Omar Anza

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

Γ Γ Γ-CONVERGENCE OF NONCONVEX UNBOUNDED INTEGRALS IN CHEEGER-SOBOLEV SPACES

Keywords: 2010 Mathematics Subject Classification. 49J45 (49Q20) Γ-convergence, Nonconvex unbounded integral, Ru-usc, General growth conditions, Metric measure space, Cheeger-Sobolev space

We study Γ-convergence of nonconvex integrals of the calculus of variations in the setting of Cheeger-Sobolev spaces when the integrands have not polynomial growth and can take infinite values. Applications to relaxation and homogenization are also developed.

Let pX, d, µq be a metric measure space, where pX, dq is separable and complete and µ is a doubling positive Radon measure on X which satisfies the annular decay property, supporting a weak p1, pq-Poincaré inequality with p ą 1. Let m, N ě 1 be two integers, let O Ă X be a bounded open set such that µpOzOq " 0 and let pΩ, F, Pq be a probability space. In this paper we consider a family of stochastic integrals E t : H1,p µ pO; R m q ˆΩ r0, 8s defined by E t pu, ωq :"

ż O L t `x, ∇ µ upxq, ω ˘dµpxq, (1.1)
where L t : O ˆM ˆΩ r0, 8s is a Borel measurable stochastic integrand 1 depending on a parameter t ą 0, not necessarily convex with respect to ξ P M, where M denotes the space of real m ˆN matrices, and possibly taking infinite values. The space H 1,p µ pO; R m q denotes the class of p-Cheeger-Sobolev functions from Ω to R m and ∇ µ u is the µ-gradient of u.

The object of the present paper is to deal with the problem of computing the almost sure Γ-convergence (see Definitions 2.1) of the stochastic family tE t u tą0 , as t 8, to a stochastic integral E lim : H 1,p µ pO; R m q ˆΩ r0, 8s of the type

E lim pu, ωq " ż O L lim `x, ∇ µ upxq, ω ˘dµpxq (1.2)
with L lim : O ˆMˆΩ r0, 8s not depending on the parameter t. When L lim is independent of the variable x, the procedure of passing from (1.1) to (1.2) is referred to as stochastic homogenization. If furthermore L lim is independent of the variable ω then E lim is said to be deterministic, otherwise E lim is said to be stochastic. When tL t u tą0 is deterministic, i.e. L t is independent of the variable ω for all t ą 0, the procedure of passing from (1.1) to (1.2) is referred to as deterministic homogenization.

In the case where L t has p-growth, this Γ-convergence problem was already studied in [START_REF] Anza | Γ-convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization[END_REF] and in [START_REF] Maione | Γ-convergence for functionals depending on vector fields. I. Integral representation and compactness[END_REF][START_REF] Maione | Γ-convergence for functionals depending on vector fields. II. Convergence of minimizers[END_REF] for functionals depending on vector fields. Here we treat the case where L t has not necessarily p-growth and can take infinite values (see Section 2 for more details).

In the Euclidean case, i.e. when pX, d, µq " pR N , | ¨´¨|, L N q where L N is the Lebesgue measure on R N , Γ-convergence of unbounded integrals was studied by Carbone and De Arcangelis in [CCDAG02, CCDAG04, CDA02] for the scalar case, i.e. when m " 1, and in [AHM11, AHMZ15, DG16, AHM21] (see also [START_REF] Hafsa | On a homogenization technique for singular integrals[END_REF][START_REF] Hafsa | Homogenization of nonconvex unbounded singular integrals[END_REF]) for the vector case, i.e. when m ą 1.

One motivation for developing Γ-convergence, and more generally calculus of variations, in the setting of metric measure spaces comes from applications to hyperelasticity. In fact, the interest of considering a general measure is that its support can be interpreted as a hyperelastic structure together with its singularities like for example thin dimensions, corners, junctions, etc. Such mechanical "singular" objects naturally lead to develop calculus of variations in the setting of metric measure spaces. Indeed, for example, a low multi-dimensional structures can be described by a finite number of smooth compact manifolds S i of dimension k i on which a superficial measure µ i " H k i | S i is attached. Such a situation leads to deal with the finite union of manifolds S i , i.e. X " Y i S i , together with the finite sum of measures µ i , i.e. µ " ř i µ i , whose mathematical framework is that of metric measure spaces (for more examples, we refer the reader to [START_REF] Bouchitte | Energies with respect to a measure and applications to low-dimensional structures[END_REF][START_REF] Zhikov | Averaging of problems in the theory of elasticity on singular structures[END_REF][START_REF] Chechkin | On homogenization of networks and junctions[END_REF] and [CPS07, Chapter 2, §10] and the references therein).

Another motivation is the development of the calculus of variations on "singular" spaces, which are of interest for geometers and physicists, like Carnot groups, glued spaces, Laakso spaces, Bourdon-Pajot spaces, Gromov-Hausdorff limit spaces, spaces satisfying generalized Ricci bounds (see [START_REF] Kleiner | Differentiable structures on metric measure spaces: a primer[END_REF] for more details). Indeed, all these spaces are examples of doubling metric measure spaces satisfying a Poincaré inequality on which the theory of Γ-convergence on Cheeger-Sobolev spaces could be applied.

The plan of the paper is as follows. In Section 2 we state the main result of the paper, see Theorem 2.5 (and also Proposition 2.8 whose proof is given at the end of Section 2). Theorem 2.5 is a Γ-convergence result of tE t u tą0 as t 8 to E lim in the setting of metric measure spaces and in a unbounded framework. Classically, its proof is a consequence of Proposition 2.6 (the lower bound) and Proposition 2.7 (the upper bound). Section 3 is devoted to several auxiliary definitions and results needed for understanding and proving our Γ-convergence result: in Subsection 3.1 we provide materials about Cheeger-Sobolev spaces; in Subsection 3.2 we recall the concept of (family of) ru-usc2 integrand(s) and its main properties that will be used in the proof of Propositions 2.6, 2.7 and 2.8; the proof of Proposition 2.7 also needs the use of the Vitali envelope of a set function which is recalled in Subsection 3.3. Section 4 is devoted to the proofs of Propositions 2.6 and 2.7 and Theorem 2.5. Finally, applications to relaxation and homogenization are developed in Section 5.

Notation. The open and closed balls centered at x P X with radius ρ ą 0 are denoted by: B ρ pxq :" ! y P X : dpx, yq ă ρ ) ;

B ρ pxq :" ! y P X : dpx, yq ď ρ

) .

For x P X and ρ ą 0 we set BB ρ pxq :" B ρ pxqzB ρ pxq " ! y P X : dpx, yq " ρ ) .

For A Ă X, the diameter of A is defined by diampAq :" sup x,yPA dpx, yq. The symbol ş stands for the mean-value integral

ż B f dµ " 1 µpBq ż B f dµ.
For F Ă M, where M denotes the space of real m ˆN matrices, the interior and the closure of F are respectively denoted by intpFq and F.

The Γ-convergence result

We begin by recalling the definition of the almost sure Γ-convergence. (For more details on the theory of Γ-convergence we refer to [START_REF] Dal | An introduction to Γ-convergence[END_REF].) Definition 2.1. We say that tE t u tą0 almost surely ΓpL p µ q-converges as t 8 to the functional E lim : H 1,p µ pO; R m q ˆΩ r0, 8s if there exists Ω 1 P F with PpΩ 1 q " 1 such that for every ω P Ω 1 , one has: Γ-lim: for every u P H 1,p µ pO; R m q, ΓpL p µ q-lim t 8 E t pu, ωq ě E lim pu, ωq with ΓpL p µ q-lim t 8

E t pu, ωq :" inf

" lim t 8 E t pu t , ωq : u t u in L p µ pO; R m q * ,
or equivalently, for every u P H 1,p µ pO; R m q and every tu t u tą0 Ă H 1,p µ pO; R m q such that u t u in L p µ pO; R m q, lim t 8 E t pu ε , ωq ě E lim pu, ωq;

Γ-lim: for every u P H 1,p µ pO; R m q, ΓpL p µ q-lim t 8 E t pu, ωq ď E lim pu, ωq with

ΓpL p µ q-lim t 8 E t pu, ωq :" inf ! lim t 8 E t pu t , ωq : u t u in L p µ pO; R m q ) ,
or equivalently, for every u P H 1,p µ pO; R m q there exists tu t u tą0 Ă H 1,p µ pO; R m q such that u t u in L p µ pO; R m q and lim t 8

E t pu t , ωq ď E lim pu, ωq.

Referring to the next section for any unfamiliar notation or definition, in what follows we state the main results of the paper. Let G : M r0, 8s be a Borel measurable integrand satisfying the following conditions:

(C 1 ) there exists γ ą 0 such that for every ξ, ζ P M and every τ Ps0, 1r, Gpτ ξ `p1 ´τ qζq ď γp1 `Gpξq `Gpζqq;

(C 2 ) 0 P intpGq, where G denotes the effective domain of G, i.e. G :" tξ P M : Gpξq ă 8u. Let Q µ G : O ˆM r0, 8s be defined by

Remark 2.2. If (C 1 ) is satisfied then G is convex, but G is not necessarily convex (see [AHMZ15, Sect. 9]). So, if moreover (C 2 )
Q µ Gpx, ξq :" lim ρ 0 inf # ż Bρpxq Gpξ `∇µ wpyqqdµpyq : w P H 1,p µ,0 pB ρ pxq; R m q + ,
where the space H 1,p µ,0 pB ρ pxq; R m q is defined as the closure of Lip 0 pB ρ pxq; R m q :" ! u P LippO; R m q : u " 0 on OzB ρ pxq ) with respect to the H 1,p µ -norm, where LippO; R m q :" rLippOqs m with LippOq denoting the algebra of Lipschitz functions from O to R. (The integrand Q µ G is called the H 1,p µquasiconvexification of G. For more details on the notion of H 1,p µ -quasiconvexity, we refer to [START_REF] Anza | Lower semicontinuity of integrals of the calculus of variations in Cheeger-Sobolev spaces[END_REF][START_REF] Anza | Integral representation and relaxation of local functionals on Cheeger-Sobolev spaces[END_REF].) Denote the effective domain of Q µ Gpx, ¨q by Q µ G x . We further suppose that:

(C 3 ) for every u P H 1,p µ pO; R m q, if ş O Q µ Gpx, ∇ µ upxqqdµ ă 8 and if ∇ µ upxq P intpQ µ G x q for µ-a.a. x P O then ş O Gp∇ µ upxqqdµ ă 8; (C 4 ) for every x P O, Q µ Gpx, ¨q is lsc 3 on intpQ µ G x q.
Remark 2.3.

(i) For every px, ξq P O ˆM, Q µ Gpx, ξq ď Gpξq, and so

G Ă Q µ G x for all x P O. (ii) Considering G, Q µ G : H 1,p
µ pO; R m q r0, 8s defined by Gpuq :" ş O Gp∇ µ upxqqdµpxq and Q µ Gpuq :" ş O Q µ Gpx, ∇ µ upxqqdµpxq and denoting their effective domains by dompGq and dompQ µ Gq, we see that (C 3 ) means that

! u P dompQ µ Gq : ∇ µ upxq P intpQ µ G x q for µ-a.a. x P O ) Ă dompGq.
(iii) If either dompQ µ Gq " dompGq or Gpuq ă 8 for all u P H 1,p µ pO; R m q such that ∇ µ upxq P intpQ µ G x q for µ-a.a. x P O, then (C 3 ) can be dropped.

(iv) Under (C 1 )-(C 2 ), if G " Q µ G, i.e. G is H 1,p
µ -quasiconvex, then (C 3 ) holds. In particular, since convexity implies H 1,p µ -quasiconvexity (see [START_REF] Anza | Lower semicontinuity of integrals of the calculus of variations in Cheeger-Sobolev spaces[END_REF]), if G is convex then (C 3 ) holds. (v) If G satisfies (C 1 ) then Q µ G verifies the same condition, i.e. for every x P O, every ξ, ζ P M and every τ Ps0, 1r,

Q µ Gpx, τ ξ `p1 ´τ qζq ď γp1 `Qµ Gpx, ξq `Qµ Gpx, ζqq,
and so Q µ G x is convex for all x P O. Hence, under (C 1 )-(C 2 ), for every x P O, τQ µ G x Ă intpQ µ G x q for all τ Ps0, 1r.

Let pX, d, µq be a metric measure space, where pX, dq is separable and complete and µ is a doubling positive Radon measure on X which satisfies the annular decay property, supporting a weak p1, pq-Poincaré inequality with p ą κ :" lnpC d q lnp2q where C d ě 1 is the doubling constant.

Let O Ă X be a bounded open set such that µpOzOq " 0 and let pΩ, F, Pq be a probability space. Throughout the paper, we consider a family tL t : O ˆM ˆΩ r0, 8su tą0 of Borel measurable stochastic integrands depending on a parameter t ą 0 and satisfying the following conditions:

(C 5 ) tL t u tą0 is p-coercive, i.e. there exists c ą 0 such that for every t ą 0, every x P O, every ξ P M and every ω P Ω, L t px, ξ, ωq ě c|ξ| p ;

(C 6 ) tL t u tą0 has G-growth, i.e. there exist α, β ą 0 such that for every x P O, every ξ P M and every ω P Ω, αGpξq ď L t px, ξ, ωq ď βp1 `Gpξqq.

