Deterministic equivalent of a random Riemannian metric defined by

Dedicated to Professor Philippe G. Ciarlet on his 70th birthday

DETERMINISTIC EQUIVALENT OF A RANDOM RIEMANNIAN METRIC DEFINED BY GEODESICS OANA IOSIFESCU and GÉRARD MICHAILLE

We give a deterministic equivalent in probability of a rapidly oscillating random Riemannian metric. This equivalent metric is obtained by geodesics dened in the space of functions of bounded variation via suitable ergodic theorems.

AMS 2000 Subject Classication: Primary 60F10; Secondary 34F05, 60F15, 73B27. Key words: stochastic homogenization, Poisson process, Gamma-convergence, function of bounded variation, ergodic theorem.

1.

INTRODUCTION

Let Ω be the set of locally nite sequences (ωi )i∈N in RN and M the set of all countable and locally nite sums of Dirac measures, equipped with their standard σ -algebra. Given a positive number λ and ε > 0 intended to tend to 0, we consider the Poisson point process Ω → M,

ω 7→ Nε (ω, ·) =

X

δωi

i∈N

with intensity λε = λ/εN LN , where LN denotes the Lebesgue measure on RN , and dene the heterogeneous random Riemannian metric on RN by ( S ds2 (ξ) if x ∈ i∈N B(ωi , εr), dsε (ω, x, ξ) := ds1 (ξ) otherwise. For every i in N, B(ω , εr) is the open ball in RN with radius εr centered at Pi k ωi and dsk (ξ) := ( i,j ai,j ξi ξj )1/2 for all ξ ∈ RN with (aki,j )i=1,...,N, j=1,...,N , k = 1, 2, two symmetric positive matrices (see Section 3 for more details). The space RN is then equipped with the highly random oscillating geodesic metric (a, b) 7→ dε (ω)(a, b) dened for any points a, b in RN by Z

1

0

dsε (ω, γ(t), γ (t)) dt : γ ∈ W

inf

1,1

N



((0, 1), R ), γ(0) = a, γ(1) = b .

0

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 479498

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Oana Iosifescu and Gérard Michaille

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Classically, there exists a probability measure P on Ω satisfying P([Nε (· , A) = k]) = λkε LN (A)k

exp(−λe LN (A)) k!

for every bounded Borel set A
in RN , so that the expectation of the random variable Nε (· , A) is E Nε (· , A) = λε LN (A). Consequently, dε is statistically like a geodesic distance in a Riemaniann space with small balls as heterogeneities, and whose number is of order λε = λ/εN per unit of volume. It will be noted that the balls can interpenetrate. It is easy to see that the random distance ω 7→ dε (ω)(a, b) has the same law as (i.e., is statistically equivalent to) the random distance given by Z

1

 γ(t) 0 1,1 N inf ds(ω, , γ (t)) dt : γ ∈ W ((0, 1), R ), γ(0) = a, γ(1) = b , ε 0
where ds(ω, x, ξ) := ds1 (ξ) + ds2 (ξ) − ds1 (ξ) min(1, N (ω, B(x, r)) and N is

the Poisson point process

Ω → M, ω 7→ N (ω, ·) =

X

δωi

i∈N

with intensity λLN . The scaling 1/ε on the space variable, well known in homogenization theory, is more adapted to a mathematical processing. In this paper, we look at this distance that we still denote by dε (ω). We would like to stress that we can treat the problem, without changing proofs, for random Riemannian metrics in general stationary ergodic media. Since we only have a statistical knowledge of dε (ω) which is moreover highly oscillating, for obtaining information on dε (ω) it is natural to identify its a.s. variational limit as ε → 0 and take advantage of the stationarity of the Poisson point process for proving the deterministic nature of the limit. In fact, we have been unable to fully carry out this program: we only establish the variational convergence in law and in probability of dε to a deterministic distance d0 . Our proof is based on the following compactness result established in Section 3: there exists a subsequence of the sequence of integral functionals Fε dening dε , which almost surely converges to a deterministic functional F0 in the sense of Gamma-convergence. We deduce the convergence in law of any sequence of relaxed functionals F¯ε to the Dirac probability measure δF0 . More precisely, we know that Gamma-convergence induces a metric on the set SC of all semicontinous functions G : BV (0, 1), RN → R∪{+∞}. Then, considering the probability images PFε−1 and PF0−1 = δF0 of F¯ε , F0 : Ω → SC

we obtain PFε−1 * δF0 . Denoting by dΓ the metric induced in SC by Gammaconvergence, we obtain as a corollary the convergence in probability of all

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Deterministic equivalent of a random Riemannian metric dened by geodesics

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sequences, (Fε )ε>0 , i.e., P{ω : dΓ (F¯ε (ω), F0 ) > η} → 0 as ε → 0.

Finally, using the continuity of the map G 7→ min G in the metric space (SC, dΓ ), we deduce our main result, the convergence in probability P{ω : |dε (ω)(a, b) − d0 (a, b)| > η} → 0 as ε → 0.

We would like to stress that we could treat this analysis for random Riemannian metrics in general stationary ergodic media, without changing proofs. This problem has already been studied in the deterministic case when the Riemannian metric is periodically distributed. Precisely, it has been solved in [2] when p > 1 and in [4] when p = 1 by using an integral representation theorem for functionals dened in W 1,p ((0, 1), RN ) or BV ((0, 1), RN ). The paper is organized as follows. The next section contains a brief summary of some results related to the ergodic theory of random subadditive processes, to Gamma-convergence and to the BV ((0, 1), RN ) space. Section 3 is devoted to the study of two suitable random processes and to the denition of the deterministic limit distance. In Subsection 4.1 we establish our compactness result using both random processes. Finally, in the last section, we conclude as to the convergence in probability of all sequences of random geodesic metrics to the deterministic metric dened in Section 3. 2. 2.1.

