DETERMINISTIC EQUIVALENT OF A RANDOM RIEMANNIAN METRIC DEFINED BY GEODESICS

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

LetΩbe the set of locally nite sequences(ω_i)i∈N inR^N andMthe set of all countable and locally nite sums of Dirac measures, equipped with their standard σ-algebra. Given a positive numberλand ε >0 intended to tend to 0, we consider the Poisson point process

Ω→ M, ω7→ N_ε(ω,·) =X

i∈N

δ_ωi

with intensityλε=λ/ε^NL_N, whereL_N denotes the Lebesgue measure onR^N, and dene the heterogeneous random Riemannian metric on R^N by

dsε(ω, x, ξ) :=

( ds2(ξ) if x∈S

i∈NB(ωi, εr), ds₁(ξ) otherwise.

For every iin N, B(ωi, εr) is the open ball in R^N with radius εr centered at ω_i and ds_k(ξ) := (P

i,ja^k_i,jξ_iξ_j)^1/2 for all ξ ∈ R^N with (a^k_i,j)i=1,...,N, j=1,...,N, k= 1,2, two symmetric positive matrices (see Section 3 for more details).

The spaceR^N is then equipped with the highly random oscillating geo- desic metric (a, b)7→d_ε(ω)(a, b) dened for any pointsa, b inR^N by

inf Z 1

0

dsε(ω, γ(t), γ⁰(t)) dt:γ ∈W^1,1((0,1),R^N), γ(0) =a, γ(1) =b

.

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

Classically, there exists a probability measurePon Ωsatisfying P([N_ε(·, A) =k]) =λ^k_εL_N(A)^k exp(−λ_eL_N(A))

k!

for every bounded Borel set A in R^N, so that the expectation of the random variable N_ε(·, A) isE N_ε(·, A)

=λ_εL_N(A). Consequently, d_ε is statistically like a geodesic distance in a Riemaniann space with small balls as hetero- geneities, and whose number is of orderλ_ε=λ/ε^N per unit of volume. It will be noted that the balls can interpenetrate.

It is easy to see that the random distanceω 7→d_ε(ω)(a, b) has the same law as (i.e., is statistically equivalent to) the random distance given by

inf Z 1

0

ds(ω,γ(t)

ε , γ⁰(t)) dt:γ ∈W^1,1((0,1),R^N), γ(0) =a, γ(1) =b

,

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, ω7→ N(ω,·) =X

i∈N

δ_ωi

with intensity λL_N. 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 d_ε(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 d_ε(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 distanced₀. Our proof is based on the following compactness result established in Section 3: there exists a subsequence of the sequence of integral functionals Fε deningdε, 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 functionalsF¯εto the Dirac probability measure δF0. More precisely, we know that Gamma-convergence induces a metric on the set SCof all semicontinous functionsG:BV(0,1),R^N →R∪{+∞}. Then, considering the probability images PF_ε⁻¹ and PF₀⁻¹ =δF0 of

F¯ε, F0 : Ω→ SC

we obtain PF_ε⁻¹ * δ_F0. Denoting byd_Γthe metric induced inSC by Gamma- convergence, we obtain as a corollary the convergence in probability of all

sequences, (Fε)ε>0, i.e.,

P{ω:dΓ( ¯Fε(ω), F0)> η} →0asε→0.

Finally, using the continuity of the map G 7→ minG in the metric space (SC, d_Γ), we deduce our main result, the convergence in probability

P{ω:|d_ε(ω)(a, b)−d0(a, b)|> η} →0 asε→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 inW^1,p((0,1),R^N) or BV((0,1),R^N).

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),R^N) space. Section 3 is devoted to the study of two suitable random processes and to the deni- tion 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. BASIC NOTIONS

2.1. Basic notions on Gamma-convergence

Let(X, d)be a metric space and{(F_n)n, F, n→+∞}functionals map- pingX intoR∪ {+∞}. The notion of convergence below is equivalent to con- vergence of the epigraph ofFnto the epigraph ofF in the Kuratovski-Painlevé sense and has been rst introduced in [11]. We give the denition in the frame- work of metric spaces. Indeed, in the sequel, (X, d) will be L¹((0,1),R^N) or BV((0,1),R^N) equipped with the strong convergence of L¹((0,1),R^N). For details, we refer the reader to [7] and [9].

Denition 1. A sequence(F_n)n∈N Gamma-converges toF atx in X i i) there exists a sequence (xn)n∈N converging to xinX such that

F(x)≥lim sup

n→+∞ Fn(xn),

ii) for every sequence(yn)n∈N converging to xinX we have F(x)≤lim inf

n→+∞Fn(yn).

When this property holds for everyxinX, we say that(Fn)nGamma-converges to F in (X, d) and writeF = Γ limF_n.

It is straightforward to show that this convergence is equivalent to Γ lim sup

n→+∞

F_n(x)≤F(x)≤Γ lim inf

n→+∞F_n(x) for any x∈X, where

Γ lim sup

n→+∞ Fn(x) := min n

lim sup

n→+∞ Fn(xn) :xn→x o

,

Γ lim inf

n→+∞F_n(x_n) := minn lim inf

n→+∞F_n(x_n) :x_n→xo .