Remark 2.4. If (C 1 ) and (C 6 ) hold then the effective domain L t,x,ω of L t px, ¨, ωq is equal to G and so is convex and does not depend on x and ω.

The p-growth case, i.e. when Gpξq " |ξ| p , was already studied in [START_REF] Anza | Γ-convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization[END_REF] (see also [START_REF] Maione | Γ-convergence for functionals depending on vector fields. I. Integral representation and compactness[END_REF][START_REF] Maione | Γ-convergence for functionals depending on vector fields. II. Convergence of minimizers[END_REF]). The object of this paper is to deal with the G-growth case. For this, in addition, we need to suppose that (C 7 ) for every ω P Ω, tL t u tą0 is ru-usc at ω, i.e. for every ω P Ω, there exists ta t p¨, ωqu tą0 For each t ą 0 and each ρ ą 0, let H ρ µ L t : O ˆM ˆΩ r0, 8s be defined by

H ρ µ L t px, ξ, ωq :" inf # ż Bρpxq L t py, ξ `∇µ wpyq, ωqdµpyq : w P H 1,p µ,0 pB ρ pxq; R m q + .
For each t ą 0, let E t : H 1,p µ pO; R m q ˆΩ r0, 8s be defined by (1.1). The main result of the paper is the following Γ-convergence result.

Theorem 2.5 (Γ-lim). Assume that p ą κ. If (C 1 )-(C 7 ) hold and if (C 8 ) there exists Ω 1 P F with PpΩ 1 q " 1 such that for every ω P Ω ´L8 px, τ ξ, ωq and, for each x P O and each ω P Ω, let L 8 px, ¨, ωq denotes the lsc envelope of L 8 px, ¨, ωq. The following proposition makes more precise the formula of the limit integrand L lim in Theorem 2.5. Proposition 2.8. Assume that (C 1 )-(C 2 ) and (C 6 )-(C 7 ) hold.

(i) For every ω P Ω,

p L 8 px, ξ, ωq " lim τ 1 ´L8 px, τ ξ, ωq " # lim τ 1 ´L8 px, τ ξ, ωq if ξ P Q µ G x 8 otherwise.
So, in Theorem 2.5 we have L lim " p L 8 . (ii) Suppose furthermore that for every ω P Ω and every x P O, L 8 px, ¨, ωq is lsc on intpQ µ G x q. Then p L 8 px, ξ, ωq " L 8 px, ξ, ωq "

$ & % L 8 px, ξ, ωq if ξ P intpQ µ G x q lim τ 1 ´L8 px, τ ξ, ωq if ξ P BQ µ G x 8
otherwise.

(2.4)

In such a case, in Theorem 2.5, L lim is given by (2.4).

Proof of Proposition 2.8. From (C 7 ) and Proposition 3.14, we can assert that for every ω P Ω, L 8 is ru-usc at ω. Moreover, by (C 6 ) it is easily seen that for every x P O and every ω P Ω, the effective domain of L 8 px, ¨, ωq is equal to Q µ G x . So, taking (C 1 )-(C 2 ) into account (see Remark 2.3(v)), Proposition 2.8 follows from Theorem 3.12.

Auxiliary results

3.1. Cheeger-Sobolev spaces. Let pX, d, µq be a separable and complete metric measure space. Here and subsequently, we assume that µ is doubling on X, i.e. there exists a constant C d ě 1 such that µ pB ρ pxqq ď C d µ ´Bρ 2 pxq ¯(3.1) for µ-a.a. x P X and all ρ ą 0, and X supports a weak p1, pq-Poincaré inequality with p ą 1, i.e. there exist C P ą 0 and σ ě 1 such that for µ-a.e. x P X and every ρ ą 0, Remark 3.1. As µ is doubling, for µ-a.e. x P X and every r ą 0, we have µpB ρ pxqq{µpB r pxqq ě 4 ´κ pρ{rq κ for all x P B r pxq and all 0 ă ρ ď r, where κ :" lnpC d q lnp2q (see [START_REF] Haj | Sobolev spaces on metric-measure spaces[END_REF]Lemma 4.7]). We further assume that pX, d, µq satisfies the annular decay property, i.e. there exist δ ą 0 and C A ě 1 such that µ pB σr pxqzB r pxqq ď C A ˆ1 ´1 σ ˙δ µpB σr pxqq (3.3) for all x P X, all r ą 0 and all σ Ps1, 8r.

ż
Remark 3.2. From [Buc99, Corollary 2.2] and [CM98, Lemma 3.3] (see also [Che99, Proposition 6.12] and [HKST15, Proposition 11.5.3 pp. 328]), under (3.1) and (3.2), if moreover pX, dq is a length space, i.e. the distance between any two points equals infimum of lengths of curves connecting the points, then (3.3) holds.

Remark 3.3. If (3.3) holds then µ `Br pxqzB r pxq ˘" 0 for all x P X and all r ą 0, i.e. the boundary of any ball is of zero measure. Indeed, given x P X and r ą 0, we have 1 ě µpBrpxqq µpBrpxqq ě µpBrpxqq µpBσrpxqq ě 1 ´CA p1 ´1 σ q δ for all σ Ps1, 8r. Hence, by letting σ 1, we obtain µpBrpxqq µpBrpxqq " 1, i.e. µpB r pxqq " µpB r pxqq. Let O Ă X be a bounded open set. Denote the algebra of Lipschitz functions from O to R by LippOq. (Note that, by Hopf-Rinow's theorem (see [BH99, Proposition 3.7, pp. 35]), the closure of O is compact, and so every Lipschitz function from O to R is bounded.) Let LippO; R m q :" rLippOqs m and let ∇ µ : LippO; R m q L 8 µ pO; Mq be given by

∇ µ u :" ¨Dµ u 1 . . . D µ u m ' with u " pu 1 , ¨¨¨, u m q,
where D µ : LippOq L 8 µ pO; R N q is the differential of Cheeger (see [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF]Theorem 4.38] and [Kei04, Definition 2.1.1 and Theorem 2.3.1] for more details). The p-Cheeger-Sobolev space H 1,p µ pO; R m q is defined as the completion of LippO; R m q with respect to the norm }u} H 1,p µ pO;R m q :" }u} L p µ pO;R m q `}∇ µ u} L p µ pO;Mq .

(3.4)

As }∇ µ u} L p µ pO;Mq ď }u} W 1,p µ pO;R m q for all u P LippO; R m q, the linear map ∇ µ from LippO; R m q to L p µ pO; Mq has a unique extension to H 1,p µ pO; R m q which will still be denoted by ∇ µ and will be called the µ-gradient. For more details on the various possible extensions of the classical theory of the Sobolev spaces to the setting of metric measure spaces, we refer to [Hei07, §10-14] (see also [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF][START_REF] Shanmugalingam | Newtonian spaces: an extension of Sobolev spaces to metric measure spaces[END_REF][START_REF] Gol | Axiomatic theory of Sobolev spaces[END_REF][START_REF] Haj | Sobolev spaces on metric-measure spaces[END_REF]).

The following proposition brings together useful known properties for dealing with calculus of variations in the metric measure setting. (For a proof we refer to [START_REF] Heinonen | Sobolev spaces on metric measure spaces[END_REF] (i) O satisfies the Vitali covering theorem, i.e. for every A Ă O and every family B of closed balls in O, if inftρ ą 0 : B ρ pxq P Bu " 0 for all x P A (we say that B is a fine cover of A) then there exists a countable disjoint subfamily

B 1 of B such that µpAz Y BPB 1 Bq " 0; in other words, A Ă `YBPB 1 B ˘Y N with µpN q " 0; (ii) the µ-gradient is closable in H 1,p
µ pO; R m q, i.e. for every u P H 1,p µ pO; R m q and every open set A Ă O, if upxq " 0 for µ-a.a. x P A then ∇ µ upxq " 0 for µ-a.a. x P A; (iii) O supports a p-Sobolev inequality, i.e. there exists C S ą 0 such that

˜żBρpxq |v| p dµ ¸1 p ď ρC S ˜żBρpxq |∇ µ v| p dµ ¸1 p
for all 0 ă ρ ď ρ 0 , with ρ 0 ą 0, and all v P H 1,p µ,0 pB ρ pxq; R m q, where, for each open set A Ă O, H 1,p µ,0 pA; R m q is the closure of Lip 0 pA; R m q with respect to H 1,p µ -norm defined in (3.4) with Lip 0 pA; R m q :" u P LippO; R m q : u " 0 on OzA ( ;

(iv) for every u P H 1,p µ pO; R m q and µ-a.e. x P O there exists u x P H 1,p µ pO; R m q such that: ∇ µ u x pyq " ∇ µ upxq for µ-a.a. y P O;

lim ρ 0 1 ρ }u ´ux } L 8 µ pBρpxq;R m q " 0 if p ą κ,
where κ :" lnpC d q lnp2q with C d ě 1 given by the inequality (3.1); (v) for every x P O, every ρ ą 0 and every λ Ps0, 1r there exists a Urysohn function ϕ P LippOq for the pair pOzB ρ pxq, B λρ pxqq5 such that

}D µ ϕ} L 8 µ pO;R N q ď θ ρp1 ´λq for some θ ą 0; (vi) for µ-a.e. x P O, lim λ 1 ´lim ρ 0 µpB λρ pxqq µpB ρ pxqq " lim λ 1 ´lim ρ 0
µpB λρ pxqq µpB ρ pxqq " 1;

(vii) for every u P H 1,p µ pO; R m q and every ϕ P LippOq, ∇ µ pϕuq " ϕ∇ µ u `Dµ ϕ b u.

Remark 3.5. As µ is a Radon measure and O satisfies the Vitali covering theorem, for every open set A Ă O and every ε ą 0 there exists a countable family tB ρ i px i qu iPI of disjoint open balls of A with x i P A, ρ i Ps0, εr such that µ `Az Y iPI B ρ i px i q ˘" 0. By the annular decay property, see (3.3), we also have µpBB ρ i px i qq " 0 for all i P I (see Remark 3.3).

In the framework of the p-Cheeger-Sobolev spaces with p ą κ :" lnpC d q{ lnp2q, where C d ě 1 is the doubling constant, we also have the following L 8 µ -compactness result. Theorem 3.6. Assume that p ą κ and µ `OzO ˘" 0.

If u P H 1,p µ pO; R m q and tu n u n Ă H 1,p µ pO; R m q are such that lim n 8 }u n ´u} L p µ pO;R m q " 0 and sup ně1 }∇ µ u n } L p µ pO;Mq ă 8, (3.5)
then, up to a subsequence, lim

n 8 }u n ´u} L 8 µ pO;R m q " 0. (3.6)
Proof of Theorem 3.6. Since pX, d, µq is a complete doubling metric space, pX, d, µq is proper, i.e. every closed ball is compact (see [HKST15, Lemma 4.1.14]), and so pO, d| OˆO q is compact. Thus, as µ `OzO ˘" 0 we can assert that pO, d| OˆO , µ| O q is a compact doubling metric measure space supporting a weak p1, pq-Poincaré inequality. In what follows, to simplify the notation we set pY, δ, νq :" pO, d| OˆO , µ| O q.

Step 1: two auxiliary lemmas. We need the following two lemmas (cf. Lemmas 3.7 and 3.8).

Lemma 3.7. If p ą κ then for every r ą 0 and ν-a.e. x P Y there exists Cpr, xq ą 0 such that |upyq ´upzq| ď Cpr, xqδpy, zq 1´κ p ˆżB 6σr pxq |∇ ν u| p dν ˙1 p for all u P H 1,p ν pY ; R m q and all y, z P B r pxq, where σ ě 1 is given by (3.2). Proof of Lemma 3.7. From [Haj03, Theorem 9.7] we can assert that there exists c ą 0 such that

|wpyq ´wpzq| ď cr κ p δpy, zq 1´κ p ˆż B 6σr pxq g p w dν ˙1 p (3.7)
for all w P H 1,p ν pY q, all x P Y , all r ą 0 and all y, z P B r pxq, where σ ě 1 is given by (3.2) and g w P L p ν pY q denotes the minimal p-weak upper gradient for w. On the other hand, from Remark 3.1 it is easy to see that for every r ą 0 and ν-a.e. x P Y there exists θpr, xq ą 0 such that νpB r pxqq ě θpr, xqr κ . But g w ď α|D ν w| with α ě 1 (see [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF]§4]) and so şB 6σr pxq g p w dν ď α p şB 6σr pxq |D ν w| p dν. Thus, for every r ą 0, ν-a.e. x P Y and every y, z P B r pxq, (3.7) can be rewritten as follows for all u P H 1,p ν pY ; R m q and all y, z P B r pxq, and the proof of Lemma 3.7 is complete. Denote the space of continuous functions from Y to R m by CpY ; R m q. As a consequence of Lemma 3.7 we have the following result.

|wpyq
Lemma 3.8. If p ą κ then H 1,p ν pY ; R m q continuously embeds into CpY ; R m q, i.e. H 1,p ν pY ; R m q Ă CpY ; R m q and there exists K 0 ą 0 such that }u} CpY ;R m q ď K 0 }u} H 1,p ν pY ;R m q (3.8)
for all u P H 1,p ν pX; R m q. Moreover, there exists K 1 ą 0 such that |upyq ´upzq| ď K 1 δpy, zq 1´κ p }∇ ν u} L p ν pY ;Mq (3.9) for all u P H 1,p ν pY ; R m q and all y, z P Y .