BASIC NOTIONS

Basic notions on Gamma-convergence

Let (X, d) be a metric space and {(Fn )n , F, n → +∞} functionals mapping X into R ∪ {+∞}. The notion of convergence below is equivalent to convergence of the epigraph of Fn to the epigraph of F in the Kuratovski-Painlevé sense and has been rst introduced in [11]. We give the denition in the framework of metric spaces. Indeed, in the sequel, (X, d) will be L1 ((0, 1), RN ) or BV ((0, 1), RN ) equipped with the strong convergence of L1 ((0, 1), RN ). For details, we refer the reader to [7] and [9]. Denition 1. A sequence (Fn )n∈N Gamma-converges to F at x in X i i) there exists a sequence (xn )n∈N converging to x in X such that F (x) ≥ lim sup Fn (xn ), n→+∞

ii) for every sequence (yn )n∈N converging to x in X we have F (x) ≤ lim inf Fn (yn ). n→+∞

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When this property holds for every x in X , we say that (Fn )n Gamma-converges to F in (X, d) and write F = Γ lim Fn . It is straightforward to show that this convergence is equivalent to Γ lim sup Fn (x) ≤ F (x) ≤ Γ lim inf Fn (x) n→+∞

n→+∞

for any x ∈ X , where n o Γ lim sup Fn (x) := min lim sup Fn (xn ) : xn → x , n→+∞

n→+∞

n

o Γ lim inf Fn (xn ) := min lim inf Fn (xn ) : xn → x . n→+∞

n→+∞

The main interest of this concept lies in the following variational property. Proposition

X be such that

1. Assume that (Fn ) Gamma-converges to F and let xn ∈

Fn (xn ) ≤ inf{ Fn (x) : x ∈ X } + εn , where εn > 0, εn → 0. Assume furthermore that {xn , n ∈ N} is relatively compact. Then any cluster point x of {xn , n ∈ N} is a minimizer of F and lim inf{ Fn (x) : x ∈ X } = F (x).

n→+∞

2.2.

Basic notions on

BV -functions

The set BV (I, RN ) is the space of all functions u ∈ L1 (I, RN ) such that the distributional derivative u0 iof u is a bounded Borel R measure with values in RN . The nite total variation of u0 is denoted by I |u0 | and BV (I, RN ) is equipped with the norm Z |u|BV (I,RN ) = |u|L1 ((0,1),RN ) +

|u0 |.

I

The unit ball of BV (I, RN ) is weakly sequentially compact in the following sense: if |un |BV (I,RN ) ≤ 1, there exist a subsequence (unk )k∈N and u in BV (I, RN ) with |u|BV (I,RN ) ≤ 1 such that unk weakly converges to u in BV (I, RN ), that is, (

unk → u strongly in L1 (I, RN ), u0nk * u0 in the sense of weak convergence of bounded measures.

In the sequel, u0a and u0s will denote the density of the regular part of the measure u0 and its singular part in the Lebesgue decomposition, i.e., u0 = u0a L1bI + u0s . The function ua is actually the approximate dierential of u in I . More precisely,

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Deterministic equivalent of a random Riemannian metric dened by geodesics Proposition

1 ρ→0 ρ

483

2. Let u be any element of BV (I, RN ). Then Z

lim

Iρ (t0 )

|u(t) − u(t0 ) − (t − t0 ) u0a (t0 )| dt = 0 |t − t0 |

for almost all t0 in I , where Iρ (t0 ) denotes the interval centered at t0 of length 2ρ.

For a general study of BV -functions we refer the reader to [6]. 2.3.

Basic notions on subadditive random processes

We consider a probability space (Ω, T , P) and a group (τs )s∈R of Ppreserving transformations of (Ω, T ). The group (τs )s∈R is said to be ergodic if every set E in T such that τs E = E , s ∈ R, has probability equal to 0 or 1. A sucient condition to ensure ergodicity of (τs )s∈R is the mixing condition: for every E and F in T , lim P (τs E ∩ F ) = P (E)P (F )

|s|→+∞

which expresses an asymptotic independence. Let E denote the expectation operator and J the set of half open intervals [a, b) of R. A random subadditive process with respect to (τs )s∈R is a set function S : J → L1 (Ω, T , P) satisfying the conditions below: (i) for every I ∈ J such thatSthere exists a nite family P(Ij )j∈J of pairwise disjoint intervals in J with I = j∈J Ij , we have SI (·) ≤ j∈J SIj (·); (ii) ∀I ∈ J , ∀s ∈ R, Ss+I = SI ◦ τs . The subadditive ergodic Theorem 1 below is due to Ackoglu-Krengel. It was rst applied in the context of stochastic homogenization in [10]. In order to dene the limit metric, it will be applied in the next section with d = 1. Theorem 1. Let S be a subadditive process with respect to (τs )z∈R that is assumed to be ergodic. Assume that

Z inf Ω

 SI (ω) P(dω) : |I| = 6 0 > −∞. |I|

Let I be any xed interval in I . Then, almost surely,   S[0,m[d StI (ω) lim = inf ∗ E . t→+∞ |tI| m∈N md

For a proof see [3] and, for some extensions, [12], [13].

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Oana Iosifescu and Gérard Michaille

3.

6

IDENTIFYING THE LIMIT METRIC

Let us give more details on the probabilistic setting. We consider two symmetric N × N -matrices (aki,j )i=1,...,N, j=1,...,N , satisfying ν|ξ|2 ≤

X

aki,j ξi ξj ≤ Λ|ξ|2 ,

k = 1, 2,

i,j

for every ξ ∈ RN , where ν, Λ are two given positive constants. Denote by ds1 and ds2 the two corresponding homogenous Riemannian metrics. On the other hand, we consider a Poisson point process ω 7→ N (ω, ·) with intensity N λ LN , λ > 0, from the probability space (Ω, T , P ) into NB(R ) equipped with the standard product σ -algebra, which classically satises: i) for every bounded Borel set A in RN , we have N (ω, A) =

X

δωi (A);

i∈N

in

ii) for every nite family (Ai )i∈I of pairwise disjoint bounded Borel sets the random variables N (· , Ai ), i ∈ I , are independent; iii) for every bounded Borel set A and every k ∈ N,

RN ,

P ([N (· , A) = k]) = λk LN (A)k

exp(−λLN (A)) . k!

Note that N (ω, A) = card(A ∩ Ω) and that E(N (· , A)) = λ LN (A) for every bounded Borel set of RN . For a given r > 0, we dene a random Riemannian metric by setting ( ds(ω, x, ξ) =

ds2 if x ∈

S

i∈N B(ωi , r),

ds1 otherwise.