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

Proposition 1. Assume that (F_n) Gamma-converges to F and let x_n∈ X be such that

F_n(x_n)≤inf{ F_n(x) :x∈X }+ε_n,

where ε_n > 0, ε_n → 0. Assume furthermore that {x_n, n ∈ N} is relatively compact. Then any cluster point x of {x_n, n∈N} is a minimizer of F and

n→+∞lim inf{F_n(x) :x∈X}=F(x).

2.2. Basic notions on BV-functions

The setBV(I,R^N)is the space of all functionsu∈L¹(I,R^N)such that the distributional derivative u⁰ iof u is a bounded Borel measure with values in R^N. The nite total variation of u⁰ is denoted byR

I|u⁰|and BV(I,R^N) is equipped with the norm

|u|_BV(I,RN)=|u|_L1((0,1),R^N)+ Z

I

|u⁰|.

The unit ball of BV(I,R^N) is weakly sequentially compact in the follow- ing sense: if |u_n|_BV(I,RN) ≤ 1, there exist a subsequence (un_k)k∈N and u in BV(I,R^N) with |u|_BV(I,RN) ≤ 1 such that u_nk weakly converges to u in BV(I,R^N), that is,

( u_nk →u strongly in L¹(I,R^N),

u⁰_nk * u⁰ in the sense of weak convergence of bounded measures.

In the sequel, u⁰_a and u⁰_s will denote the density of the regular part of the measure u⁰ and its singular part in the Lebesgue decomposition, i.e., u⁰ = u⁰_aL_1bI+u⁰_s. The functionuais actually the approximate dierential ofuinI. More precisely,

Proposition 2. Letu be any element of BV(I,R^N). Then

ρ→0lim 1 ρ

Z

Iρ(t0)

|u(t)−u(t₀)−(t−t₀) u⁰_a(t₀)|

|t−t₀| dt= 0

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

For a general study ofBV-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 (τ_s)s∈R of P- preserving transformations of (Ω,T). The group (τ_s)s∈R is said to be ergodic if every set E inT such thatτsE=E,s∈R, has probability equal to0 or 1. A sucient condition to ensure ergodicity of (τ_s)s∈R is the mixing condition:

for every E andF inT,

|s|→+∞lim P(τsE∩F) =P(E)P(F) which expresses an asymptotic independence.

LetEdenote the expectation operator andJ 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 everyI ∈ J such that there exists a nite family(Ij)j∈J of pairwise disjoint intervals in J withI =S

j∈JI_j, we haveS_I(·)≤P

j∈JS_Ij(·);

(ii)∀I ∈ J, ∀s∈R,S_s+I=S_I◦τ_s.

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. LetS be a subadditive process with respect to (τ_s)z∈R that is assumed to be ergodic. Assume that

inf Z

Ω

S_I(ω)

|I| P(dω) :|I| 6= 0

>−∞.

Let I be any xed interval in I. Then, almost surely,

t→+∞lim S_tI(ω)

|tI| = inf

m∈N^∗

ES_[0,m[d

m^d

.

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

3. IDENTIFYING THE LIMIT METRIC

Let us give more details on the probabilistic setting. We consider two symmetric N×N-matrices(a^k_i,j)i=1,...,N, j=1,...,N, satisfying

ν|ξ|²≤X

i,j

a^k_i,jξiξj ≤Λ|ξ|², k= 1,2,

for every ξ ∈ R^N, 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 λL_N,λ >0, from the probability space(Ω,T, P)intoN^B(RN)equipped with the standard productσ-algebra, which classically satises:

i) for every bounded Borel setA inR^N, we have N(ω, A) =X

i∈N

δωi(A);

ii) for every nite family(Ai)i∈I of pairwise disjoint bounded Borel sets inR^N, the random variablesN(·, Ai), i∈I, are independent;

iii) for every bounded Borel setA and everyk∈N, P([N(·, A) =k]) =λ^kL_N(A)^k exp(−λL_N(A))

k! .

Note that N(ω, A) = card(A∩Ω) and that E(N(·, A)) =λL_N(A) for every bounded Borel set of R^N.

For a givenr >0, we dene a random Riemannian metric by setting ds(ω, x, ξ) =

( ds2 if x∈S

i∈NB(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 F_ν,Λ of the functions f fromR^N×R^N intoRthat 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 denoted by T˜ of the Borel σ-eld of R^RN×RN, that is the smallest σ-algebra on F_ν,Λ such that all evaluation maps f 7→ f(x, ξ), x ∈ R^N, ξ ∈ R^N, are measurable. According to (1), it is readily seen that the map ds : Ω→ F_ν,Λ, ω 7→ ω˜ := ds(ω) is (T,T˜)-measurable. Throughout the paper we will reason in the probability space( ˜Ω,T˜,P) := (F˜ _ν,Λ,T˜,P˜)whereP˜ is the law ofds, i.e.

the probability image of Pby the measurable map ds.