Proof of Lemma 3.8. Applying Lemma 3.7 with r " diampY q and for a fixed x " x 0 P Y , where diampY q " suptδpy, zq : y, z P Y u ă 8 because pY, δq is compact, we see that |upyq ´upzq| ď C pdiampY q, x 0 q δpy, zq 1´κ p }∇ ν u} L p ν pY ;Mq ď C pdiampY q, x 0 q diampY q 1´κ p }∇ ν u} L p ν pY ;Mq (3.10) for all u P H 1,p ν pY ; R m q and all y, z P Y . Hence (3.9) holds with K 1 " C pdiampY q, x 0 q and every u P H 1,p ν pY ; R m q is p1 ´κ p q-Hölder continuous. In particular, it follows that H 1,p ν pY ; R m q Ă CpY ; R m q. On the other hand, given any u P H 1,p ν pY ; R m q and any y P Y , we have |upyq| p ď 2 p p|upyq ´upzq| p `|upzq| p q for all z P Y , and consequently νpY q

1 p |upyq| ď 2 1`1 p ˆżY |upyq ´upzq| p dνpzq ˙1 p `21`1 p }u} L p ν pY ;R m q . (3.11)
But, by (3.10) we have

ˆżY |upyq ´upzq| p dνpzq ˙1 p ď νpY q 1 p C pdiampY q, x 0 q diampY q 1´κ p }∇ ν u} L p ν pY ;Mq . (3.12)
Hence, combining (3.11) and (3.12) we deduce that for every y P Y ,

|upyq| ď 2 1`1 p C pdiampY q, x 0 q diampY q 1´κ p }∇ ν u} L p ν pY ;Mq `21`1 p νpY q 1 p }u} L p ν pY ;R m q ď K 0 }u} H 1,p
ν pY ;R m q with K 0 " sup " 2 1`1 p C pdiampY q, x 0 q diampY q 1´κ p , 2 1`1 p νpY q 1 p * , and (3.8) follows.

Step 2: end of the proof of Theorem 3.6. As µ `OzO ˘" 0, from (3.5) we deduce that lim n 8

}u n ´u} L p ν pY ;R m q " 0 and sup

ně1 }∇ ν u n } L p ν pY ;Mq ă 8,
and so sup ně1 }u n } H 1,p ν pY ;R m q ă 8. By Lemma 3.8 we can assert that sup ně1 }u n } CpY ;R m q ă 8, i.e. tu n u n is bounded in CpY ; R m q with pY, δq a compact metric space. Moreover, using (3.9) we see that tu n u n is equicontinuous. Consequently, up to a subsequence, Theorem 3.12. Let ω P Ω. If L is ru-usc at ω with ap¨, ωq and if for every x P O, τ L x,ω Ă intpL x,ω q for all τ Ps0, 1r, where L x,ω denotes the effective domain of Lpx, ¨, ωq, then:

lim n 8 }u n ´u} L 8 ν pY ;R m q " 0 by Arzelà-Ascoli'
(i) p Lpx, ξ, ωq :" lim τ 1 ´Lpx, τ ξ, ωq " # lim τ 1 ´Lpx, τ ξ, ωq if ξ P L x,ω 8 otherwise; (ii) p
L is ru-usc at ω with ap¨, ωq.

If moreover Lpx, ¨, ωq is lsc on intpL x,ω q then:

(iii) p Lpx, ξ, ωq " $ & % Lpx, ξ, ωq if ξ P intpL x,ω q lim τ 1 ´Lpx, τ ξ, ωq if ξ P BL x,ω 8
otherwise; (iv) for every x P O, p Lpx, ¨, ωq is the lsc envelope of Lpx, ¨, ωq.

The following definition extends Definition 3.9 to a family tL t u tą0 of Borel measurable stochastic integrands L t : OˆMˆΩ r0, 8s. (When L t " L for all t ą 0 we retrieve Definition 3.9.) Definition 3.13. Let ω P Ω. We say that tL t u tą0 is ru-usc at ω if there exists ta t p¨, ωqu tą0 Ă L 1 µ pO; s0, 8sq, satisfying (2.1) and (2.2), such that lim

τ 1 ´sup tą0 ∆ at Lt pτ, ωq ď 0.
For each t ą 0 and each ρ ą 0, let H ρ µ L t : O ˆM ˆΩ r0, 8s be defined by

H ρ µ L t px, ξ, ωq :" inf # ż Bρpxq L t py, ξ `∇µ wpyq, ωqdµpyq : w P H 1,p µ,0 pB ρ pxq; R m q + ,
where the space H 1,p µ,0 pB ρ pxq; R m q is defined as the closure of Lip 0 pB ρ pxq; R m q :" ! u P LippO; R m q : u " 0 on OzB ρ pxq

)
with respect to the H 1,p µ -norm, where LippO; R m q :" rLippOqs m with LippOq denoting the algebra of Lipschitz functions from O to R. Let L 8 : O ˆM ˆΩ r0, 8s be given by

L 8 px, ξ, ωq :" lim ρ 0 lim t 8 H ρ µ L t px, ξ, ωq. (3.13)
The following proposition shows that ru-usc is conserved under the operation characterized by (3.13).

Proposition 3.14. Let ω P Ω. If tL t u tą0 is ru-usc at ω with ta t p¨, ωqu tą0 then L 8 is ru-usc at ω with a 8 p¨, ωq given by (2.2).

Proof of Proposition 3.14. Fix any τ P r0, 1s, any x P O and any ξ P L 8,x,ω , where L 8,x,ω is the effective domain of L 8 px, ¨, ωq. Then L 8 px, ξ, ωq " lim ρ 0 lim t 8 H ρ µ L t px, ξ, ωq ă 8 and without loss of generality we can suppose that H ρ µ L t px, ξ, ωq ă 8 for all ρ ą 0 and all t ą 0. Fix any ρ ą 0 and any t ą 0. By definition, there exists tw n u n Ă H 1,p µ,0 pB ρ pxq; R m q such that: for all ρ ą 0 and all t ą 0, where ∆pτ, ωq :" sup są0 ∆ as Ls pτ, ωq. By letting t 8 and ρ 0 in (3.17), we get L 8 px, τ ξ, ωq ´L8 px, ξ, ωq ď ∆pτ, ωq `a8 px, ωq `L8 px, ξ, ωq with a 8 p¨, ωq P L 1 µ pO; s0, 8sq given by (2.2), which implies that ∆ a8 L8 pτ, ωq ď ∆pτ, ωq for all τ P r0, 1s. As tL t u tą0 is ru-usc at ω with ta t p¨, ωqu tą0 , i.e. lim τ 1 ´∆pτ , ωq ď 0, we conclude that lim τ 1 ´∆a8 L8 pτ, ωq ď 0 which means that L 8 is ru-usc at ω with a 8 p¨, ωq. Remark 3.15. In the proof of Proposition 3.14 we do not need (2.1). In fact, (2.1) will be used in the proof of the Γ-convergence result (see Section 4). For each ε ą 0 and each A P OpOq, we denote the class of countable families tB i :" B ρ i px i qu iPI of disjoint open balls of A with x i P A, ρ i Ps0, εr and µpBB i q " 0 such that µpAz Y iPI B i q " 0 by V ε pAq. Definition 3.16. Given S : OpOq r0, 8s, for each ε ą 0 we define S ε : OpOq r0, 8s by

H ρ µ L t px
S ε pAq :" inf # ÿ iPI SpB i q : tB i u iPI P V ε pAq + .
By the Vitali envelope of S we call the set function S ˚: OpOq r´8, 8s defined by S ˚pAq :" sup Proof of Proposition 2.6. Fix ω P Ω. Let u P H 1,p µ pO; R m q and let tu t u tą0 Ă H 1,p µ pO; R m q be such that }u t ´u} L p µ pO;R m q 0. We have to prove that lim Step 1: localization. For each t ą 0, we define the (positive) Radon measure ν t on O by ν t :" L t p¨, ∇ µ u t p¨q, ωqdµ.

From (4.2) we see that sup tą0 ν t pOq ă 8, and so there exists a (positive) Radon measure ν on O such that, up to a subsequence, ν t á ν weakly. By Lebesgue's decomposition theorem, we have ν " ν a `νs where ν a and ν s are (positive) Radon measures on O such that ν a ! µ and ν s K µ. Thus, to prove (4.1) it suffices to show that

ν a ě lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t p¨, τ ∇ µ up¨q, ωqdµ. (4.7)
From Radon-Nikodym's theorem we have ν a " f p¨qdµ with

f p¨q :" lim ρ 0 νpB ρ p¨qq µpB ρ p¨qq P L 1 µ pO; r0, 8rq, (4.8)
and so to prove (4.7) it is sufficient to establish that for µ-a.e. x 0 P O,

f px 0 q " lim ρ 0 νpB ρ px 0 qq µpB ρ px 0 qq ě lim τ 1 ´lim ρ 0 lim t 8
H ρ µ L t px 0 , τ ∇ µ upx 0 q, ωq. (4.9) Fix x 0 P OzN where N Ă O is a suitable set such that µpN q " 0. As νpOq ă 8, without loss of generality we can assume that νpBB ρ px 0 qq " 0 for all ρ ą 0, which implies, by Alexandrov's theorem, that νpB ρ px 0 qq " lim t 8 ν t pB ρ px 0 qq. Consequently, to prove (4.9) it suffices to show that lim

ρ 0 lim t 8 ż Bρpx 0 q L t px, ∇ µ u t pxq, ωqdµ ě lim τ 1 ´lim ρ 0 lim t 8
H ρ µ L t px 0 , τ ∇ µ upx 0 q, ωq.

(4.10)

On the other hand, as G is convex, see (C 1 ) and Remark 2.2, and 0 P intpGq, see (C 2 ), from (4.3) we can assert for every σ Ps0, 1r, σ∇ µ u t pxq P G for all t ą 0 and for µ-a.a. x P O.

Hence, given any τ Ps0, 1r, we see that for every t ą 0 and every ρ ą 0, H ρ µ L t px 0 , τ ∇ µ upx 0 q, ωq. (4.11)

ż Bρpx 0 q L t px, τ ∇ µ u t pxq, ωqdµ ď `1 `∆pτ, ωq ˘ż Bρpx 0 q L t px, ∇ µ u t pxq
Step 2: cut-off method. Fix any t ą 0, any τ Ps0, 1r, any σ Psτ, 1r, any λ Ps0, 1r and any ρ ą 0. By Proposition 3.4(v) there is a Uryshon function ϕ P LippOq for the pair pOzB ρ px 0 q, B λρ px 0 qq such that }D µ ϕ} L 8 µ pO;R N q ď θ ρp1´λq for some θ ą 0 (which does not depend on ρ). Define v t P H 1,p µ pO; R m q by v t :" ϕu t `p1 ´ϕqu x 0 " ϕpu t ´ux 0 q `ux 0 with u x 0 P H 1,p µ pO; R m q given by Proposition 3.4(iv). (Note that ∇ µ u x 0 pxq " ∇ µ upx 0 q for µ-a.a. x P O.) Then τ v t ´τ u x 0 P H 1,p µ,0 pB ρ px 0 q; R m q (4.12) and, using Proposition 3.4(vii),

τ ∇ µ v t " " τ ∇ µ u t in B λρ px 0 q τ σ `ϕσ∇ µ u t `p1 ´ϕqσ∇ µ upx 0 q ˘``1 ´τ σ ˘Ψt,ρ in B ρ px 0 qzB λρ px 0 q (4.13)
with Ψ t,ρ :" τ 1´τ σ D µ ϕ b pu t ´ux 0 q. Using the right inequality in (C 6 ) it follows that

ż Bρpx 0 q L t px, τ ∇ µ v t , ωqdµ " 1 µpB ρ px 0 qq ż B λρ px 0 q L t px, τ ∇ µ u t , ωqdµ `1 µpB ρ px 0 qq ż Bρpx 0 qzB λρ px 0 q L t px, τ ∇ µ v t , ωqdµ ď ż Bρpx 0 q L t px, τ ∇ µ u t , ωqdµ `β µpB ρ px 0 qzB λρ px 0 qq µpB ρ px 0 qq `β µpB ρ px 0 qq ż Bρpx 0 qzB λρ px 0 q Gpτ ∇ µ v t qdµ. (4.14)
On the other hand, taking (4.13) into account and using (C 1 ) and the left inequality in (C 6 ), we have

Gpτ ∇ µ v t q ď c 1 p1 `Gpσ∇ µ u t q `Gpσ∇ µ upx 0 qq `GpΨ t,ρ qq ď c 1 ˆ1 `1 α L t px, σ∇ µ u t , ωq `Gpσ∇ µ upx 0 qq `GpΨ t,ρ q ˙(4.15)

with c 1 :" 2pγ `γ2 q ą 0. Note that from (C 2 ) and (4.5) we can assert that σ∇ µ upx 0 q P G, and so Gpσ∇ µ upx 0 qq ă 8.