It may be interesting to note that (1)


ds(ω, x, ξ) := ds1 (ξ) + ds2 (ξ) − ds1 (ξ) min(1, N (ω, B(x, r)).

For all ω in Ω, ds takes on values in the class Fν,Λ of the functions f from RN × RN into R that are convex with respect to the second variable and satisfy ν|ξ| ≤ f (x, ξ) ≤ Λ(1 + |ξ|). This class will be equipped with the trace N N denoted by T˜ of the Borel σ -eld of RR ×R , that is the smallest σ -algebra on Fν,Λ such that all evaluation maps f 7→ f (x, ξ), x ∈ RN , ξ ∈ RN , are measurable. According to (1), it is readily seen that the map ds : Ω → Fν,Λ , ω 7→ ω ˜ := ds(ω) is (T , T˜ )-measurable. Throughout the paper we will reason ˜ := (Fν,Λ , T˜ , P˜ ) where P˜ is the law of ds, i.e. ˜ T˜ , P) in the probability space (Ω, the probability image of P by the measurable map ds.

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Deterministic equivalent of a random Riemannian metric dened by geodesics

485

For every xed point a in RN we introduce the ergodic group (Ts )s∈R of P˜ -preserving transformations by setting Ts ω˜ (x, ξ) = ω˜ (x + sa, ξ) and dene the subadditive random process with respect to (Ts )s∈R by setting Z SI (˜ ω , a) = inf

0

ω ˜ (γ(t), γ (t)) dt : γ ∈ W

1,1

◦

N



(I , R ), γ(α) = αa, γ(β) = βa .

I

for every open half interval I = [α, β) in R. Conditions (i) and (ii) dening a subadditive random process are easily checked as well as measurability and integrability of SI (· , a). Ergodicity of (Ts )s∈R is a straightforward consequence of condition (ii) in the denition of the Poisson point process which ensure the mixing condition introduced in Subsection 2.3. In what follows, ε denotes a sequence of positive numbers going to 0. Applying Theorem 1, we obtain Theorem 2. Let I be an arbitrary interval of R. Then there exists a ˜ 0 of full probability such that for all ω ˜ 0 and for all a ∈ QN subset Ω ˜ in Ω the limit

ds0 (a) = lim

ω , a) S 1 I (˜

ε→0

ε

| 1ε I|

  S]0,n[ (· , a) S]0,n[ (˜ ω , a) = inf ∗ E = lim n→+∞ n∈N n n

exists. Proof. For every xed a in QN , using Theorem 1 we deduce the existence 0 ˜ ˜ 0 the limit of Ωa of full probability such that for every ω˜ ∈ Ω a S 1 I (˜ ω , a)

S]0,n[ (˜ ω , a) n→+∞ ε→0 n T ˜0 = ˜0 exists, and it is sucient to set Ω  a∈QN Ωa . lim

ε

| 1ε I|

= lim

We are going to establish the existence of the limit for every a ∈ RN ˜ 0 . This will be an easy consequence of the equilipschitzian and every ω˜ ∈ Ω property below. es

Proposition

3. The above normalized subadditive random process satis|

S 1 I (˜ ω , a) ε

| 1ε I|

−

S 1 I (˜ ω , b) ε

| 1ε I|

| ≤ Λ|a − b|.

for every ω˜ and every a and b in QN . Consequently, |ds0 (a) − ds0 (b)| ≤ Λ|a − b| ˜ 0 , and ds0 may be extended to RN by a function still denoted for every ω˜ in Ω ds0 , satisfying the same lipschitzian property and which is still the almost sure limit of the previous normalized subadditive process.

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Oana Iosifescu and Gérard Michaille

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Proof. We assume without loss of generality that ε = (1/n)n∈N∗ and I = (0, 1). Let s ∈ R, s > 0 intended to go to 0 and γ ∈ W 1,1 ((0, n), RN ) an admissible function in the denition of S]0,n[ (˜ω , a)/n, which requires γ(0) = 0 and γ(n) = na. We modify the trajectory γ in [n − s, n) by the segment [γ(n − s), nb] in RN so that the new trajectory γs satises γs (0) = 0 and γs (n) = nb and is an admissible function in the denition of S]0,n[ (˜ ω , b)/n.

More precisely, we set

( γs (t) =

γ(t) nb +

t−n s

in [0, n − s]
nb − γ(n − s) in [n − s, n].

According to the growth condition, we have Z S]0,n[ (˜ ω , b) 1 n ω ˜ (γs (t), γs0 (t)) dt ≤ n n 0 Z Λ 1 n ω ˜ (γ(t), γ 0 (t)) dt + |nb − γ(n − s)|. ≤ n 0 n Letting s → 0 and n → +∞ in this order, and noticing that every element of W 1,1 ((0, n), RN ] is continuous in [0, n], we obtain S]0,n[ (˜ ω , b) S]0,n[ (˜ ω , a) ≤ + Λ|b − a|, n n which completes the proof.  ˜ 0 denotes various subsets of full In what follows, the same notation Ω ˜ -preserving transformaprobability. In the proofs below we will consider the P N ˜ dened for every ξ ∈ R by τξ ω tion τξ on Ω ˜ (x, ζ) = ω ˜ (x + ξ, ζ). We also 1 ˜ ˜ ˜ introduce a second process S from J into L (Ω, T , P ) dened by Z  ◦ 0 1,1 N S˜I (˜ ω , a) = inf ω ˜ (u(t), u (t)) dt : u ∈ W (I , R ), u(α)−u(β) = (α−β)a . I

Clearly, S˜ is invariant under translations of R and satises SI ≥ S˜I for all I ∈ I . Although S˜ is not subadditive we have 3. For every t0 ∈ R, ρ > 0 and a ∈ RN , S˜1 Iρ (t0 ) (· , a)/| 1ε Iρ (t0 )| ε tends in probability to ds0 (a) as ε → 0. ˜ 0 of Therefore, if a belongs to QN and ρ to Q+ , there exists a subset Ω ˜ of full probability which does not depend on a, ρ and t0 , and a subsequence, Ω not relabelled, possibly depending on a and ρ, such that Theorem

lim

ε→0

˜ 0. for all ω˜ in Ω

ω , a) S˜1 Iρ (t0 ) (˜ ε

| 1ε Iρ (t0 )|

= ds0 (a)