For every xed point a in R^N we introduce the ergodic group (Ts)s∈R

of P˜-preserving transformations by setting T_sω(x, ξ) = ˜˜ ω(x+sa, ξ)and dene the subadditive random process with respect to (Ts)s∈R by setting

S_I(˜ω, a) = inf Z

I

˜

ω(γ(t), γ⁰(t)) dt:γ ∈W^1,1(I,^◦ R^N), γ(α) =αa, γ(β) =βa

.

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 ofS_I(·, a). Ergodicity of(Ts)s∈Ris 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 subset Ω˜⁰ of full probability such that for all ω˜ in Ω˜⁰ and for all a ∈ Q^N the limit

ds₀(a) = lim

ε→0

S1 εI(˜ω, a)

|¹_εI| = lim

n→+∞

S_]0,n[(˜ω, a)

n = inf

n∈N^∗

ES_]0,n[(·, a) n

exists.

Proof. For every xedainQ^N, using Theorem 1 we deduce the existence of Ω˜⁰_a of full probability such that for everyω˜ ∈Ω˜⁰_a the limit

ε→0lim S1

εI(˜ω, a)

|¹_εI| = lim

n→+∞

S_]0,n[(˜ω, a) n exists, and it is sucient to set Ω˜⁰=T

a∈Q^NΩ˜⁰_a.

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

Proposition 3. The above normalized subadditive random process satis- es

| S1

εI(˜ω, a)

|¹_εI| − S1

εI(˜ω, b)

|¹_εI| | ≤Λ|a−b|.

for every ω˜ and everya andb in Q^N. Consequently,

|ds₀(a)−ds0(b)| ≤Λ|a−b|

for every ω˜ in Ω˜⁰, andds0 may be extended to R^N by a function still denoted ds₀, satisfying the same lipschitzian property and which is still the almost sure limit of the previous normalized subadditive process.

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 0and γ ∈W^1,1((0, n),R^N) 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 R^N 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

γ_s(t) =

( γ(t) in[0, n−s]

nb+^t−n_s nb−γ(n−s)

in[n−s, n].

According to the growth condition, we have S_]0,n[(˜ω, b)

n ≤ 1

n Z n

0

˜

ω(γs(t), γ_s⁰(t)) dt

≤ 1 n

Z n 0

˜

ω(γ(t), γ⁰(t)) dt+Λ

n|nb−γ(n−s)|.

Letting s→ 0 andn→+∞ in this order, and noticing that every element of W^1,1((0, n),R^N]is continuous in [0, n], we obtain

S_]0,n[(˜ω, b)

n ≤ S_]0,n[(˜ω, a)

n + Λ|b−a|, which completes the proof.

In what follows, the same notation Ω˜⁰ denotes various subsets of full probability. In the proofs below we will consider theP˜-preserving transforma- tion τξ on Ω˜ dened for every ξ ∈ R^N by τξω(x, ζ) = ˜˜ ω(x+ξ, ζ). We also introduce a second process S˜from J intoL¹( ˜Ω,T,P˜) dened by

S˜_I(˜ω, a) = inf Z

I

˜

ω(u(t), u⁰(t)) dt:u∈W^1,1(I,^◦ R^N), u(α)−u(β) = (α−β)a

.

Clearly, S˜ is invariant under translations of R and satises S_I ≥ S˜_I for all I ∈ I. Although S˜is not subadditive we have

Theorem 3. For everyt0 ∈R,ρ >0anda∈R^N,S˜1

εIρ(t0)(·, a)/|¹_εIρ(t0)|

tends in probability to ds₀(a) as ε→0.

Therefore, if a belongs to Q^N and ρ to Q⁺, there exists a subset Ω˜⁰ of Ω˜ of full probability which does not depend on a, ρ and t0, and a subsequence, not relabelled, possibly depending on a andρ, such that

ε→0lim S˜1

εIρ(t0)(˜ω, a)

|¹_εIρ(t0)| =ds0(a) for all ω˜ in Ω˜⁰.

Proof. AsS˜_I+s(˜ω, a) = ˜S_I(˜ω, a)for everysinR, it is sucient to establish the proof for the interval I_ρ(0). To simplify notation, we denote by I this interval. Let γε be an ε-minimizer of S˜1

εI, i.e., S˜1

εI(˜ω, a)

|¹_εI| ≥ 1

|¹_εI|

Z

1 εI

˜ ω

γε

ε, γ_ε⁰

dt−ε,

and set ˜γε =γε−bεwherebε=γε(αε)−αεaandαε is the lower bound of the interval (1/ε)I. Since ˜γ_ε is an admissible function for S1

εI, we have S˜1

εI(˜ω, a)

|¹_εI| ≥ 1

|¹_εI|

Z

1 εI

τ_bεω˜ γ˜_ε

ε,γ˜_ε⁰

dt−ε≥ S1

εI(τ_bεω, a)˜

|¹_εI| −ε.