Moreover, it is easy to see that }Ψ t,ρ } L 8 µ pBρpx 0 q;Mq ď θτ p1 ´τ σ qp1 ´λq

1 ρ }u ´ux 0 } L 8 µ pBρpx 0 q;R m q `θτ ρp1 ´τ σ qp1 ´λq }u t ´u} L 8 µ pO;R m q ,
where lim ρ 0 θτ p1 ´τ σ qp1 ´λq 1 ρ }u ´ux 0 } L 8 µ pBρpx 0 q;R m q " 0 (4.16) by Proposition 3.4(iv), i.e., lim ρ 0 1 ρ }u ´ux 0 } L 8 µ pBρpx 0 q;R m q " 0, and lim t 8 θτ ρp1 ´τ σ qp1 ´λq }u t ´u} L 8 µ pO;R m q " 0 (4.17) by (4.6), i.e., lim t 8 }u t ´u} L 8 µ pO;R m q " 0. From (C 1 )-(C 2 ) there exists r ą 0 such that c 2 :" sup |ξ|ďr Gpξq ă 8

(see Remark 2.2). By (4.16) there exists ρ ą 0 such that

θτ p1´τ σ qp1´λq 1 ρ }u´u x 0 } L 8 µ pBρpx 0 q;R m q ă r 2
for all ρ Ps0, ρr. Fix any ρ Ps0, ρr. Taking (4.17) into account we can assert that there exists t ρ ą 0 such that GpΨ t,ρ q ď c 2 for all t Ps0, t ρ r.

(4.18) Thus, from (4.14), (4.15) and (4.18) we deduce that

ż Bρpx 0 q L t px, τ ∇ µ v t , ωqdµ ď ż Bρpx 0 q L t px, τ ∇ µ u t , ωqdµ `c3 pσqγ ρ,λ `βc 1 α Γ t,ρ,λ,σ
for all t Ps0, t ρ r with:

c 3 pσq :" βc 1 ˆ1 `1 c 1 `Gpσ∇ µ upx 0 qq `c2 ˙Ps0, 8r ;
γ ρ,λ :" µpB ρ px 0 qzB λρ px 0 qq µpB ρ px 0 qq ; Γ t,ρ,λ,σ :" 1 µpB ρ px 0 qq ż Bρpx 0 qzB λρ px 0 qq L t px, σ∇ µ u t , ωqdµ.

But, taking (4.12) into account, we see that

H ρ µ L t px 0 , τ ∇ µ upx 0 q, ωq ď ż Bρpx 0 q L t px, τ ∇ µ v t , ωqdµ,
hence, for every ρ ą 0, every t Ps0, t ρ r, every λ Ps0, 1r, every τ Ps0, 1r and every σ Psτ, 1r, we have

H ρ µ L t px 0 , τ ∇ µ upx 0 q, ωq ď ż Bρpx 0 q L t px, τ ∇ µ u t , ωqdµ `c3 pσqγ ρ,λ `βc 1 α Γ t,ρ,λ,σ . (4.19)
Step 3: passing to the limit. Letting t 8, ρ 0, λ

1 ´, σ 1 ´and τ 1 ´in (4.19), we obtain lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px 0 , τ ∇ µ upx 0 q, ωq ď lim τ 1 ´lim ρ 0 lim t 8 ż Bρpx 0 q L t px, τ ∇ µ u t , ωqdµ `lim σ 1 ´c3 pσq lim λ 1 ´lim ρ 0 γ ρ,λ `βc 1 α lim σ 1 ´lim λ 1 ´lim ρ 0 lim t 8 Γ t,ρ,λ,σ . (4.20)
Substep 3-1: proving that lim λ 1 ´lim ρ 0 γ ρ,λ " 0 lim λ 1 ´lim ρ 0 γ ρ,λ " 0 lim λ 1 ´lim ρ 0 γ ρ,λ " 0. As the boundary of any ball is of zero measure (see Remark 3.3), we have Since pX, d, µq is a complete doubling metric space, pX, d, µq is proper, i.e. every closed ball is compact. Hence B ρ px 0 q is compact, and so B ρ px 0 qzB λρ px 0 q is compact. As ν t á ν weakly, by Alexandrov's theorem, we have lim t 8 ν t `Bρ px 0 qzB λρ px 0 q ˘ď ν `Bρ px 0 qzB λρ px 0 q ˘, hence lim t 8 ν t `Bρ px 0 qzB λρ px 0 q ˘ď ν `Bρ px 0 q ˘´ν pB λρ px 0 qq , and consequently, since the boundary of any ball is of zero measure with respect to µ, lim t 8 ν t `Bρ px 0 qzB λρ px 0 q μpB ρ px 0 qq ď ν `Bρ px 0 q μpB ρ px 0 qq ´µ pB λρ px 0 qq µpB ρ px 0 qq ν pB λρ px 0 qq µpB λρ px 0 qq . For each ε ą 0 and each A P OpOq, we denote the class of countable families tB i :" B ρ i px i qu iPI of disjoint open balls of A with x i P A and ρ i Ps0, εr such that µpAz Y iPI B i q " 0 by V ε pAq, and we consider m ε u,ω : OpOq r0, 8s given by m ε u,ω pAq :" inf

γ ρ,

It

# ÿ iPI m u,ω pB i q : tB i u iPI P V ε pAq + ,
and we define m ů,ω : OpOq r0, 8s by m ů,ω pAq :" sup

εą0 m ε u,ω pAq " lim ε 0 m ε u,ω pAq.
The set function m ů,ω is called the Vitali envelope of m u,ω (see §3.3).

Step 1: link between Γ-lim Γ-lim Γ-lim and Vitali envelope. Let u P H 1,p µ pO; R m q. We are going to prove that ΓpL p µ q-lim t 8

E t pu, ωq ď m ů,ω pOq. (4.27)

Without loss of generality we can assume that m ů,ω pOq ă 8. Fix any ε ą 0. By definition of m ε u,ω pOq there exists tB i u iPI P V ε pOq such that ÿ iPI m u,ω pB i q ď m ε u,ω pOq `ε 2 . (4.28) Fix any t ą 0. For each i P I, by definition of m t u,ω pB i q there exists v i t P H 1,p µ pO; R m q such that v i t ´u P H 1,p µ,0 pB i ; R m q and ż On the other hand, we have

B i L t `x, ∇ µ v i t pxq, ω ˘dµpxq ď m t u,ω pB i q `εµpB i q 2µpOq . (4.29) Define u ε t : O R m by u ε t :" # u in Oz Y iPI B i v i t in B i . Then u ε t ´u P H 1,p µ,0 pO; R m q. Moreover,
}u ε t ´u} p L p µ pO;R m q " ż O ˇˇu ε t ´uˇˇp dµ " ÿ iPI ż B i ˇˇv i t ´uˇˇp dµ.
As O supports a p-Sobolev inequality, see Proposition 3.4(iii), and ρ i Ps0, εr for all i P I, we have

ÿ iPI ż B i ˇˇv i t ´uˇˇp dµ ď ε p C p S ÿ iPI ż B i ˇˇ∇ µ v i t ´∇µ u ˇˇp dµ
with C S ą 0, and so Step 2: differentiation with respect to µ µ µ. Let u P H 1,p µ pO; R m q be such that Gpuq :" ş O Gp∇ µ upxqqdµpxq ă 8. We are going to prove that m ů,ω pOq " m u,ω pAq ď m u,ω pBq `mu,ω pCq, which shows the subadditivity of m u,ω . On the other hand, given any t ą 0, by using the right inequality in (C 6 ) we have m t u,ω pAq ď ż A β `1 `Gp∇ µ upxqq ˘dµpxq for all A P OpOq. Thus (4.36) holds with the Radon measure ν :" β `1 `Gp∇ µ up¨qq ˘µ which is necessarily finite since Gpuq ă 8.

}u ε t ´u} p L p µ pO;R m q ď 2 p ε p C p S ˜ÿ iPI ż B i |∇ µ v i t |
ż O lim ρ 0 m u,
Step 3: cut-off method. Let τ Ps0, 1r, let σ Psτ, 1r and let u P H 1,p µ pO; R m q be such that Gpσuq ă 8. We are going to prove that for µ-a.e. }D µ ϕ} L 8 µ pΩ;R N q ď θ ρp1 ´λq for some θ ą 0 (which does not depend on ρ). Define v P H 1,p µ pO; R m q by v :" ϕu x `p1 ´ϕqu " ϕpu x ´uq `u.

Then

τ v ´τ u P H 1,p µ,0 pB ρ pxq; R m q (4.40)

and, using Proposition 3.4(iv), i.e. ∇ µ u x pyq " ∇ µ upxq for µ-a.a. y P O, and Proposition 3.4(vii),

τ ∇ µ v " " τ ∇ µ upxq in B λρ px 0 q τ σ `ϕσ∇ µ upxq `p1 ´ϕqσ∇ µ u ˘``1 ´τ σ ˘Ψρ in B ρ px 0 qzB λρ px 0 q (4.41) with Ψ ρ :" τ 1´τ σ D µ ϕ b pu x ´uq.
From (4.38) and (4.40) we have τ v `pτ w ´τ u x q ´τ u P H 1,p µ,0 pB ρ pxq; R m q. Noticing that µpBB λρ pxqq " 0 (see Remark 3.3) and, because of Proposition (3.4)(ii), ∇ µ pτ w ´τ u x qpyq " τ ∇ µ w ´τ ∇ µ u x " 0 for µ-a.a. y P B ρ pxqzB λρ pxq, we see that On the other hand, taking (4.41) into account and using (C 1 ), we have

m t τ u,ω pB ρ pxqq µpB λρ pxqq ď 1 µpB λρ pxqq ż Bρpxq L t py, τ ∇ µ v `τ ∇ µ w ´τ ∇ µ u x , ωq dµ " 1 µpB λρ pxqq ż B λρ pxq L t py, τ ∇ µ upxq `τ ∇ µ w ´τ ∇ µ upxq, ωq dµ `1 µpB λρ pxqq ż BρpxqzB λρ pxq L t py, τ ∇ µ v, ωqdµ " 1 µpB λρ pxqq ż B λρ pxq L t py, τ ∇ µ w, ωq dµ `1 µpB λρ pxqq ż BρpxqzB λρ pxq L t py, τ ∇ µ v,
Gpτ ∇ µ vq ď c 1 p1 `Gpσ∇ µ upxqq `Gpσ∇ µ uq `GpΨ ρ qq (4.43)
with c 1 :" 2pγ `γ2 q ą 0. Moreover, it is easy to see that

}Ψ ρ } L 8 µ pBρpxq;Mq ď θτ p1 ´τ σ qp1 ´λq 1 ρ }u ´ux } L 8 µ pBρpxq;R m q ,
where lim ρ 0 θτ p1 ´τ σ qp1 ´λq

1 ρ }u ´ux } L 8 µ pBρpxq;R m q " 0 (4.44)
by Proposition 3.4(iv), i.e. lim ρ 0 1 ρ }u ´ux } L 8 µ pBρpxq;R m q " 0. From (C 1 )-(C 2 ) there exists r ą 0 such that c 2 :" sup |ξ|ďr Gpξq ă 8

(see Remark 2.2). By (4.44) there exists ρ ą 0 such that

θτ p1´τ σ qp1´λq 1 ρ }u ´ux } L 8
µ pBρpxq;R m q ă r for all ρ Ps0, ρr. Fix any ρ Ps0, ρr. We then have GpΨ ρ q ď c 2 .

(4.45)

From (4.43) and (4.45) it follows that Conclusion of the steps 1, 2 and 3. As a direct consequence of (4.27), (4.35) and (4.37) together with Remarks 4.1 and 4.2, we have the following lemma.

β µpB λρ pxqq ż BρpxqzB λρ pxq Gpτ ∇ µ vqdµ ď βc 1 `1 `Gpσ∇ µ upxqq `c2 ˘µpB ρ pxqzB
Lemma 4.3. For every τ Ps0, 1r and every u P H 1,p µ pO; R m q such that Gpτ uq ă 8 and Gpσuq ă 8 for some σ Psτ, 1r, one has

ΓpL p µ q-lim t 8 E t pτ u, ωq ď ż O lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq.
Step 4: end of the proof. Let u P H 1,p µ pO; R m q. We have to prove that Substep 4-1: proving (4.53) under the constraint ∇ µ upxq P intpQ µ G x q ∇ µ upxq P intpQ µ G x q ∇ µ upxq P intpQ µ G x q for µ µ µ-a.a. x P O x P O x P O. Assume that ∇ µ upxq P intpQ µ G x q for µ-a.a. x P O. (4.57) Then, since (C 1 )-(C 2 ) implies that τ ∇ µ upxq P intpQ µ G x q for all τ Ps0, 1r and for µ-a.a. x P O (see Remark 2.3(v)), by (C 4 ) we have lim

ΓpL p µ q-lim t 8 E t pu, ωq ď ż O lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq,
τ 1 ´Qµ Gpx, τ ∇ µ upxqq ě Q µ Gpx, ∇ µ upxqq for µ-a.a. x P O.
(4.58) Using (4.58) and the left inequality in (C 6 ) we see that H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq for all u P H 1,p µ pO; R m q.