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Deterministic equivalent of a random Riemannian metric dened by geodesics

487

Proof. As S˜I+s (˜ ω , a) = S˜I (˜ ω , a) for every s in R, it is sucient to establish the proof for the interval Iρ (0). To simplify notation, we denote by I this interval. Let γε be an ε-minimizer of S˜1 I , i.e., ε

S˜1 I (˜ ω , a) ε

≥

| 1ε I|

1 1 | ε I|

Z ω ˜

γ

1 I ε

ε

ε

 , γε0 dt − ε,

and set γ˜ε = γε − bε where bε = γε (αε ) − αε a and αε is the lower bound of the interval (1/ε)I . Since γ˜ε is an admissible function for S 1 I , we have ε

ω , a) S˜1 I (˜ ε

| 1ε I|

Thus

≥

1 1 | ε I|

ω , a) S 1 I (˜ ε

| 1ε I| so that, for every η > 0,



Z 1 I ε

τbε ω ˜

≥

˜ , a) S 1 I (τbε ω γ˜ε 0 , γ˜ε dt − ε ≥ ε 1 − ε. ε | ε I|

S˜1 I (˜ ω , a) ε

| 1ε I|



≥

S 1 I (τbε ω ˜ , a) ε

| 1ε I|

− ε,

 S˜1 (· , a)  εI − ds0 (a) > η | 1ε I|  S˜1 (· , a) S 1 (τ · , a) η   S 1 I (τbε · , a) I bε I ⊆ ε 1 − ε 1 + ε > ∪ ε 1 − ε − ds0 (a) > 2 | ε I| | ε I| | ε I|  S 1 (· , a) S 1 (τ · , a)   S 1 (τ · , a) I bε I I bε η ⊆ ε 1 ∪ ε 1 − ε 1 + ε > − ε − ds0 (a) > 2 | ε I| | ε I| | ε I|

η 2



 η . 2

˜ preserving transformation, Theorem 2 implies that S˜1 (· , a)/| 1 I| Since τbε is a P ε  εI η ˜ tends in probability to ds0 (a) (use the Markov inequality P |fε | > 2 ≤ R 2 ˜  η Ω |fε | dP).

4.

THE MAIN COMPACTNESS RESULT

As noted in the introduction, ds(ω, x/ε, ξ) denes a random metric which has the same law as the strongly oscillating random Riemannian metric dened by
dsε (ω, x, ξ) := ds1 (ξ) + ds2 (ξ) − ds1 (ξ) min(1, Nε (ω, B(x, εr))

where Nε is a Poisson process with intensity ελN (reason by using the Laplace transform of N ). The next two subsections are devoted to the proof of the following compactness theorem and its corollary.

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Oana Iosifescu and Gérard Michaille

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Theorem 4. There exists a subsequence of ε such that the length dened in L1 ((0, 1), RN ) by

Fε (˜ ω )(γ) :=

 Z  

1

 ω ˜

0

 

 γ(t) 0 , γ (t) dt if γ ∈ W 1,1 ((0, 1), RN ), ε

otherwise,

+∞

Gamma-converges almost surely to the deterministic function dened in L1 ((0, 1), RN ) by F0 (γ) =

 Z  

1

ds0 (γa0 (t))

0

 

Z dt +

1

ds∞ 0



0

γs0 |γs0 |



|γs0 | if γ ∈ BV ((0, 1), RN ),

otherwise,

+∞

where γa0 is the density of the regular part of γ 0 in its Lebesgue decomposition N and ds∞ 0 is the recession function dened for every a ∈ R by ds∞ 0 (a) = lim

t→+∞

ds0 (ta) . t

Before proving Theorem 4, let us take into account the boundary conditions. For every (a, b) in R2N , consider the functionals Ia,b , I¯a,b : L1 ((0, 1), RN ) → R+ ∪ {+∞}

dened by ( 0 if γ(0) = a and γ(1) = b, Ia,b (γ) = +∞ otherwise, ( ∞ N ds∞ 0 (γ(0) − a) + ds0 (b − γ(1)) if γ ∈ BV ((0, 1), R ), I¯a,b (γ) = +∞ otherwise.

It is well known that I¯a,b is the relaxed functional of Ia,b . As a consequence of Theorem 4, we obtain the following result whose proof is a straightforward adaptation of that of Theorem 4 (see [8]). Corollary 1. There exists a subsequence of ε such that Fε +Ia,b Gammaconverges almost surely to the deterministic function F0 + I¯a,b .

We are going to establish Theorem 4. For proving the upper bound, we use the subadditive ergodic process. The lower bound will be established using the second ergodic process.

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Deterministic equivalent of a random Riemannian metric dened by geodesics

4.1.

489

The upper bound

With the notation and denitions of previous sections, we establish, for a subsequence, the upper bound in the denition of Gamma-convergence. More precisely, we have ˜ 0 of Ω ˜ of full probability such 4. There exists a subset Ω 0 ˜ that for all ω˜ in Ω there exists a subsequence still denoted ε such that Proposition

Γ lim sup Fε (˜ ω ) ≤ F0 . ε→0

Proof. We proceed in two steps. Step 1. We prove the result in a dense countable subset D of C 1 ((0, 1), RN ) for the strong topology of L1 ((0, 1), RN ). Let γ ∈ D. Subdivise (0, 1) in intervals Ii = (ti−1 , ti ), i = 1, . . . , m, t0 = 0, tm = 1, of length h intended to go to 0, and consider the continuous piecewise linear function γh taking the values γ(ti ) for every i = 0, . . . , m. Denote by ai the gradient of γh in Ii . It is easily seen that γh strongly converges to γ in W 1,1 ((0, 1), RN ). According to ˜ 0 of full probability such that, for Theorem 2 and Proposition 3, there exists Ω 0 ˜ and i = 1, . . . , m, all ω˜ ∈ Ω ω , ai ) S 1 Ii (˜ ε

ds0 (ai ) = lim

| 1ε Ii |

ε→0

1 = lim ε→0 |Ii |

Z ω ˜( Ii

i (˜ ω , t) γh,ε

ε

0

i , γh,ε (˜ ω , t)) dt,

i (˜ ω , ·) is an ε/m-minimizer of the where we performed a change of scale and γh,ε problem