Thus

S1 εI(˜ω, a)

|¹_εI| ≥ S˜1

εI(˜ω, a)

|¹_εI| ≥ S1

εI(τ_bεω, a)˜

|¹_εI| −ε, so that, for every η >0,

S˜1 εI(·, a)

|¹_εI| −ds0(a) > η

⊆

S˜1 εI(·, a)

|¹_εI| − S1

εI(τ_bε·, a)

|¹_εI| +ε > η

2

∪

S1

εI(τ_bε·, a)

|¹_εI| −ε−ds₀(a) > η

2

⊆

S1 εI(·, a)

|¹_εI| − S1

εI(τ_bε·, a)

|¹_εI| +ε > η

2

∪

S1

εI(τ_bε·, a)

|¹_εI| −ε−ds₀(a) > η

2

.

Sinceτbεis aP˜ preserving transformation, Theorem 2 implies thatS˜1

εI(·, a)/|¹_εI|

tends in probability to ds0(a) (use the Markov inequality P˜

|f_ε| > ^η₂

≤

2 η

R

Ω|f_ε|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 de- ned by

ds_ε(ω, x, ξ) :=ds₁(ξ) + ds₂(ξ)−ds₁(ξ)

min(1,N_ε(ω, B(x, εr)) where N_ε is a Poisson process with intensity _ε^λN (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.

Theorem 4. There exists a subsequence ofεsuch that the length dened in L¹((0,1),R^N) by

F_ε(˜ω)(γ) :=





 Z 1

0

˜ ω

γ(t) ε , γ⁰(t)

dt if γ ∈ W^1,1((0,1),R^N),

+∞ otherwise,

Gamma-converges almost surely to the deterministic function dened in L¹((0,1),R^N) by

F₀(γ) =





 Z 1

0

ds0(γ_a⁰(t)) dt+ Z 1

0

ds^∞₀ γ_s⁰

|γ_s⁰|

|γ_s⁰| ifγ ∈BV((0,1),R^N),

+∞ otherwise,

where γ_a⁰ is the density of the regular part of γ⁰ in its Lebesgue decomposition and ds^∞₀ is the recession function dened for every a∈R^N by

ds^∞₀ (a) = lim

t→+∞

ds0(ta) t .

Before proving Theorem 4, let us take into account the boundary condi- tions. For every (a, b) inR^2N, consider the functionals

I_a,b,I¯_a,b:L¹((0,1),R^N)→R⁺∪ {+∞}

dened by

I_a,b(γ) =

(0 if γ(0) =aand γ(1) =b, +∞ otherwise,

I¯_a,b(γ) =

(ds^∞₀ (γ(0)−a) +ds^∞₀ (b−γ(1)) if γ ∈BV((0,1),R^N),

+∞ otherwise.

It is well known that I¯_a,b is the relaxed functional of I_a,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 ofεsuch thatFε+I_a,bGamma- converges 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.

4.1. 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

Proposition 4. There exists a subset Ω˜⁰ of Ω˜ of full probability such that for all ω˜ in Ω˜⁰ there exists a subsequence still denoted ε such that

Γ lim sup

ε→0

Fε(˜ω)≤F0. Proof. We proceed in two steps.

Step 1. We prove the result in a dense countable subsetDofC¹((0,1),R^N) for the strong topology of L¹((0,1),R^N). Let γ ∈ D. Subdivise (0,1)in in- tervals I_i = (ti−1, t_i), i = 1, . . . , m, t₀ = 0, t_m = 1, of length h intended to go to 0, and consider the continuous piecewise linear function γh taking the values γ(t_i)for every i= 0, . . . , m. Denote bya_i the gradient ofγ_h inI_i. It is easily seen that γh strongly converges to γ inW^1,1((0,1),R^N). According to Theorem 2 and Proposition 3, there exists Ω˜⁰ of full probability such that, for all ω˜ ∈Ω˜⁰ and i= 1, . . . , m,

ds0(ai) = lim

ε→0

S1

εIi(˜ω, ai)

|¹_εI_i| = lim

ε→0

1

|I_i| Z

Ii

˜

ω(γ_h,εⁱ (˜ω, t)

ε , γ_h,ε⁰ⁱ (˜ω, t)) dt, where we performed a change of scale andγⁱ_h,ε(˜ω,·)is anε/m-minimizer of the problem

(2) inf

v∈W^1,1(Ii,R^N)

Z

Ii

˜ ω

v(t) ε , v⁰(t)

dt:v(ti−1) =ti−1ai, v(ti) =tiai

.