1 α E lim pu, ωq ě ż O lim τ 1 ´Qµ Gpx, τ ∇ µ upxqqdµ ě ż O Q µ Gpx, ∇ µ upxqqdµ ": Q µ Gpuq

Applications

In this section we give some applications of Theorem 2.5 and Proposition 2.8.

Note that by Remark 3.3 we have µpBBq " 0 for all B P BapXq, and so BapXq Ă B 0 . In order to deal with homogenization in the framework of metric measure spaces, we need to introduce the quadruple `G, tτ g u gPG , U, th t u tą0 ˘with:

(O 1 ) G a subgroup of HomeopXq such that µ is G-invariant, i.e. g 7 µ " µ for all g P G which means that for every g P G and every A P BpXq, µpg ´1pAqq " µpAq;

(O 2 ) tτ g u gPG a group of P-preserving transformations on pΩ, F, Pq, i.e.

$ & % τ g is F-mesurable for all g P G τ g oτ f " τ gof and τ g ´1 " τ ´1 g for all g, f P G Ppτ g pAqq " PpAq for all A P F and all g P G;

(O 3 ) U P B 0 such that µpUq ą 0; (O 4 ) th t u tą0 Ă HomeopXq, where U can be interpretated as the "unit cell" with respect to X and th t u tą0 as a family of "dilations" in X. Let J Ă B 0 be given by (Theses sets are of interest for the development of subadditive theorems in the setting on measure spaces with acting group, see [START_REF] Anza | On subadditive theorems for group actions and homogenization[END_REF] for more details.) To obtain homogenization results in the framework of metric measure spaces, we need to refine it by assuming that:

(F 1 ) pX, BpXq, µq is G-meshable with respect to h k pUq ( kPN ˚, i.e. for each k P N ˚there is G k P Dph k pUqq with the property that for each q P N ˚there exist G q,k Ă G q,k P

P f pG k q such that $ ' & ' % Y gPG q,k g ´1ph k pUqq Ă h q pUq Ă Y gPG q,k g ´1ph k pUqq lim q 8
ˇˇG q,k zG q,k ˇˇµph k pUqq µph q pUqq " 0;

(F s 1 ) pX, BpXq, µq is strongly G-meshable with respect to h k pUq ( kPN ˚, i.e. pX, BpXq, µq is G-meshable with respect to h k pUq ( kPN ˚with the additional property that G " Y kPN ˚Gk where every G k is a countable discrete amenable 7 subgroup of G;

7 Let H Ă G be a subgroup and let P f pH q denote the class of finite subsets of H . We say that H is amenable if for each E P P f pH q and each δ ą 0 there exists F P P f pH q such that,|F ∆EF | ď δ|F |, where | ¨| denotes the counting measure on G, ∆ is the symmetric difference of sets and EF " tgof : pg, f q P E ˆF u. (For more details on the theory of amenable groups, see [START_REF] Krengel | Ergodic theorems[END_REF]§6.4 Let I :" A P F : Ppτ g pAq∆Aq " 0 for all g P G ( be the σ-algebra of invariant sets with respect to pΩ, F, P, tτ g u gPG q. Recall that pΩ, F, P, tτ g u gPG q is said to be ergodic if PpAq P t0, 1u for all A P I.

Theorem 5.4 (stochastic case). Under (O 1 )-(O 4 ), (F s 1 )-(F s 2 ), (S 1 )-(S 2 ) and (S s 3 ), there exists Ω 1 P F with PpΩ 1 q " 1 such that for every ω P Ω 1 and every where ErSph k pUq, ¨qs denotes the expectation of Sph k pUq, ¨q with respect to P.

Homogenization theorems.

In what follows, we establish deterministic and stochastic homogenization theorems of nonconvex unbounded integrals in the setting of metric measure spaces according to our framework in §5.2.1 (see Theorems 5.9 and 5.10). Let L : X ˆM ˆΩ r0, 8s be a Borel measurable stochastic integrand such that: (H 1 ) L is p-coercive, i.e. there exists c ą 0 such that for every x P X, every ξ P M and every ω P Ω, Lpx, ξ, ωq ě c|ξ| p ; (H 2 ) L has G-growth, i.e. there exist α, β ą 0 such that for every x P X, every ξ P M and every ω P Ω, αGpξq ď Lpx, ξ, ωq ď βp1 `Gpξqq with G : M r0, 8s satisfying (C 1 )-(C 4 ); (H s 3 ) L is G-stationary with respect to tτ g u gPG , i.e. for every x P X, every ξ P M, every g P G and every ω P Ω, Lpg ´1pxq, ξ, ωq " Lpx, ξ, τ g pωqq and, when L is deterministic, i.e. Lpx, ξ, ωq " Lpx, ξq, (H 3 ) L is G-periodic, i.e. for every x P X, every ξ P M and every g P G,

Lpg ´1pxq, ξq " Lpx, ξq.

For each t ą 0, we consider L t : X ˆM ˆΩ r0, 8s given by L t px, ξ, ωq :" Lph t pxq, ξ, ωq.

(5.2) L pτ, ωq ď 0, and so, letting τ 1 ´in (5.4), we obtain lim τ 1 ´sup tą0 ∆ at Lt pτ, ωq ď 0 which means that tL t u tą0 is ru-usc at ω with ta t p¨, ωqu tą0 " taph t p¨q, ωqu tą0 . Lemma 5.5 shows that (H 4 ) implies (C 7 ). So, according to Theorem 2.5 and Proposition 2.8, to prove Theorems 5.9 and 5.10 below, it is sufficient to establish the condition (C 8 ) in Theorem 2.5. For this, we consider the following assumption:

(H 5 ) for every ξ P Y yPO intpQ µ G y q, there exists C ξ ą 0 such that for every A P B 0 , inf "ż Å Gpξ `∇µ wpyqqdµpyq : w P H 1,p µ,0 p Å; R m q * ď C ξ µpAq.

Remark 5.6. It is clear that if intpQ µ G y q Ă G for all y P O, then (H 5 ) is satisfied with C ξ " Gpξq. (In particular, (H 5 ) holds when G is H 1,p µ -quasiconvex.)

For each ξ P M, let S ξ L : B 0 ˆΩ r0, 8s be defined by

S ξ L pA, ωq :" inf "ż Å Lpy, ξ `∇µ wpyq, ωqdµpyq : w P H 1,p µ,0 p Å; R m q * .
As M is separable, also is Y yPO intpQ µ G y q. Let D Ă Y yPO intpQ µ G y q be a countable set such that D " Y yPO intpQ µ G y q. In the stochastic case, we need the following two additional assumptions: (H s 6 ) for every ξ P Y yPO intpQ µ G y q and every A P B 0 , S ), there exists Ω 1 P F with PpΩ 1 q " 1 such that for every ω P Ω 1 , every ρ ą 0, every x P O and every ξ P intpQ µ G x q, one has (5.7)

lim
Finally, fix x P O and ξ P intpQ µ G x q (then ξ P Y yPO intpQ µ G y q). Taking (5.2) into account, from (O 3 ) and (F 3 ), we see that for every ω P Ω 1 , every B P BapXq, and every t ą 0, one has and the proposition follows by using (5.7) with B " B ρ pxq.

S
For each t ą 0, let E t : H 1,p µ pO; R m q ˆΩ r0, 8s be defined by (1.1) with L t given by (5.2). Taking Theorem 2.5 and Proposition 2.8 into account, from Propositions 5.7 and 5.8 respectively, we deduce the following two homogenization theorems. Theorem 5.9 (deterministic case). Assume that p ą κ. where E I denotes the conditional expectation over I with respect to P, with I being the σ-algebra of invariant sets with respect to pΩ, F, P, tτ g u gPG q. If moreover pΩ, F, P, tτ g u gPG q is ergodic then L hom is deterministic and is given by

L hom pξq " $ & % lim τ 1 ´inf kPN ˚ErS τ ξ L ph k pUq, ¨qs µph k pUqq if ξ P Y yPO Q µ G y 8 otherwise,
where E denotes the expectation with respect to P.

  1. Introduction 2. The Γ-convergence result 3. Auxiliary results 3.1. Cheeger-Sobolev spaces 3.2. Ru-usc integrands 3.3. Integral representation of the Vitali envelope of a set function 4. Proofs 4.1. Proof of the lower bound 4.2. Proof of the upper bound 4.3. Proof of the Γ-convergence result 5

  holds then τ G Ă intpGq for all τ Ps0, 1r, and there exists r ą 0 such that sup |ξ|ďr Gpξq ă 8, see [AHM12b, Lemma 4.1].

  and [AHM20a, §7].) Proposition 3.4. Under (3.1), (3.2) and (3.3) the following properties hold:

3. 3 .

 3 Integral representation of the Vitali envelope of a set function. What follows was first developed in [BFM98, BB00] (see also [AHM17, AHCM17, AHM18]). Let pO, dq be a metric space, let OpOq be the class of open subsets of O and let µ be a positive finite Radon measure on O. We begin with the concept of the Vitali envelope of a set function.

εą0S ε pAq " lim ε 0 Slim ρ 0

 00 ε pAq. The interest of Definition 3.16 comes from the following integral representation result. (For a proof we refer to [AHM18, §3.3] or [AHCM17, §A.4].) Theorem 3.17. Let S : OpOq r0, 8s be a set function satisfying the following two conditions: (i) there exists a finite Radon measure ν on O which is absolutely continuous with respect to µ such that SpAq ď νpAq for all A P OpOq; (ii) S is subadditive, i.e. SpAq ď SpBq `SpCq for all A, B, C P OpOq with B, C Ă A, B X C " H and µpAzpB Y Cqq " 0. Then lim ρ 0 SpBρp¨qq µpBρp¨qq P L 1 µ pOq and for every A P OpOq, one has S ˚pAq " ż A SpB ρ pxqq µpB ρ pxqq dµpxq. 4. Proofs 4.1. Proof of the lower bound. Here we prove Proposition 2.6.

k

  pUq ˘: k P N ˚, H P D `hk pUq ˘and |H| ă 8 * with D `hk pUq ˘:" ! H Ă G : g ´1ph k pUqq ( gPH is disjoint ) .

  Assume that p ą κ. If (C 1 )-(C 7 ) hold then, for every ω P Ω, one has

	then tE t u tą0 almost surely ΓpL p µ q-converges as t	8 to the functional E lim : H 1,p µ pO; R m q Ω
	r0, 8s defined by (1.2) with L lim : O ˆM ˆΩ	r0, 8s given by
		L lim px, ξ, ωq " lim τ 1 ´lim ρ 0	lim t 8	H ρ µ L t px, τ ξ, ωq
	Theorem 2.5 is a consequence of the following two propositions.
	Proposition 2.6 (Γ-lim). Assume that p ą κ. If (C 1 )-(C 2 ) and (C 5 )-(C 7 ) hold then, for
	every ω P Ω, one has
					ż
	ΓpL p µ q-lim t 8	E t pu, ωq ě	O	lim τ 1 ´lim ρ 0	lim t 8	H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq
	for all u P H 1,p µ pO; R m q.
	Proposition 2.7 (Γ-lim). ΓpL p µ q-lim t 8 E t pu, ωq ď	ż O	lim τ 1 ´lim ρ 0	lim t 8	H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq
	for all u P H 1,p µ `O; R m	˘.
	Let L 8 : O ˆM ˆΩ	r0, 8s be defined by
			L 8 px, ξ, ωq :" lim ρ 0	lim t 8	H ρ µ L t px, ξ, ωq.
	(Note that if (C 8 ) is satisfied then L 8 p¨, ¨, ωq " lim ρ 0 lim t 8 H ρ µ L t p¨, ¨, ωq for P-a.e. ω P Ω.)
	Let p L 8 : O ˆM ˆΩ	r0, 8s be given by
			p L 8 px, ξ, ωq :" lim
						τ 1
							1 , one has
		lim ρ 0	lim t 8	H ρ µ L t px, ξ, ωq ě lim ρ 0	lim t 8	H ρ µ L t px, ξ, ωq
	for all x P O and all ξ P intpQ µ G x q,

  Definition 3.9. Let ω P Ω. We say that L is radially uniformly upper semicontinuous (ru-usc) at ω if there exists ap¨, ωq P L 1 Remark 3.10. If L is ru-usc at ω P Ω then lim τ 1 ´Lpx, τ ξ, ωq ď Lpx, ξ, ωq for all x P O and all ξ P L x,ω . On the other hand, given ω P Ω, if there exist x P O and ξ P L x,ω such that Lpx, ¨, ωq is lsc at ξ then, for each ap¨, ωq P L 1 µ pO; s0, 8sq, lim τ 1 ´∆a L pτ, ωq ě 0, and so if in addition L is ru-usc at ω then lim τ 1

	µ pO; s0, 8sq such that
	lim τ 1 ´∆a L pτ, ωq ď 0.
	The concept of ru-usc integrand was introduced in [AH10] and then developed in [AHM11,
	AHM12a, AHM12b, Man13, AHM14, AHMZ15, AHM18].
	´∆a L pτ, ωq " 0 for some ap¨, ωq P L 1 µ pO; s0, 8sq.
	r0, 8s be a Borel measurable
	stochastic integrand. For each tap¨, ωqu ω Ă L 1 µ pO; s0, 8sq we define ∆ a L : r0, 1sˆΩ s´8, 8s
	by
	∆ a

s theorem, and (3.6) follows because µ `OzO ˘" 0. 3.2. Ru-usc integrands. Let pX, d, µq be a metric measure space, let O Ă X be an open set, let pΩ, F, Pq be a probability space and let L : OˆMˆΩ L pτ, ωq :" sup xPO sup ξPLx,ω Lpx, τ ξ, ωq ´Lpx, ξ, ωq apx, ωq `Lpx, ξ, ωq , where L x,ω denotes the effective domain of Lpx, ¨, ωq. Remark 3.11. Given ω P Ω, if, for every x P O, Lpx, ¨, ωq is convex and 0 P L x,ω , then L is ru-usc at ω. The interest of Definition 3.9 comes from the following theorem. (For a proof we refer to [AHM11, Theorem 3.5] and also [AHM12b, §4.2].) Let p L : O ˆM ˆΩ r0, 8s be defined by p Lpx, ξ, ωq :" lim τ 1 ´Lpx, τ ξ, ωq.