(2)



Z ω ˜

inf

v∈W 1,1 (Ii ,RN )

Ii

  v(t) 0 , v (t) dt : v(ti−1 ) = ti−1 ai , v(ti ) = ti ai . ε

Set bi = γ(ti ) − ti ai for i = 1, . . . , m and denote by χIi the characteristic function of the interval Ii . The function γh,ε (˜ ω , ·) =

m X


i (τ bi ω γh,ε ˜ , ·) + bi χIi ε

i=1

belongs to W 1,1 ((0, 1), RN ). The measure preserving transformation τ bi is ε introduced for technical reasons which will appear below. We have Z lim sup E ε→0

1

ds0 (γh0 )

Z dt −

0

≤ lim sup ε→0

m X i=1

1

 ω ˜

0 

 γh,ε (˜ ω , t) 0 , γh,ε (˜ ω , t) dt ε

 i (τ ω Z γ ˜ , t) + b b i i h,ε 0i ε   E ds0 (ai )|Ii | − ω ˜ , γh,ε (τ bi ω ˜ , t) dt ε ε Ii

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Oana Iosifescu and Gérard Michaille

= lim sup

m X

ε→0

i=1

12

  i (τ ω Z γh,ε bi ˜ , t) 0i ε E ds0 (ai )|Ii | − ˜ , γh,ε (τ bi ω ˜ , t) dt τ bi ω ε ε ε Ii

Z m X = lim sup E ds0 (ai )|Ii | − ω ˜ ε→0

i (˜ γh,ε ω , t)

ε

Ii

i=1

!

, γh,ε (˜ ω , t) dt = 0, 0i

where we have used the Lebesgue dominated convergence theorem. According to Proposition 3, we obtain (3)

Z lim lim E h→0 ε→0

1

1

Z

0

ds0 (γ (t)) dt −

 ω ˜

0

0

 γh,ε (˜ ω , t) 0 , γh,ε (˜ ω , t) dt ε

= 0.

i (˜ On the other hand, using Poincaré's inequality (note that γ(ti ) = γh,ε ω , ti ) + bi ), we obtain

Z

1

|γ(t) − γh,ε (˜ ω , t)| dt ≤

E

m X

0

=

m X

E

i=1

Z E

i ˜ , t) − bi | dt (τ bi ω |γ(t) − γh,ε ε

Ii m

i (˜ ω , t) − bi | dt ≤ |γ(t) − γh,ε

Ii

i=1

Z

1 X h E 2 i=1

Z

0

i (˜ ω , t)| dt. |γ 0 (t) − γh,ε

Ii

i (˜ ω , ·) is an ε/m-minimizing From coercivity, growth condition, and since γh,ε sequence of (2), we have

Z

1 ω , t)| dt ≤ |γh,ε (˜ ν Ii 0i



Z ω ˜ Ii

 t ε Λ ε ai , ai dt + ≤ |ai | |Ii | + , ε mν ν mν

so that the previous inequality yields Z

(4)

E 0

1

ε |γ(t) − γh,ε (˜ ω , t)| dt ≤ Ch + , ν

where C depends only on Λ, ν and sup |γ 0 |. Let ε → 0 and h → 0 in this order [0,1]

in (3), (4). Using a simultaneous diagonalisation argument, we deduce that there exists a map ε 7→ h(ε) such that Z lim E ε→0

1

Z ds0 (γ) dt −

0

 γh(ε),ε (˜ ω , t) 0 , γh(ε),ε) (˜ ω , t) dt = 0, ε

1

|γ(t) − γh(ε),ε (˜ ω , t)| dt = 0.

lim E

ε→0

 ω ˜

0

Z

1

0

˜ 0 of full Set γε (˜ω , ·) = γh(ε),ε (˜ω , ·). By the above, there exist a set Ω probability that we may assume independent on γ and, by a classical Cantor diagonalization argument, a subsequence of ε independent on γ such that

13

Deterministic equivalent of a random Riemannian metric dened by geodesics

491

(γε (˜ ω , ·))ε strongly converges in L1 ((0, 1), RN ) to γ with lim sup Fε (˜ ω )(γε (˜ ω , ·)) ≤ F0 (γ), ε→0

so that Γ lim sup Fε (˜ ω ) ≤ F0 ε→0

in D. Step 2. Let us consider the functional dened in L1 ((0, 1), RN ) by F˜0 (γ) =

1

 Z  0



ds0 (γ 0 (t)) dt if γ ∈ D,

otherwise.

+∞

According to Step 1, the estimate Γ lim sup Fε (˜ ω ) ≤ F˜0 ε→0

˜ 0 . We conclude by taking the lower holds in for every ω˜ in Ω semicontinuous envelope for the strong convergence in L1 ((0, 1), RN ) of the two functionals in the inequality above. Indeed, from the semicontinuity of Γ lim sup Fε (˜ ω ) (see [7]) and classical integral representation results on the L1 ((0, 1), RN )

ε→0

lower semicontinuous envelope of convex functions with linear growth (see [8] or [5] in the quasi-convex case), we obtain Γ lim sup Fε (˜ω ) ≤ F0 .  ε→0

4.2.