Set bi = γ(ti)−tiai for i = 1, . . . , m and denote by χ_Ii the characteristic function of the interval I_i. The function

γ_h,ε(˜ω,·) =

m

X

i=1

γ_h,εⁱ (τ_bi

ε

˜

ω,·) +b_i χ_Ii

belongs to W^1,1((0,1),R^N). The measure preserving transformation τbi ε

is introduced for technical reasons which will appear below. We have

lim sup

ε→0

E

Z 1 0

ds0(γ_h⁰) dt− Z 1

0

˜ ω

γ_h,ε(˜ω, t)

ε , γ_h,ε⁰ (˜ω, t)

dt

≤lim sup

ε→0 m

X

i=1

E

ds₀(a_i)|I_i| − Z

Ii

˜ ω





γ_h,εⁱ (τ_bi

ε

˜

ω, t) +bi

ε , γ_h,ε⁰ⁱ (τ_bi

ε

˜ ω, t)



dt

= lim sup

ε→0 m

X

i=1

E

ds₀(a_i)|I_i| − Z

Ii

τ_bi

ε

˜ ω





γ_h,εⁱ (τ_bi

ε

˜ ω, t)

ε , γ_h,ε⁰ⁱ (τ_bi

ε

˜ ω, t)



dt

= lim sup

ε→0 m

X

i=1

E

ds0(ai)|I_i| − Z

Ii

˜

ω γ_h,εⁱ (˜ω, t)

ε , γ_h,ε⁰ⁱ (˜ω, t)

! dt

= 0, where we have used the Lebesgue dominated convergence theorem. According to Proposition 3, we obtain

(3) lim

h→0lim

ε→0 E

Z 1 0

ds0(γ⁰(t)) dt− Z 1

0

˜ ω

γ_h,ε(˜ω, t)

ε , γ_h,ε⁰ (˜ω, t)

dt

= 0.

On the other hand, using Poincaré's inequality (note that γ(ti) =γ_h,εⁱ (˜ω, ti) + b_i), we obtain

E Z 1

0

|γ(t)−γ_h,ε(˜ω, t)|dt≤

m

X

i=1

E Z

Ii

|γ(t)−γ_h,εⁱ (τ_bi

ε

˜

ω, t)−b_i|dt

=

m

X

i=1

E Z

Ii

|γ(t)−γ_h,εⁱ (˜ω, t)−b_i|dt≤ 1 2h

m

X

i=1

E Z

Ii

|γ⁰(t)−γ_h,ε⁰ⁱ (˜ω, t)|dt.

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

Z

Ii

|γ_h,ε⁰ⁱ (˜ω, t)|dt≤ 1 ν

Z

Ii

˜ ω

t εai, ai

dt+ ε mν ≤ Λ

ν|a_i| |I_i|+ ε mν, so that the previous inequality yields

(4) E

Z 1 0

|γ(t)−γ_h,ε(˜ω, t)|dt≤Ch+ ε ν, whereCdepends only on Λ, ν andsup

[0,1]

|γ⁰|. Letε→0andh→0in this order in (3), (4). Using a simultaneous diagonalisation argument, we deduce that there exists a map ε7→h(ε) such that

ε→0limE

Z 1 0

ds0(γ) dt− Z 1

0

˜ ω

γ_h(ε),ε(˜ω, t)

ε , γ_h(ε),ε)⁰ (˜ω, t)

dt

= 0,

ε→0limE Z 1

0

|γ(t)−γ_h(ε),ε(˜ω, t)|dt= 0.

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

(γε(˜ω,·))_ε strongly converges inL¹((0,1),R^N) to γ with lim sup

ε→0

F_ε(˜ω)(γ_ε(˜ω,·))≤F₀(γ), so that

Γ lim sup

ε→0

F_ε(˜ω)≤F₀ inD.

Step 2. Let us consider the functional dened inL¹((0,1),R^N)by F˜0(γ) =





 Z 1

0

ds₀(γ⁰(t)) dt if γ ∈D,

+∞ otherwise.

According to Step 1, the estimate Γ lim sup

ε→0

F_ε(˜ω)≤F˜₀

holds in L¹((0,1),R^N) for every ω˜ in Ω˜⁰. We conclude by taking the lower semicontinuous envelope for the strong convergence in L¹((0,1),R^N) of the two functionals in the inequality above. Indeed, from the semicontinuity of Γ lim sup

ε→0

Fε(˜ω) (see [7]) and classical integral representation results on the lower semicontinuous envelope of convex functions with linear growth (see [8]

or [5] in the quasi-convex case), we obtain Γ lim sup

ε→0

F_ε(˜ω)≤F₀.

4.2. The lower bound

We establish now the lower bound in the denition of Gamma-convergence.

More precisely, we have

Proposition 5. There exists a subsetΩ˜⁰ ofΩ˜ of full probability such that Γ lim inf

ε→0 Fε(˜ω)≥F0. for all ω˜ in Ω˜⁰.

Proof. It is enough to prove that there existsΩ˜⁰of full probability such that F0(γ)≤lim inf

ε→0 Fε(˜ω)(γε) for every γ_ε strongly converging toγ inL¹((0,1),R^N).

