connectivity, Big Data and the Internet of Things. Political, social and cultural dimensions are brought into the equation, when addressing the question of the relevance and adequateness of IS theory and practice to respond to contemporary challenges. The message is that what has already been achieved is but a shadow, a pale vision, of what might be achieved in the age of the new Renaissance.

value chain of signification and of meaning-making [START_REF] Nobre | Knowledge processes and organizational learning[END_REF]). Artificial Intelligence (AI) is a central knowledge area to take into account in order to situate, position and interpret contemporary societies. From understanding the power of AI it is possible to acknowledge the need for semiotic-based information systems' theories, such as Ronald Stamper's Organisational Semiotics (OS). Bernard Stiegler and Bruno [START_REF] Bachimont | Herméneutique matérielle et artéfacture[END_REF] are interesting authors whose contributions have helped to understand the importance of the epistemology of data. There are four paradigms that help to explain current contexts, in terms of historical evolution: first, the empirical age, where meaning emerged from practice; second, the theoretical age, where meaning emerged from ideas, such as Descartes' contribution to modern science; third, the period between 1950 and 1990, forty years of development of applied calculus; and forth, the present age of data manipulation. [START_REF] Pédauque | Document et modernités. Version finale dite[END_REF] addresses the role of documentation in the context of postmodernity. The digital world and its capacity to cut the connection to the heterogeneous nature of reality, is both its strength and its weakness. Whilst digital media agglomerates everything creating an homogeneous set of data, semiotics maintains the heterodox nature of reality, therefore it does not loose meaning neither the meaningmaking capacity.

Epistemological and phenomenological perspectives are needed in order to trace, to map and to explore the different dimensions of the complexity of IS contexts. Baranauskas and Bonacin (2018) address the role of design and of its relation to signs. IS governance will be part of business leaders' agendas once the full impact of their power to bring much needed change at global level is understood. Issues related to sustainability and to human development have much to gain from IS engagement in social innovation and in global change. The digital era has found in manufacture a stronghold for the creation of new possibilities of human realization. Lu (2017) calls attention to the open research importance of industry 4.0. The core idea is that new categories of thought and action, new cognitive structures, are the product of technological evolution. Such technical change, visible through realities such as industry 4.0 or augmented reality, is understood as an enabler of human capacity to read, to interpret and to intervene upon the world. This new digital world is both a product of human endeavors and a process through which humans create new realities, the new real.

1

  Escola Superior de Ciências Empresariais do Instituto Politécnico de Setúbal ESCE-IPS, Campus do IPS Estefanilha, 2910-761 Setúbal, Portugal angela.nobre@esce.ips.pt 2 Escola Superior de Tecnologia do Instituto Politécnico de Setúbal ESTS-IPS,

	Campus do IPS, Estefanilha, 2910-761 Setúbal, Portugal
	rogerio.duarte@estsetubal.ips.pt
	3 Universidade Aberta Rua da Escola Politécnica,
	141-147, 1269-001 Lisbon, Portugal
	mjacquinet@uab.pt