The lower bound

We establish now the lower bound in the denition of Gamma-convergence. More precisely, we have Proposition

˜ 0 of Ω ˜ of full probability such that 5. There exists a subset Ω Γ lim inf Fε (˜ ω ) ≥ F0 . ε→0

˜ 0. for all ω˜ in Ω ˜ 0 of full probability such that Proof. It is enough to prove that there exists Ω F0 (γ) ≤ lim inf Fε (˜ ω )(γε ) ε→0

for every γε strongly converging to γ in L1 ((0, 1), RN ). ˜ 0 of full probability obtained in Theorem 3 and assume Fix ω˜ in the set Ω that lim inf Fε (˜ω )(γε ) < +∞. There exists a (not relabelled) subsequence such ε→0

492

Oana Iosifescu and Gérard Michaille

14

that the bounded Radon measure  µε (˜ ω ) := ω ˜

 γε (·) 0 , γε L1b(0,1) ε

˜ 0 . Our method weakly converges to a radon measure µ(˜ω ) for any ω˜ ∈ Ω consists in analyzing the Lebesgue decomposition µ(˜ω ) = µa (˜ω ) + µs (˜ω ) of the limit measure, where µa (˜ω ) and µs (˜ω ) are respectively the regular and the singular part of µ(˜ω ) with respect to L1b(0,1) . More precisely, using the ergodic Theorem 3, we establish the existence of a subset of full probability ˜ 0 such that still denoted Ω

(5)

µa (˜ ω) ≥

ds0 (γa0 (·))L1b(0,1) ,

µs (˜ ω) ≥

ds∞ 0



 γs0 (·) |γs0 | |γs0 |

˜ 0 , and the conclusion of Theorem 5 will follow from the classical for all ω˜ ∈ Ω Alexandro inequality µ(˜ ω )(O) ≤ lim inf µε (˜ ω )(O) ε→0

for every open subset of (0, 1), applied here for O = (0, 1). To simplify notation ˜ 0 appearing in µε and µ. The next two lemmas we drop the xed variable ω˜ ∈ Ω are devoted to the proof of the two estimates in (5). Lemma

1. For L1b(0,1) -almost all t0 in (0, 1), we have lim

ρ→0

µ(Iρ (t0 )) ≥ ds0 (γa0 (t0 )). |Iρ (t0 )|

Proof. By the classical dierentiation theorem, the limit lim µ(Iρ (t0 ))/ ρ→0

|Iρ (t0 )| exists for L1b(0,1) -almost all t0 in (0, 1). We x such a t0 . Clearly, for ρ ∈ (t0 , 1 − t0 ) \ D, where D is a countable set, we have   Z µ(Iρ (t0 )) µε (Iρ (t0 )) 1 γε (t) 0 lim = lim lim = lim lim ω ˜ , γε (t) dt. ρ→0 |Iρ (t0 )| ρ→0 ε→0 |Iρ (t0 )| ρ→0 ε→0 |Iρ (t0 )| Iρ (t ) ε 0

Assume for a moment that the trace of γε agrees with the ane function γ0 (t) = γ(t0 ) + (t − t0 )a with a ∈ QN , |γa0 (t0 ) − a| ≤ η , where η > 0 is intended to go to 0. It follows that µε (Iρ (t0 )) µ(Iρ (t0 )) = lim lim ρ→0 ε→0 |Iρ (t0 )| |Iρ (t0 )|    Z 1 v(t) 0 ≥ lim sup lim sup inf ω ˜ , v (t) dt : |Iρ (t0 )| Iρ (t0 ) ε ρ→0 ε→0 lim

ρ→0

v∈W

1,1

 (Iρ (t0 ), R ), v(t0 − ρ/2) − v(t0 + ρ/2) = ρa N

15

Deterministic equivalent of a random Riemannian metric dened by geodesics

S˜1 Iρ (t0 ) (˜ ω , a) ε

≥ lim sup lim sup ρ→0

| 1ε Iρ (t0 )|

ε→0

493

= ds0 (a) ≥ ds0 (γa0 (t0 )) − Λη,

where we have used Theorem 3 giving the expression of ds0 . The conclusion follows by letting η → 0. The idea consists now in modifying γε in a neighborhood of the boundary of Iρ (t0 ) by a function γ˜ε which belongs to γ0 + W01,1 (Iρ (t0 ), RN ) and to follow the above procedure. We obtain (for the details, we refer the reader to [1]) µ(Iρ (t0 )) Λ ≥ ds0 (a) − 2 |Iρ (t0 )| ρ

(6)

Z |γ(t) − γ0 (t)| dt. Iρ (t0 )

By the denition of γ0 , (6) yields µ(Iρ (t0 )) Λ ≥ ds0 (γa0 (t0 )) − |Iρ (t0 )| ρ

Z Iρ (t0 )

|γ( t) − γ(t0 ) − (t − t0 )γa0 (t0 )| dt − 2Λη. |t − t0 |

Letting ρ → 0, according to Proposition 2 we conclude as previously.



  γ0 s (·) |γs0 |. We establish now the second estimate µs ≥ ds∞ 0 |γs0 | Lemma

2. For γs0 -almost all t0 in (0, 1), we have  0  µ(Iρ (t0 )) ∞ γ ≥ ds (·) . 0 ρ→0 |γs0 |(Iρ (t0 )) |γ 0 | lim

Proof. We rst introduce some notation and state a few technical results established in [5]. Dene in BV (I, RN ) and W 1,1 (I, RN ), respectively, the rescaled  functions Z   1 1   γ(t + ρt) − γ(s) ds , γ = 0 ρ   |γ 0 |(Iρ (t0 ))| |Iρ (t0 )| Iρ (t0 ) Z    1 1   γ = γ (t + ρt) − γ (s) ds , ε 0 ε  ε,ρ |γ 0 |(Iρ (t0 ))| |Iρ (t0 )| Iρ (t0 )

where I is the interval (−1/2, 1/2).
In the sequel, we x t0 in the Borel subset E of (0, 1) satisfying |γs0 | (0, 1) \ E = 0, for which the limit γ 0 (Iρ (t0 )) γ0 = (t0 ) ρ→0 |γ 0 |(Iρ (t0 )) |γ 0 | lim

0

exists and lim |γ 0 |(Iρρ (t0 )) = 0. Set tρ = |γ |(Iρρ (t0 )) , which thus tends to +∞. ρ→0 From now on, ρ denotes a sequence of positive rational numbers. Note that γε,ρ tends to γρ strongly in L1 ((0, 1), RN ) as ρ → 0 and that, according to the Poincaré-Wirtinger inequality, Z

Z |γρ | dt ≤ C

I

I

|γρ0 |.