Fixω˜ in the set Ω˜⁰ of full probability obtained in Theorem 3 and assume thatlim inf

ε→0 F_ε(˜ω)(γ_ε)<+∞. There exists a (not relabelled) subsequence such

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

γε(·) ε , γ_ε⁰

L_1b(0,1)

weakly converges to a radon measure µ(˜ω) for any ω˜ ∈ Ω˜⁰. Our method consists in analyzing the Lebesgue decomposition µ(˜ω) = µ_a(˜ω) +µ_s(˜ω) of the limit measure, where µ_a(˜ω) and µ_s(˜ω) are respectively the regular and the singular part of µ(˜ω) with respect to L_1b(0,1). More precisely, using the ergodic Theorem 3, we establish the existence of a subset of full probability still denoted Ω˜⁰ such that

(5) µa(˜ω)≥ds0(γ_a⁰(·))L_1b(0,1), µs(˜ω)≥ds^∞₀ γ_s⁰

|γ_s⁰|(·)

|γ_s⁰|

for all ω˜ ∈Ω˜⁰, and the conclusion of Theorem 5 will follow from the classical Alexandro inequality

µ(˜ω)(O)≤lim inf

ε→0 µ_ε(˜ω)(O)

for every open subset of(0,1), applied here forO= (0,1). To simplify notation we drop the xed variableω˜ ∈Ω˜⁰ appearing inµεandµ. The next two lemmas are devoted to the proof of the two estimates in (5).

Lemma 1. For L_1b(0,1)-almost all t0 in (0,1), we have

ρ→0lim

µ(Iρ(t0))

|I_ρ(t₀)| ≥ds0(γ_a⁰(t0)).

Proof. By the classical dierentiation theorem, the limit lim

ρ→0µ(I_ρ(t₀))/

|I_ρ(t₀)| exists for L_1b(0,1)-almost allt₀ in (0,1). We x such a t₀. Clearly, for ρ∈(t₀,1−t₀)\D, whereDis a countable set, we have

ρ→0lim

µ(I_ρ(t₀))

|I_ρ(t0)| = lim

ρ→0lim

ε→0

µ_ε(I_ρ(t₀))

|I_ρ(t0)| = lim

ρ→0lim

ε→0

1

|I_ρ(t0)|

Z

Iρ(t0)

˜ ω

γ_ε(t) ε , γ_ε⁰(t)

dt.

Assume for a moment that the trace of γ_ε agrees with the ane function γ₀(t) =γ(t₀) + (t−t₀)awitha∈Q^N,|γ_a⁰(t₀)−a| ≤η, whereη >0is intended to go to 0. It follows that

ρ→0lim

µ(Iρ(t0))

|I_ρ(t₀)| = lim

ρ→0lim

ε→0

µε(Iρ(t0))

|I_ρ(t₀)|

≥lim sup

ρ→0

lim sup

ε→0

inf 1

|I_ρ(t₀)|

Z

Iρ(t0)

˜ ω

v(t) ε , v⁰(t)

dt: v∈W^1,1(I_ρ(t₀),R^N), v(t₀−ρ/2)−v(t₀+ρ/2) =ρa

≥lim sup

ρ→0

lim sup

ε→0

S˜1

εIρ(t0)(˜ω, a)

|¹_εI_ρ(t₀)| =ds0(a)≥ds0(γ_a⁰(t0))−Λη,

where we have used Theorem 3 giving the expression of ds₀. The conclu- sion follows by letting η → 0. The idea consists now in modifying γε in a neighborhood of the boundary of Iρ(t0) by a function ˜γε which belongs to γ0+W₀^1,1(Iρ(t0),R^N) and to follow the above procedure. We obtain (for the details, we refer the reader to [1])

(6) µ(I_ρ(t₀))

|I_ρ(t₀)| ≥ds₀(a)− Λ ρ²

Z

Iρ(t0)

|γ(t)−γ₀(t)|dt.

By the denition of γ0, (6) yields µ(I_ρ(t₀))

|I_ρ(t0)| ≥ds₀(γ_a⁰(t₀))−Λ ρ

Z

Iρ(t0)

|γ₍t)−γ(t0)−(t−t0)γ_a⁰(t0)|

|t−t0| dt−2Λη.

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

We establish now the second estimateµs≥ds^∞₀ γ_s⁰

|γ_s⁰|(·)

|γ_s⁰|. Lemma 2. For γ_s⁰-almost all t₀ in (0,1), we have

ρ→0lim

µ(Iρ(t0))

|γ_s⁰|(I_ρ(t₀)) ≥ds^∞₀ γ⁰

|γ⁰|(·) .