  Without loss of generality we can assume that lim t 8 E t pu t , ωq " lim t 8 E t pu t , ωq ă 8, and so supIn particular, sup tą0 }∇ µ u t } L p µ pO;Mq ă 8 because tL t u tą0 is p-coercive, see (C 5 ). Then ∇ µ u t pxq P G for all t ą 0 and µ-a.a. x P O

			ż
	t 8	E t pu t , ωq ě	O	lim τ 1 ´lim ρ 0	lim t 8	H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq.	(4.1)
				E t pu t , ωq ă 8.	(4.2)
				tą0
								(4.3)
	and, up to a subsequence,
			u 6 we deduce
	that, up to a subsequence,
				}u t ´u} L 8 µ pO;R m q	0.	(4.6)

t á u in H 1,p µ pO; R m q.

(4.4)

As G is convex, see (C 1 ) and Remark 2.2, from (4.3) and (4.4) it follows that

∇ µ upxq P G for µ-a.a. x P O. (4.5)

As }u t ´u} H 1,p µ pO;R m q 0, sup tą0 }∇ µ u t } L p µ pO;Mq ă 8 and p ą κ, from Theorem 3.

  Bρpx 0 q L t px, τ ∇ µ u t pxq, ωqdµ ď lim Bρpx 0 q L t px, ∇ µ u t pxq, ωqdµ. Bρpx 0 q L t px, τ ∇ µ u t pxq, ωqdµ ě lim

								8 and
	ρ	0 we obtain
	lim ρ 0	lim t 8 ż Bρpx 0 q	L `lim ρ 0	lim t 8	∆pτ, ωqż	Bρpx 0 q	a t px, ωqdµ.
	But, from (C 7 ) we have
							lim τ 1 ´∆pτ , ωq ď 0
	and, by (2.2),
				lim ρ 0	lim t 8 ż Bρpx 0 q	a t px, ωqdµ ă 8
	with şBρpx 0 q a t px, ωqdµ ě 0, hence
				lim τ 1 ´lim ρ 0	lim t 8	∆pτ, ωqż	Bρpx 0 q	a t px, ωqdµ ď 0,
	and consequently
	lim τ 1 ´lim ρ 0 t 8 ż Thus, to prove (4.10) it is sufficient to show that lim t 8 ż ρ 0 lim
	lim τ 1 ´lim ρ 0	lim t 8 ż τ 1	´lim ρ 0	lim t 8

, ωqdµ `∆pτ, ωqż Bρpx 0 q a t px, ωqdµ with ∆pτ, ωq :" sup tą0 ∆ at Lt pτ, ωq, where ∆ at Lt pτ, ωq is given by (2.3). Letting t t px, τ ∇ µ u t pxq, ωqdµ ď `1 `∆pτ, ωq ˘lim ρ 0 lim t 8 ż Bρpx 0 q L t px, ∇ µ u t pxq, ωqdµ

  Substep 3-2: proving that lim σ 1 ´lim λ 1 ´lim ρ 0 lim t 8 Γ t,ρ,λ,σ " 0 lim σ 1 ´lim λ 1 ´lim ρ 0 lim t 8 Γ t,ρ,λ,σ " 0 lim σ 1 ´lim λ 1 ´lim ρ 0 lim t 8 Γ t,ρ,λ,σ " 0. For every t Ps0, t ρ r, we have Γ t,ρ,λ,σ ď `1 `∆pσ, ωq ˘νt `Bρ px 0 qzB λρ px 0 qq Bρpx 0 qzB λρ px 0 q Bρpx 0 qzB λρ px 0 q a t px, ωqdµ ě 0, hence

								λ " 1	´µpB λρ px 0 qq µpB ρ px 0 qq	,
	hence
								lim ρ 0	γ ρ,λ " 1 ´lim ρ 0	µpB λρ px 0 qq µpB ρ px 0 qq	,
	and so, by using Proposition (3.4)(vi),
								lim λ 1 ´lim ρ 0	γ ρ,λ " 0.	(4.21)
								μpB
								ρ px 0 qq
								`∆pσ, ωq	µpB ρ px 0 qq 1	ż
		´lim ρ 0	lim t 8	1 µpB ρ px 0 qq	ż Bρpx 0 qzB λρ px 0 q	a t px, ωqdµ ď lim ρ 0	lim t 8 ż Bρpx 0 q	a t px, ωqdµ ă 8
	with	µpBρpx 0 qq 1	ş
		lim σ 1 ´lim λ 1 ´lim ρ 0	lim t 8	∆pσ, ωq	1 µpB ρ px 0 qq	ż Bρpx 0 qzB λρ px 0 q	a t px, ωqdµ ď 0.	(4.23)

a t px, ωqdµ. (4.22) But lim σ 1 ´∆pσ, ωq ď 0 by (C 7 ), and by (2.2) we have lim λ 1

  because of Proposition 3.4(ii), ∇ µ u ε t pxq " ∇ µ v i t pxq for µ-a.a. x P B i . From (4.29) we see that

	E t pu ε t , ωq ď	ÿ iPI	m t u,ω pB i q	`ε 2	,
	hence lim t 8 Epu ε t , ωq ď m ε u,ω pOq `ε by using (4.28), and consequently
	lim ε 0	lim t 8	E t pu ε t , ωq ď m ů,ω pOq.	(4.30)

  OpOq. For each t ą 0, from the definition of m t u,ω in (4.26), it is easy to see that for every A, B, C P OpOq with B, C Ă A, B X C " H and µpAzpB Y Cqq " 0, one has

			ω pB ρ pxqq µpB ρ pxqq	dµpxq.	(4.35)
	According to Theorem 3.17, to prove (4.35) it suffices to establish that m u,ω is subadditive
	and there exists a finite Radon measure ν on O which is absolutely continuous with respect
	to µ such that
		m u,ω pAq ď νpAq		(4.36)
	for all A P m t u,ω pAq ď m t u,ω pBq `mt u,ω pCq,
	and so
	lim t 8	m t u,ω pAq ď lim t 8	m t u,ω pBq `lim t 8	m t u,ω pCq,
	i.e.

  x P O, B λρ pxq L t py, τ ∇ µ wpyq, ωqdµpyq ď m t τ ux,ω pB λρ pxqq `εµpB λρ pxqq. Urysohn function ϕ P LippΩq for the pair pΩzB ρ pxq, B λρ pxqq such that

										(4.39)
	By Proposition 3.4(v) there is a
				lim ρ 0	m τ u,ω pB ρ pxqq µpB ρ pxqq	ď lim ρ 0	m τ ux,ω pB ρ pxqq µpB ρ pxqq	,	(4.37)
	where u x P H 1,p µ pΩ; R m q is given by Proposition 3.4(iv).
	Remark 4.1. For µ-a.e. x P O, one has
		lim ρ 0	m τ ux,ω pB ρ pxqq µpB ρ pxqq	" lim ρ 0	lim t 8	H ρ µ L t px, τ ∇ µ upxq, ωq.
	Remark 4.2. If Gpτ uq ă 8 then Gpτ u x q ă 8 for µ-a.a. x P O, and so, by the step 2,
	lim ρ 0	m τ u,ω pB ρ pxqq µpB ρ pxqq	" lim ρ 0	m τ u,ω pB ρ pxqq µpB ρ pxqq	and lim ρ 0	m τ ux,ω pB ρ pxqq µpB ρ pxqq	" lim ρ 0	m τ ux,ω pB ρ pxqq µpB ρ pxqq	.

Fix any t ą 0, any λ Ps0, 1r, any ρ ą 0 and any ε ą 0. By definition of m t τ ux,ω pB λρ pxqq in (4.26), there exists w P H 1,p µ pO; R m q such that τ w ´τ u x P H 1,p µ,0 pB λρ pxq; R m q (4.38) and ż

  ωqdµ.

	From (4.39) and the right inequality in (C 6 ) it follows that
	m t τ u,ω pB ρ pxqq µpB ρ pxqq	ď	m t τ u,ω pB ρ pxqq µpB λρ pxqq	ď	m t τ ux,ω pB λρ pxqq µpB λρ pxqq	`ε	`β µpB ρ pxqzB λρ pxqq µpB λρ pxqq
						ż
					`β µpB λρ pxqq	BρpxqzB λρ pxq	Gpτ ∇ µ vqdµ.	(4.42)

  BρpxqzB λρ pxq Gpτ ∇ µ vqdµ ď βc 1 `1 `2Gpσ∇ µ upxqq `c2 ˘µpB ρ pxqzB λρ pxqq µpB λρ pxqq `βc 1 µpB ρ pxqq µpB λρ pxqq ż Bρpxq |Gpσ∇ µ upyqq ´Gpσ∇ µ upxqq|dµpyq. (4.46) Bρpxq |Gpσ∇ µ upyqq ´Gpσ∇ µ upxqq|dµpyq. (4.47) As Gpσuq ă 8, i.e. Gpσ∇ µ up¨qq P L 1 µ pOq, (and µ is a doubling measure) we can assert that:

	hence β µpB λρ pxqq From (4.42) and (4.46) we deduce that ż m t τ u,ω pB ρ pxqq µpB ρ pxqq ď m t τ ux,ω pB λρ pxqq µpB λρ pxqq `βc 1 ˆ1 `1 c 1 `2Gpσ∇ µ upxqq `c2 `ε `βc 1 µpB ρ pxqq lim ρ 0 ż Bρpxq |Gpσ∇ µ upyqq ´Gpσ∇ µ upxqq|dµpyq " 0. ˙µpB ρ pxqzB λρ pxqq µpB λρ pxqq As the boundary of any ball is of zero measure (see Remark 3.3), we have lim ρ 0 µpB ρ pxqzB λρ pxqq µpB λρ pxqq " lim ρ 0 ˆ1 ´µpB ρ pxqq µpB λρ pxqq ˙" 1 ´lim ρ 0 µpB ρ pxqq µpB λρ pxqq , and so, by using Proposition (3.4)(vi), lim λ 1 ´lim ρ 0 µpB ρ pxqzB λρ pxqq µpB λρ pxqq " 0. Moreover, we have: lim ρ 0 lim t 8 m t τ u,ω pB ρ pxqq µpB ρ pxqq " lim ρ 0 m τ u,ω pB ρ pxqq µpB ρ pxqq ; lim ρ 0 lim t 8 m t τ ux,ω pB λρ pxqq µpB λρ pxqq ď lim ρ 0 lim t 8 m t τ ux,ω pB ρ pxqq µpB ρ pxqq " lim ρ 0 m τ ux,ω pB ρ pxqq µpB ρ pxqq Letting t 8, ρ 0 and λ 1 ´in (4.47) and using (4.48), (4.49), (4.50), (4.51) and (4.48) (4.49) (4.50) (4.51) . (4.52) (4.52) we conclude that lim ρ 0 m τ u,ω pB ρ pxqq µpB ρ pxqq ď lim ρ 0 m τ ux,ω pB ρ pxqq µpB ρ pxqq `ε, µpB λρ pxqq ż Gpσ∇ µ upxqq ă 8; and (4.37) follows by letting ε 0.