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16

Since I |γρ0 | dt ≤ 1, there exists a function γ˜ in BV (I, RN ) and a subsequence in ρ that we shall not relabel such that γρ tends to γ˜ weakly in BV (I, RN ). A carefull analysis of this convergence leads to the following property of γ˜ : there exists a nondecreasing scalar function ψ in BV (I) such that γ˜ admits the representation γ˜ (t) = ψ(t) γ 0 /|γ 0 | (t0 ). Moreover, for every δ > 0, δ ∈]0, 1[\D, where D is a countable subset of ]0, 1[, γρ0 (δI) → γ˜ (δI). For a proof, we refer the reader to [5], Theorem 2.3. We are now in a position to complete the proof of Lemma 2. We want to ρ (t0 )) estimate from below lim |γµ(I 0 |(I (t )) for the subsequence invoked above. We have ρ 0 R

ρ→0

µ(Iρ (t0 )) µε (Iρ (t0 )) = lim lim 0 0 ρ→0 ε→0 |γ |(Iρ (t0 )) |γ |(Iρ (t0 ))   Z γε (t) 0 1 ω ˜ = lim lim 0 , γε (t) dt. ρ→0 ε→0 |γ |(Iρ (t0 )) Iρ (t ) ε 0 lim

ρ→0

In order to reason on the xed space BV (I), we change the scale and set vε,ρ =


1 γε (t0 + tρ) − rρ , ρ

where rρ approximates in QN the mean value mρ of γρ in Iρ (t0 ). More precisely, |mρ − rρ | ≤ η , where η is a positive number intended to go to 0. The reason for which we substitute rρ to this mean value will be explained later. Note that vε,ρ /tρ approximates the function γε,ρ previously dened. More precisely, we have and we obtain (7)

|mε,ρ − mρ | vε,ρ η tρ − γε,ρ ≤ |γ 0 |(Iρ (t0 )) + |γ 0 |(Iρ (t0 ))

µ(Iρ (t0 )) 1 = lim lim lim 0 ρ→0 ε→0 tρ ρ→0 |γ |(Iρ (t0 ))

Z τ rρ ω ˜ I

ε

 ρv

ε,ρ

ε

 0 , vε,ρ dt.

In order to apply the ergodic Theorem 3 with a = tρ 1δ γ˜ 0 (δI), we modify vε,ρ near the boundary of I and adapt an argument of [1]. More precisely, we slice I near its boundary by small intervals of length αρ /n, where αρ = 1/2 |γρ − γ˜ |L1 (I,RN ) and n ∈ N is intended to go to +∞: I0 = (1 − αρ )I, . . . ,

 αρ  Ii = 1 − αρ + i I, n

and consider the cut-o functions

ϕi ∈ C0∞ (Ii ), 0 ≤ ϕ ≤ 1, ϕ = 1 in Ii−1 , |ϕ0i |L∞ (Ii ) ≤

We nally dene vε,ρ,i = tρ γ0 + ϕi (vε,ρ − tρ γ0 ),

i = 1, . . . , n, n , i = 1, . . . , n. αρ

17

Deterministic equivalent of a random Riemannian metric dened by geodesics

495

where γ0 is the ane function γ0 (t) =

tρ 0 ψ(δ/2− ) + ψ(δ/2+ ) γ 0 γ˜ (δI) + (t0 ) δ 2 |γ 0 |

with gradient 1δ γ˜ 0 (δI). The constant in the denition of γ0 will guarantee the control of the defect at the end of the proof. We then obtain for i = 1, . . . , n the estimate (8)



Z

1 tρ

τ rρ ω ˜ I

ε

   Z vε,ρ 0 vε,ρ,i 0 1 τ rρ ω ˜ ,v dt ≥ ,v dt − Ri (ρ, ε, n, η), ε/ρ ε,ρ tρ I ε ε/ρ ε,ρ,i

whose proof is an easy consequence of the growth condition (see [1]), where Λ Ri (ρ, ε, n, η) = O(ρ) + tρ

Z Ii \Ii−1

0 |vε,ρ

−

tρ γ00 | dt

nΛ + tρ αρ

Z |vε,ρ − tρ γ0 | dt Ii \Ii−1

and O(ρ) does not depend on ε, n, η and tends to 0 as ρ → 0. As vε,ρ,i are admissible functions in the denition of S˜, for i = 1, . . . , n we have 1 tρ



Z τ rρ ω ˜ I

ε

vε,ρ,i 0 ,v ε/ρ ε,ρ,i



  1 0 ˜ 1 τ rρ ω S ˜ , t γ ˜ (δI) ρ δ 1 ε/ρ I ε dt ≥ . 1 tρ | ε/ρ I|

Averaging these n inequalities, from (8) we obtain   tρ 0 ˜ 1 τ rρ ω γ ˜ (δI) S ˜ , δ ve,ρ 0 1 1 ε/ρ I ε τ rρ ω ˜ , vε,ρ dt ≥ − O(ρ) − O(1/n) 1 tρ I ε ε/ρ tρ | ε/ρ I|   tρ 0 ˜ 1 τ rρ ω Z S γ ˜ (δI) ˜ , I δ Λ ε vε,ρ − γ0 dt ≥ 1 ε/ρ − O(ρ) − O(1/n) − 1 αρ I\(1−αρ )I tρ tρ | ε/ρ I| Z Λ|mε,ρ − mρ | Λη Λ − 0 − − |γε,ρ − γ0 | dt, |γ |(Iρ (t0 )) |γ 0 |(Iρ (t0 )) αρ I\(1−αρ )I Z





where O(1/n) tends to 0 as n goes to +∞. We let successively n → +∞, ˜ 0 of full probability ε → 0, η → 0. According to Theorem 3, there exists a set Ω such that   Z ve,ρ 0 1 τ rρ ω ˜ , vε,ρ dt lim sup ε/ρ ε→0 tρ I ε   Z tρ 0 1 Λ ≥ ds0 γ˜ (δI) − |γρ − γ0 | dt − O(ρ) tρ δ αρ I\(1−αρ )I   Z √ tρ 0 1 Λ γ˜ (δI) − Λ αρ − ≥ ds0 |˜ γ − γ0 | dt − O(ρ) tρ δ αρ I\(1−αρ )I