Proof. We rst introduce some notation and state a few technical results established in [5]. Dene in BV(I,R^N) and W^1,1(I,R^N), respectively, the rescaled functions











γρ= 1

|γ⁰|(I_ρ(t₀))|

γ(t0+ρt)− 1

|I_ρ(t₀)|

Z

Iρ(t0)

γ(s) ds

,

γ_ε,ρ= 1

|γ⁰|(I_ρ(t₀))|

γ_ε(t₀+ρt)− 1

|I_ρ(t₀)|

Z

Iρ(t0)

γ_ε(s) ds ,

whereI is the interval(−1/2,1/2). In the sequel, we xt0 in the Borel subset E of(0,1)satisfying |γ_s⁰| (0,1)\E

= 0, for which the limit

ρ→0lim

γ⁰(I_ρ(t₀))

|γ⁰|(I_ρ(t₀)) = γ⁰

|γ⁰|(t₀) exists and lim

ρ→0 ρ

|γ⁰|(Iρ(t0)) = 0. Set tρ = ^|γ0|(I_ρ^ρ(t0)), which thus tends to +∞. From now on, ρ denotes a sequence of positive rational numbers. Note that γε,ρ tends toγρstrongly inL¹((0,1),R^N)asρ→0 and that, according to the Poincaré-Wirtinger inequality,

Z

I

|γ_ρ|dt≤C Z

I

|γ⁰_ρ|.

SinceR

I|γ_ρ⁰|dt≤1, there exists a function˜γ inBV(I,R^N)and a subsequence inρthat we shall not relabel such that γρtends toγ˜ weakly inBV(I,R^N). A carefull analysis of this convergence leads to the following property ofγ˜: there exists a nondecreasing scalar function ψ in BV(I) such that ˜γ admits the representation ˜γ(t) =ψ(t) γ⁰/|γ⁰|(t0). Moreover, for every δ >0,δ ∈]0,1[\D, where Dis a countable subset of ]0,1[,γ_ρ⁰(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 estimate from below lim

ρ→0

µ(Iρ(t0))

|γ⁰|(Iρ(t0)) for the subsequence invoked above. We have

ρ→0lim

µ(I_ρ(t₀))

|γ⁰|(I_ρ(t0)) = lim

ρ→0lim

ε→0

µ_ε(I_ρ(t₀))

|γ⁰|(I_ρ(t0))

= lim

ρ→0lim

ε→0

1

|γ⁰|(I_ρ(t0)) Z

Iρ(t0)

˜ ω

γε(t) ε , γ_ε⁰(t)

dt.

In order to reason on the xed space BV(I), we change the scale and set v_ε,ρ= 1

ρ γ_ε(t₀+tρ)−r_ρ ,

whererρapproximates inQ^N the mean valuemρofγρinIρ(t0). More precisely,

|m_ρ−r_ρ| ≤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 thatv_ε,ρ/t_ρapproximates the functionγ_ε,ρ previously dened. More precisely,

we have

v_ε,ρ t_ρ −γ_ε,ρ

≤ |m_ε,ρ−m_ρ|

|γ⁰|(I_ρ(t₀)) + η

|γ⁰|(I_ρ(t₀)) and we obtain

(7) lim

ρ→0

µ(I_ρ(t₀))

|γ⁰|(I_ρ(t₀)) = lim

ρ→0lim

ε→0

1 t_ρ

Z

I

τ^rρ

ε ω˜ρv_ε,ρ ε , v_ε,ρ⁰

dt.

In order to apply the ergodic Theorem 3 witha=t_ρ¹_δ˜γ⁰(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 α_ρ/n, where α_ρ =

|γ_ρ−˜γ|^1/2

L¹(I,R^N) and n∈Nis intended to go to +∞: I₀ = (1−α_ρ)I, . . . , I_i =

1−α_ρ+iαρ

n

I, i= 1, . . . , n, and consider the cut-o functions

ϕ_i ∈ C₀^∞(I_i), 0≤ϕ≤1, ϕ= 1 inIi−1, |ϕ⁰_i|_L^∞_(Ii)≤ n αρ

, i= 1, . . . , n.

We nally dene

vε,ρ,i=tργ0+ϕi(vε,ρ−tργ0),

whereγ0 is the ane function γ₀(t) = t_ρ

δγ˜⁰(δI) +ψ(δ/2⁻) +ψ(δ/2⁺) 2

γ⁰

|γ⁰|(t₀)

with gradient ¹_δγ˜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) 1 tρ

Z

I

τ^rρ

ε ω˜ v_ε,ρ

ε/ρ, v_ε,ρ⁰

dt≥ 1 tρ

Z

I

τ^rρ

ε ω˜ v_ε,ρ,i

ε/ρ , v_ε,ρ,i⁰

dt−R_i(ρ, ε, n, η), whose proof is an easy consequence of the growth condition (see [1]), where

Ri(ρ, ε, n, η) =O(ρ) + Λ t_ρ

Z

Ii\Ii−1

|v⁰_ε,ρ−tργ₀⁰|dt+ nΛ t_ρα_ρ

Z

Ii\Ii−1

|v_ε,ρ−tργ0|dt and O(ρ) does not depend onε, n, η and tends to0 asρ→0.