λρ pxqq µpB λρ pxqq `βc 1 µpB λρ pxqq ż BρpxqzB λρ pxq Gpσ∇ µ upyqqdµpyq. But ż BρpxqzB λρ pxq Gpσ∇ µ upyqqdµpyq ď µpB ρ pxqqż Bρpxq |Gpσ∇ µ upyqq ´Gpσ∇ µ upxqq|dµpyq `µpB ρ pxqzB λρ pxqqGpσ∇ µ upxqq,

  L t px, τ ∇ µ upxq, ωq for µ-a.a. x P O. (4.56)

	and
	lim τ 1 ´lim ρ 0	lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωq" lim τ 1 ´lim ρ 0	lim t 8 H ρ µ
						ωqdµpxq.	(4.53)
	Without loss of generality we can assume that
		ż
		O	lim τ 1 ´lim ρ 0	lim t 8	H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq ": E lim pu, ωq ă 8.	(4.54)
	Then, by Proposition 2.8(i) we have
				∇ (4.55)

µ upxq P Q µ G x for µ-a.a. x P O

  Substep 4-2: proof of (4.53). First of all, from (C 7 ) and Proposition 3.14 we can assert that L 8 :" lim ρ 0 lim t 8 H ρ µ L t is ru-usc at ω with a 8 p¨, ωq given by (2.2). Moreover, by (C 6 ) we see that for every x P O, the effective domain of L 8 px, ¨, ωq is equal to Q µ G x . Taking (C 1 )-(C 2 ) into account (see Remark 2.3(v)), from Theorem 3.12(ii) it follows that Proof of the Γ-convergence result. Here we prove Theorem 2.5 Proof of Theorem 2.5. Fix ω P Ω 1 . By (C 6 ) we see that αQ µ Gpx, ξq ď lim lim ρ 0 lim t 8 H ρ µ L t px, ¨, ωq ˘and dom `lim ρ 0 lim t 8 H ρ µ L t px, ¨, ωq ˘denotes the effective domain of lim ρ 0 lim t 8 H ρ µ L t px, ¨, ωq and lim ρ 0 lim t 8 H ρ µ L t px, ¨, ωq respectively. Let px, ξq P O ˆM. If ξ R Q µ G x then there exists τ ξ Ps0, 1r such that τ ξ R Q µ G x for all τ P rτ ξ , 1r. Hence:' if ξ R Q µ G xthen, by (4.67), L t px, τ ξ, ωq " 8 for all τ P rτ ξ , 1r;' if ξ P Q µ G x then, from (C 1 )-(C 2 ) (see Remark 2.3(v)), we have τ ξ P intpQ µ G x q for all τ Ps0, 1r, and so, by (C 8 ),

	4.3. ρ 0	lim t 8	H ρ µ L t px, ξ, ωq ď lim ρ 0	lim t 8	H ρ µ L t px, ξ, ωq ď β `1 `Qµ Gpx, ξq	for
	, µ L ď Q µ Gpuq ă 8. (4.59) Taking (C 3 ) into account, from (4.57) and (4.59) it follows that hence, by (4.54), Gpuq ă 8. (4.60) ΓpL p µ q-lim t 8 E t pτ u, ωq ď ż O lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq for all τ Ps0, 1r. (4.61) On the other hand, from the right inequality in (C 6 ) we see that for every τ Ps0, 1r, lim ρ 0 lim t 8 H ρ β `1 `Gpτ ∇up¨qq ˘, and consequently, by using (C 1 ), lim ρ 0 lim t 8 H ρ µ L with f P L 1 µ pOq by (C 2 ) and (4.60). Taking (4.56) into account, from Lebesgue's dominated convergence theorem we deduce that lim τ 1 ´żO lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq " ż O lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq. From (4.61) we conclude that lim τ 1 ´ΓpL p µ q-lim t 8 E t pτ u, ωq ď ż O lim τ 1 ´lim ρ 0 lim t 8 H ρ L 8 :" lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t is ru-usc at ω with a 8 p¨, ωq. (4.62) From (4.54) we see that ∇ µ upxq P p L 8,x,ω for µ-a.a. x P O, where p L 8,x,ω denotes the effective domain of p L 8 px, ¨, ωq. Hence, for every τ Ps0, 1r, ż O p L 8 px, τ ∇ µ upxq, ωqdµ ď `1 `∆a8 p L8 pτ, ωq ˘żO p L 8 px, ∇ µ upxq, ωqdµ `∆a8 p L8 pτ, ωq ż O a 8 px, ωqdµ with ∆ a8 p L8 pτ, ωq :" sup xPO sup ξP p L8,x,ω p L8px,τ ξ,ωq´p L8px,ξ,ωq a8px,ωq`p L8px,ξ,ωq , i.e. E lim pτ u, ωq ď `1 `∆a8 p L8 pτ, ωq ˘Elim pu, ωq `∆a8 p L8 pτ, ωq ż O a 8 px, ωqdµ (4.63) for all τ Ps0, 1r. Using (4.54) and (2.2), i.e. a 8 P L 1 µ pOq, we see that E lim pτ u, ωq ă 8 for all τ Ps0, 1r. (4.64) On the other hand, from (4.55) and (C 1 )-(C 2 ) (see Remark 2.3(v)) we deduce that ∇ µ pτ uqpxq P intpQ µ G x q for all τ Ps0, 1r and µ-a.a. x P O. (4.65) According to (4.65) and (4.64), from the substep 4-1 we can assert that ΓpL p µ q-lim t 8 E t pτ u, ωq ď E lim pτ u, ωq for all τ Ps0, 1r, and so, taking (4.63) into account, ΓpL p µ q-lim t 8 E t pτ u, ωq ď `1 `∆a8 p L8 pτ, ωq ˘Elim pu, ωq `∆a8 p L8 pτ, ωq ż O a 8 px, ωqdµ (4.66) lim τ 1 ´ΓpL p µ q-lim t 8 E t pτ u, ωq ď E lim pu, ωq, all x P O and all ξ P M. So, for every x P O, one has dom ˆlim ρ 0 lim t 8 H ρ µ L t px, ¨, ωq ˙" dom ˆlim ρ 0 lim t 8 H ρ µ L t px, ¨, ωq ˙" Q µ G x , (4.67) where dom `lim ρ 0 lim t 8 H ρ µ L t px, τ ξ, ωq " lim ρ 0 lim t 8 H ρ µ lim ρ 0 lim t 8 H ρ µ L t px, τ ξ, ωq ě lim ρ 0 lim t 8 H ρ µ L t px, τ ξ, ωq for all τ Ps0, 1r. It follows that lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px, ξ, ωq ě lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px, ξ, ωq for all px, ξq P O ˆM. From Propositions 2.6 and 2.7 we deduce that ΓpL p µ q-lim t 8 E t pu, ωq ě ż O lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq ě ż O lim τ 1 ´lim ρ 0 lim t 8 H ρ µ L t px, τ ∇ µ upxq, ωqdµpxq ě ΓpL p µ q-lim t 8 E t pu, ωq. for all u P H 1,p pO; R m q. Hence µ L p for all τ Ps0, 1r. Moreover, by (4.62) we have lim τ 1 ´∆a8 p L8 pτ, ωq ď 0. Hence, letting τ 1 ż ín (4.66) we conclude that ΓpL p µ q-lim t 8 E t pu, ωq " O lim τ 1 ´lim ρ 0 lim t 8
	and (4.53) follows because ΓpL p µ q-lim t 8 E t p¨, ωq is L p µ -lsc and τ u	u in L p µ pO; R m q as
	τ	1 ´.

But, by (C 1 ) we see that for every τ Ps0, 1r, Gpτ uq ď γµpOqp1 `Gp0qq `γGpuq, hence, by (C 2 ) and (4.60), Gpτ uq ă 8 for all τ Ps0, 1r, and so, by Lemma 4.3 we have t px, τ ∇ µ up¨q, ωq ď β `1 `Qµ Gpx, τ ∇up¨qq t px, τ ∇ µ up¨q, ωq ď β `1 `Gp0q `Gp∇ µ up¨q ˘:" f p¨q for all τ Ps0, 1r t px, τ ∇ µ upxq, ωqdµpxq, and (4.53) follows because ΓpL p µ q-lim t 8 E t p¨, ωq is L p µ -lsc and τ u u in L p µ pO; R m q as τ 1

´.

  ] and [DZ15, §2].) Theorem 5.3 (deterministic case). Under (O 1 ), (O 3 )-(O 4 ), (F 1 )-(F 2 ) and (S 1 )-(S 3 ), for every B P BapXq, one has

	lim t 8	Sph t pBqq µph t pBqq	" inf kPN ˚Sph k pUqq µph k pUqq	.

  B P BapXq, one has where E I rSph k pUq, ¨qs denotes the conditional expectation of Sph k pUq, ¨q over I with respect to P. If moreover pΩ, F, P, tτ g u gPG q is ergodic then

	lim t 8	Sph t pBq, ωq µph t pBqq	" inf kPN ˚EI rSph k pUq, ¨qspωq µph k pUqq	,
	lim t 8	Sph t pBq, ωq µph t pBqq	" inf kPN ˚ErSph k pUq, ¨qs µph k pUqq	,

  Gpξq " |ξ| p . Here we are concerned with the G-growth case. For this, we need to suppose that (H 4 ) for every ω P Ω, L is ru-usc at ω with respect to th t u tą0 , i.e. for every ω P Ω, there exists ap¨, ωq P L 1 P r0, 1s. But L is ru-usc at ω with ap¨, ωq, i.e. lim τ 1

	µ pX; s0, 8sq with
			ż
		lim t 8	X	aph t pxq, ωqdµpxq ă 8
	and
	lim ρ 0	lim t 8 ż Bρp¨q	aph t pyq, ωqdµpyq P L 1 µ pXq
	such that
		lim τ 1 ´∆a L pτ, ωq ď 0,
	where ∆ a L : r0, 1s ˆΩ s ´8, 8s is given by
	∆ a L pτ, ωq :" sup
					yPX	sup ξPLy,ω	Lpy, τ ξ, ωq ´Lpy, ξ, ωq apy, ωq `Lpy, ξ, ωq	" ∆ a L pτ, ωq,
	and from (5.3) we deduce that
		sup tą0	∆ at Lt pτ, ωq ď ∆ a L pτ, ωq	(5.4)
	for all τ ´∆a

Then, under (H 1 )-(H 2 ), it is easy to see that tL t u tą0 satisfies (C 5 )-(C 6 ). Homogenization of integrals was already studied in

[START_REF] Anza | Γ-convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization[END_REF]

and

[START_REF] Anza | On subadditive theorems for group actions and homogenization[END_REF] §3]

in the p-growth case, i.e. when xPX sup ξPLx,ω Lpx, τ ξ, ωq ´Lpx, ξ, ωq apx, ωq `Lpx, ξ, ωq with L x,ω denoting the effective domain of Lpx, ¨, ωq. Lemma 5.5. Let ω P Ω. If L is ru-usc at ω with respect to th t u tą0 with ap¨, ωq, then tL t u tą0 is ru-usc at ω with taph t p¨q, ωqu tą0 . Proof of Lemma 5.5. Set a t p¨, ωq :" aph t p¨q, ωq for all t ą 0. For any τ P r0, 1s, any t ą 0, any x P Ω and any ξ P L t,x,ω , one has L t px, τ ξ, ωq ´Lt px, ξ, ωq a t px, ωq `Lt px, ξ, ωq " Lph t pxq, τ ξ, ωq ´Lph t pxq, ξ, ωq aph t pxq, ωq `Lph t pxq, ξ, ωq . (5.3) As L t,x,ω " L htpxq,ω and h t pxq P X we see that Lph t pxq, τ ξ, ωq ´Lph t pxq, ξ, ωq aph t pxq, ωq `Lph t pxq, ξ, ωq ď sup

  such that for every A P B 0 , every ω P Ω, every ξ P Y yPO intpQ µ G y q and every ζ P D,The following two propositions are consequences of Theorems 5.3 and 5.4 respectively. We only give the proof of the stochastic proposition. The deterministic proposition can be proved by the same method.