496

Oana Iosifescu and Gérard Michaille

18

˜ 0 . Note that Ω ˜ 0 a priori depends on rρ and tρ /δ γ˜ 0 (δI). Actually, for every ω˜ ∈ Ω ˜ r,a , where (r, a) belongs to ˜0 = T Ω we consider the set of full probability Ω N N Q × Q . Without any loss of generality, we suppose here that tρ /δ γ˜ 0 (δI) belongs to QN . If not, is sucient to reason as in the previous subsection. Recalling (7), we nally obtain µ(Iρ (t0 )) 1 lim ≥ lim sup ds0 ρ→0 |γ 0 |(Iρ (t0 )) ρ→0 tρ



 Z tρ 0 Λ |˜ γ − γ0 | dt. γ˜ (δI) − lim sup δ ρ→0 αρ I\(1−αρ )I

By a straightforward computation, it is easily seen that γ˜ −γ0 has zero trace on − +) γ0 ∂δI . This is the reason for which we chose the constant ψ(δ/2 )+ψ(δ/2 2 |γ 0 | (t0 ) in the denition of γ0 . Therefore, by the properties of trace in BV (I), the last term on the right hand side tends to 0. For the rst term we have    tρ 0 tρ 0 1δ γ˜ (δI) = lim sup ds0 γ˜ (δI) δ δ ρ ρ→0 δ tρ     |γ 0 |(Iδρ (t0 ) tρ γ 0 1δ ds0 (t0 ) − lim inf Λ 1 − 0 = lim sup ρ→0 δ |γ 0 | |γ |(Iρ (t0 ) ρ→0 δ tρ  0  1 γ ≥ ds∞ (t0 ) − (1 − δ). 0 δ |γ 0 | 1 lim sup ds0 ρ→0 tρ



We conclude after letting δ → 1. The proof of Proposition 2 is now complete. 5.



CONVERGENCE IN PROBABILITY

In order to simplify notation, we write Fε and F0 to denote the restrictions to BV ((0, 1), RN ) of Fε + Ia,b and F0 + I¯a,b , respectively. We want to establish the convergence in probability of all sequences (Fε )ε>0 of random functionals in a suitable metric space. We equip the space BV ((0, 1), RN ) with the strong topology of L1 ((0, 1), RN ) and denote by SC the space of real valued lower semicontinuous functions G dened on BV ((0, 1), RN ) that satisfy G(u) ≥ R ν Ω |Du|. It is well known (see for instance [9]) that SC is metrizable by the Gamma-convergence. We denote by dΓ a metric induced by the latter in SC and by BΓ the Borel eld of the metric space (SC, dΓ ). Recall that the law of ˜ T˜ ) → (SC, BΓ ) is the image of P ˜ by G on the metric a measurable map G : (Ω, space (SC, dΓ ). ˜ let F¯ε (˜ 5. For every ω˜ ∈ Ω ω ) denote the lower semicontinuous envelope of Fε (˜ ω ). Then all sequences F¯ε converge in law and in probability to F0 in the metric space (SC, dΓ ). Theorem

19

Deterministic equivalent of a random Riemannian metric dened by geodesics

497

Proof. Clearly (see for instance [7]), for a subsequence Fε that Gammaconverges almost surely to F0 , the corresponding subsequence of F¯ε Gammaconverges almost surely to the same limit F0 which belongs to SC . Therefore, any sequence of laws of random functions F¯ε weakly converges to the Dirac measure δF0 . Then, by classical probabilistic arguments, any sequence F¯ε con˜ ω∈Ω ˜ : dΓ (F¯ε (˜ verges in probability to the constant F0 in SC , i.e., P{˜ ω ), F0 ) > η} → 0 as ε → 0 for all η > 0.  Corollary 2. The random geodesic metric dε converges in probability to the deterministic metric d0 in the following sense: for all η > 0 and all points (a, b) in RN ,

P{ω : |dε (ω)(a, b) − d0 (a, b)| > η} → 0 as ε → 0. ˜ ω : |dε (˜ Proof. It is equivalent to establish that P{˜ ω )(a, b) − d0 (a, b)| > η} tends to 0 as ε → 0, which is a straightforward consequence of ˜ ω∈Ω ˜ : dΓ (F¯ε (˜ P{˜ ω ), F0 ) > η} → 0

and the continuity of the real valued map G 7→ min G dened in (SC, dΓ ).



REFERENCES [1] Y. Abddaimi, G. Michaille and C. Licht, Stochastic homogenization for an integral functional of a quasiconvex function with linear growth. Asymptotic Anal. 15(2) (1997), 183+. [2] E. Acerbi and G. Buttazzo, On the limit of periodic Riemannian metrics. J. Anal. Math. 43(4) (1983), 183201. [3] M.A. Ackoglu and U. Krengel, Ergodic theorem for superadditive processes. J. Reine Angew. Math. 323 (1981), 5367. [4] M. Amar and E. Vitali, Homogenization of periodic Finsler metric. J. Convex Anal. 5(1) (1998), 171186. [5] L. Ambrosio and G. Dal Maso, On the relaxation in BV (Ω; Rm ) of quasi-convex integrals. J. Funct. Anal. 109 (1992), 7697. [6] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Univ. Press, New York, 2000. [7] H. Attouch, Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984. [8] G. Bouchitté, Convergence et relaxation des fonctionnelles du calcul des variations á croissance linéaire. Application à l'homogénéisation en plasticité. Ann. Fac. Sci. Toulouse Math (5) 8 (1986/87), 1, 736. [9] G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser, Boston, 1993. [10] G. Dal Maso and L. Modica, Non linear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 363 (1986), 2743. [11] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), 842850.

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[12] U. Krengel, Ergodic Theorems. De Gruyter Studies in Mathematics 6. Walter de Gruyter, Berlin, 1985. [13] C. Licht and G. Michaille, Global-local subadditive ergodic theorems and application to homogenization in elasticity. Prépublication, Université Montpellier II, 1997.

Received 3 April 2008

I3M, UMR-CNRS 5149 Université Montpellier II Case courier 051 Place Eugène Bataillon 34095 Montpellier Cedex 5, France [email protected] and

I3M, UMR-CNRS 5149 Université Montpellier II Case courier 051 Place Eugène Bataillon 34095 Montpellier Cedex 5, France & AVA, Université de Nîmes Place Gabriel Péri 30021 Nîmes Cedex 0, France [email protected]