	ξ L pA, ¨q is F-mesurable; 7 ) there exist φ : r0, 8r r0, 8s and θ : Y yPO intpQ µ G y q ˆD (H s r0, 8s with
		# lim r 0	φprq " 0 !		)
		sup	θpξ, ζq : ζ P D and |ζ| ď M	ă 8
		ˇˇˇˇS L pA, ωq ξ µpAq	´Sζ L pA, ωq µpAq	ˇˇˇˇď θpξ, ζqφp|ξ ´ζ|q.
	lim t 8	H ρ µ L t px, ξq " lim t 8	S ξ L ph t pB ρ pxqqq µ ph t pB ρ pxqqq	" inf kPN ˚Sξ L ph k pUqq µph k pUqq	,
	which implies (C 8 ).
	Proposition 5.8 (Stochastic case). Under (O 1 )-(O 4 ), (F s 1 )-(F s 2 ), (F 3 )-(F 5 ), the right in-equality in (H 2 ), (H s 3 ), (H 4 )-(H 5 ), (H s 6 )-(H s 7

Proposition 5.7 (deterministic case). Under (O 1 ), (O 3 )-(O 4 ), (F 1 )-(F 5 ), the right inequality in (H 2 ) and (H 3 )-(H 5 ), for every ρ ą 0, every x P O and every ξ P intpQ µ G x q, one has

  I denotes the conditional expectation over I with respect to P, with I being the σ-algebra of invariant sets with respect to pΩ, F, P, tτ g u gPG q. If moreover pΩ, F, P, tτ g u gPG q is ergodic then Proof of Proposition 5.8. First of all, from (H s 7 ) we see that for every ω P Ω, every k P N ˚, every ξ P Y yPO intpQ µ G y q and every ζ P D, and so, passing to the conditional expectation E I and then to the infimum on k, P Ω and all ξ P Y yPO intpQ µ G y q. Fix any ζ P D. From the right inequality in (H 2 ) and (H 5 ) we have S ζ L pA, ωq ď βp1 `Cζ qµpAq for all ω P Ω and all A P B 0 , and so, by (H s 6 ), S ζ L pA, ¨q P L 1 pΩ, F, Pq for all A P B 0 . Moreover, from (O 1 ), (F 3 ) and (H s 3 ) it easily seen that the set function S ζ L is G-stationary, and S ζ L is also subadditive because, for each A, B P B 0, µ `{ A Y Bzp Å Y Bq ˘" 0 since { A Y Bzp Å Y Bq Ă BA Y BBand µpBAq " µpBBq " 0. Then, by Theorem 5.4, there exists Ω 1 ζ P F with PpΩ 1 ζ q " 1 such that for every ω P Ω 1 ζ and every B P BapXq, one has Now, set Ω 1 :" X ζPD Ω 1 ζ . Since D is countable, PpΩ 1 q " 1. Fix any ω P Ω 1 , any ξ P Y yPO intpQ µ G y q and any B P BapXq. From (H s 7 ) we see that for every t ą 0 and every ζ P D, Consequently, for every ω P Ω 1 , every ξ P Y yPO intpQ µ G y q and every B P BapXq,

	t 8 where E lim H ρ µ L t px, ξ, ωq " lim t 8 t 8 H ρ ˇˇˇˇi nf kPN ˚EI rS ξ L ph k pUq, ¨qspωq µph k pUqq Taking the properties of φ and θ in (H s S ξ L ph t pB ρ pxqq , ωq µ ph t pB ρ pxqqq ´inf kPN ˚EI rS ζ L ph k pUq, ¨qspωq " inf kPN ˚EI rS ξ L ph k pUq, ¨qspωq µph k pUqq µph k pUqq ˇˇˇˇď θpξ, ζqφp|ξ ´ζ|q. , 7 ) into account, we obtain lim DQζ ξ inf kPN ˚EI rS ζ L ph k pUq, ¨qspωq µph k pUqq " inf kPN ˚EI rS ξ L ph k pUq, ¨qspωq µph k pUqq for all ω lim t 8 S ζ L ph t pBq, ωq µph t pBqq " inf kPN ˚EI rS ζ L ph k pUq, ¨qspωq µph k pUqq . S ζ L ph t pBq, ωq µph t pBqq ´θpξ, ζqφp|ξ ´ζ|q ď S ξ L ph t pBq, ωq µph t pBqq ď S ζ L ph t pBq, ωq µph t pBqq `θpξ, ζqφp|ξ ´ζ|q, (5.5) (5.6) and so, letting t 8 and using (5.6), we obtain: inf kPN ˚EI rS ζ L ph k pUq, ¨qspωq µph k pUqq ´θpξ, ζqφp|ξ ´ζ|q ď lim t 8 S ξ L ph t pBq, ωq µph t pBqq ; lim t 8 S ξ L ph t pBq, ωq µph t pBqq ď inf kPN ˚EI rS ζ L ph k pUq, ¨qspωq µph k pUqq `θpξ, ζqφp|ξ ´ζ|q. By the properties of φ and θ in (H s 7 ) and (5.5), letting D Q ζ ξ, it follows that: inf kPN ˚EI rS ξ L ph k pUq, ¨qspωq µph k pUqq ď lim t 8 S ξ L ph t pBq, ωq µph t pBqq ; lim t 8 S ξ L ph t pBq, ωq µph t pBqq ď inf kPN ˚EI rS ξ µ L S ξ L ph k pUq, ωq µph k pUqq ´θpξ, ζqφp|ξ ´ζ|q ď S ζ L ph k pUq, ωq µph k pUqq ď S ξ L ph k pUq, ωq µph k pUqq `θpξ, ζqφp|ξ ´ζ|q, lim t 8 S ξ L ph t pBq, ωq µph t pBqq " inf kPN ˚EI rS ξ L ph k pUq, ¨qspωq µph k pUqq .

t px, ξ, ωq " lim t 8 S ξ L ph t pB ρ pxqq , ωq µ ph t pB ρ pxqqq " inf kPN ˚ErS ξ L ph k pUq, ¨qs µph k pUqq , where E denotes the expectation with respect to P. Consequently (C 8 ) holds. L ph k pUq, ¨qspωq µph k pUqq .

  ξ L ph t pBq, ωq " inf "ż htpBq Lpy, ξ `∇µ wpyq, ωqdµpyq : w P H 1,p µ,0 ph t pBq; R m q Lph t pyq, ξ `∇µ wph t pyqq, ωqdph ´1 t q 7 µpyq : w P H 1,p µ,0 ph t pBq; R m q py, ξ `∇µ wph t pyqq, ωqdµpyq : w P H 1,p µ,0 ph t pBq; R m q * .But µph t pUqqµpBq " ph ´1 t q 7 µpBq " µph t pBqq by using (F 3 ), and so from (F 5 ) we obtain S ξ L ph t pBq, ωq " µph t pBqq inf " ż B L t py, ξ `∇µ wpyq, ωqdµpyq : w P H 1,p µ,0 pB; R m q

	*
	"ż
	" inf

B * " µph t pUqq inf "ż B L t * for all ω P Ω 1 , all B P BapXq and all t ą 0. Consequently, for every ω P Ω 1 and every ρ ą 0, we have: lim t 8 H ρ µ L t px, ξ, ωq " lim t 8 S ξ L ph t pB ρ pxqq, ωq µ ph t pB ρ pxqq ; lim t 8 H ρ µ L t px, ξ, ωq " lim t 8 S ξ L ph t pB ρ pxqq, ωq µ ph t pB ρ pxqqq ,

  Under (O 1 ), (O 3 )-(O 4 ), (F 1 )-(F 5 ) and (H 1 )-(H 5 ), one has Theorem 5.10 (stochastic case). Assume that p ą κ.Under (O 1 )-(O 4 ), (F s 1 )-(F s 2 ), (F 3 )-(F 5 ), (H 1 )-(H 2 ),(H s3 ), (H 4 )-(H 5 ), (H s 6 )-(H s 7 ), there exists Ω 1 P F with PpΩ 1 q " 1 such that for every ω P Ω 1 , one has H 1,p µ pO; R m q with L hom : M ˆΩ r0, 8s given by L hom pξ, ωq "

	$ &	lim τ 1 ´inf kPN ˚EI rS τ ξ L ph k pUq, ¨qspωq µph k pUqq	if ξ P Y yPO	Q µ G y
	%	8				otherwise,
				ż
	ΓpL p µ q-lim t 8	E t puq "
				r0, 8s given by
	L hom pξq "	$ &	lim τ 1 ´inf kPN ˚Sτ ξ L ph k pUqq µph k pUqq	if ξ P Y yPO	Q µ G y
		%	8	otherwise.
	ΓpL p µ q-lim

O L hom `∇µ upxq ˘dµpxq for all u P H 1,p µ pO; R m q with L hom : M t 8 E t pu, ωq " ż O L hom `∇µ upxq, ω ˘dµpxq for all u P

Throughout the paper, by a Borel measurable stochastic integrand L : O ˆM ˆΩ r0, 8s we mean that L is pBpXq b BpMq b F, BpRqq-measurable, where BpXq, BpMq and BpRq denote the Borel σ-algebra on X, M and R respectively.

The abbreviation ru-usc means radially uniformly upper semicontinuous.

The abbreviation lsc means lower semicontinuous.

Given a metric space pO, dq, by a Urysohn function from O to R for the pair pOzV, Kq, where K Ă V Ă O with K compact and V open, we mean a continuous function ϕ : O R such that ϕpxq P r0, 1s for all x P O, ϕpxq " 0 for all x P OzV and ϕpxq " 1 for all x P K.

When L t px, ξ, ωq " Lpx, ξq we have H ρ µ L t px, ξ, ωq " Q µ Lpx, ξq, and (C 8 ) is trivially satisfied.

5.1. Relaxation. In case L t px, ξ, ωq " Lpx, ξq, and so E t p¨, ωq " Ep¨q, we retrieve the relaxation theorem established in [START_REF] Anza | Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger-Sobolev spaces[END_REF]Theorem 2.7]. More precisely, denoting the lsc envelope of E with respect to the strong topology of L p µ pΩ; R m q by E, as a direct consequence of Theorem 2.5 and Proposition 2.8 we have the following result.

Corollary 5.1. Assume that the hypotheses of Theorem 2.5 are satisfied with L t px, ξ, ωq " Lpx, ξq 6 . Then

where Q µ L : O ˆM r0, 8s is defined by

where, for each x P O, Q µ Lpx, ¨q denotes the lsc envelope of Q µ Lpx, ¨q.

5.2. Homogenization. Homogenization of integrals of the calculus of variations in noneuclidean settings has been studied for the first time in [START_REF] Anza | Γ-convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization[END_REF] (see also [START_REF] Dirr | Γ-convergence and homogenisation for a class of degenerate functionals[END_REF][START_REF] Anza | On subadditive theorems for group actions and homogenization[END_REF]) for integrands having p-growth. In this paragraph, we attempt to develop a framework to deal with integrands which have not necessarily p-growth and can take infinite values, by using (the Γ-convergence result) Theorem 2.5 together with Proposition 2.8 and subadditive theorems that we proved in [START_REF] Anza | On subadditive theorems for group actions and homogenization[END_REF] (see Theorems 5.3 and 5.4). (5.1) (F s 2 ) for every B P BapXq, th t pBqu tą0 is asymptotically strongly G-regular, i.e. there exist two strongly G-regular families 9 tJ t u tą0 , tJ t u tą0 Ă J satisfying (5.1); (F 3 ) for each t ą 0, pph t q ´1q 7 µ " µph t pUqqµ, i.e. for every A P BpXq, µph t pAqq " µph t pUqqµpAq;

5.

(F 4 ) for every g P G and every B P BapXq, there exists a bijective map T g,B from H 1,p µ,0 pg ´1pBq; R m q to H 1,p µ,0 pB; R m q such that ∇ µ T g,B pwq " ∇ µ w o g ´1 for all w P H 1,p µ,0 pg ´1pBq; R m q and ∇ µ pT g,B q ´1pvq " ∇ µ v o g for all v P H 1,p µ,0 pB; R m q; (F 5 ) for every t ą 0 and every B P BapXq, there exists a bijective map H t,B from H 1,p µ,0 ph t pBq; R m q to H 1,p µ,0 pB; R m q such that ∇ µ H t,B pwq " ∇ µ w o h t for all w P H 1,p µ,0 ph t pBq; R m q and ∇ µ pH t,B q ´1pvq " ∇ µ v o ph t q ´1 for all v P H 1,p µ,0 pB; R m q. Remark 5.2. From (F 3 ) we see that µph t pUqq ą 0 and ph t q 7 µ " pµph t pUqqq ´1µ for all t ą 0. Moreover, as µpUz Ůq " 0 we have µp Ůq " µpUq and so µph t p Ůqq " µph t pUqq for all t ą 0.

Subadditive theorems.

In what follows, we recall subadditive theorems that we proved in [AHM20b, Theorem 2.19]. Let S : B 0 ˆΩ r0, 8s be such that SpA, ¨q P L 1 pΩ, F, Pq for all A P B 0 . In Theorems 5.3 and 5.4 below we need the following properties on S:

(S 1 ) there exists C ą 0 such that for every A P B 0 and every ω P Ω, SpA, ωq ď CµpAq;

(S 2 ) S is subadditive, i.e. for every A, B P B 0 with A X B " H and every ω P Ω, SpA Y B, ωq ď SpA, ωq `SpB, ωq;

(S s 3 ) S is G-stationary, i.e. for every A P B 0 , every g P G and every ω P Ω, S `g´1 pAq, ω ˘" S `A, τ g pωq ȃnd, when S is deterministic, i.e. Sp¨, , ωq " Sp¨q, (S 3 ) S is G-invariant, i.e. for every A P B 0 and every g P G, S `g´1 pAq ˘" SpAq.

8 Given tJ t u tą0 Ă J, for each t ą 0, J t " Y gPHt g ´1ph qt pUqq with H t P Dph qt pUqq. We say that tJ t u tą0 is G-regular if lim t 8 q t " 8 and for every t ą 0 and every k P N ˚there exist qt,k P N ˚, ḡt,k P G and F t,k P P f pG k q such that G qt,k H t Ă F t,k , ḡ´1