As v_ε,ρ,i are admissible functions in the denition of S˜, for i= 1, . . . , n we have

1 t_ρ

Z

I

τ^rρ

ε ω˜ vε,ρ,i

ε/ρ , v⁰_ε,ρ,i

dt≥ 1 t_ρ

S˜ 1 ε/ρI

τ^rρ

ε ω, t˜ ρ1 δ˜γ⁰(δI)

|_ε/ρ¹ I| . Averaging these ninequalities, from (8) we obtain

1 tρ

Z

I

τ^rρ

ε ω˜ v_e,ρ

ε/ρ, v_ε,ρ⁰

dt≥ 1 tρ

S˜1 ε/ρI

τ^rρ

ε ω,˜ ^t_δ^ργ˜⁰(δI)

|_ε/ρ¹ I| −O(ρ)−O(1/n)

−Λ αρ

Z

I\(1−αρ)I

v_ε,ρ tρ

−γ₀

dt≥ 1 tρ

S˜ 1 ε/ρI

τ^rρ

ε ω,˜ ^t_δ^ργ˜⁰(δI)

|_ε/ρ¹ I| −O(ρ)−O(1/n)

− Λη

|γ⁰|(I_ρ(t0))− Λ|m_ε,ρ−m_ρ|

|γ⁰|(I_ρ(t0)) − Λ αρ

Z

I\(1−αρ)I

|γ_ε,ρ−γ₀|dt,

where O(1/n) tends to 0 as n goes to +∞. We let successively n → +∞, ε→0,η→0. According to Theorem 3, there exists a setΩ˜⁰ of full probability such that

lim sup

ε→0

1 t_ρ

Z

I

τ^rρ

ε ω˜ v_e,ρ

ε/ρ, v_ε,ρ⁰

dt

≥ 1 tρ

ds₀ t_ρ

δγ˜⁰(δI)

− Λ αρ

Z

I\(1−αρ)I

|γ_ρ−γ₀|dt−O(ρ)

≥ 1 tρ

ds0

t_ρ δγ˜⁰(δI)

−Λ√

α_ρ− Λ αρ

Z

I\(1−αρ)I

|˜γ−γ0|dt−O(ρ)

for everyω˜ ∈Ω˜⁰. Note thatΩ˜⁰ a priori depends onrρandtρ/δγ˜⁰(δI). Actually, we consider the set of full probability Ω˜⁰ = TΩ˜_r,a, where (r, a) belongs to Q^N ×Q^N. Without any loss of generality, we suppose here that t_ρ/δ ˜γ⁰(δI) belongs to Q^N. If not, is sucient to reason as in the previous subsection.

Recalling (7), we nally obtain

ρ→0lim

µ(Iρ(t0))

|γ⁰|(I_ρ(t₀)) ≥lim sup

ρ→0

1 t_ρds0

tρ

δ ˜γ⁰(δI)

−lim sup

ρ→0

Λ α_ρ

Z

I\(1−αρ)I

|˜γ−γ0|dt.

By a straightforward computation, it is easily seen thatγ˜−γ₀ has zero trace on

∂δI. This is the reason for which we chose the constant ^{ψ(δ/2−)+ψ(δ/2}₂ ⁺⁾_|γ^γ00|(t₀) in the denition ofγ₀. Therefore, by the properties of trace inBV(I), the last term on the right hand side tends to 0. For the rst term we have

lim sup

ρ→0

1 t_ρds₀

tρ

δγ˜⁰(δI)

= lim sup

ρ→0

1 δ

δ t_ρds₀

tρ

δ ˜γ_ρ⁰(δI)

= lim sup

ρ→0

1 δ

δ t_ρds0

tρ

δ γ⁰

|γ⁰|(t0)

−lim inf

ρ→0 Λ

1−|γ⁰|(I_δρ(t0)

|γ⁰|(I_ρ(t₀)

≥ 1 δ ds^∞₀

γ⁰

|γ⁰|(t0)

−(1−δ).

We conclude after letting δ →1.

The proof of Proposition 2 is now complete.

5. CONVERGENCE IN PROBABILITY

In order to simplify notation, we writeF_εandF₀to denote the restrictions to BV((0,1),R^N)ofFε+Ia,bandF0+ ¯Ia,b, respectively. We want to establish the convergence in probability of all sequences (F_ε)_ε>0 of random functionals in a suitable metric space. We equip the spaceBV((0,1),R^N)with the strong topology of L¹((0,1),R^N) and denote by SC the space of real valued lower semicontinuous functions G dened on BV((0,1),R^N) that satisfy G(u) ≥ νR

Ω|Du|. It is well known (see for instance [9]) that SC is metrizable by the Gamma-convergence. We denote by d_Γ a metric induced by the latter in SC and by B_Γ the Borel eld of the metric space (SC, d_Γ). Recall that the law of a measurable mapG: ( ˜Ω,T˜)→(SC,B_Γ)is the image ofP˜ byGon the metric space (SC, d_Γ).

Theorem 5. For everyω˜ ∈Ω˜ letF¯ε(˜ω)denote the lower semicontinuous envelope of F_ε(˜ω). Then all sequences F¯_ε converge in law and in probability to F0 in the metric space(SC, d_Γ).

Proof. Clearly (see for instance [7]), for a subsequenceFε that Gamma- converges almost surely to F₀, the corresponding subsequence of F¯_ε Gamma- converges almost surely to the same limitF0 which belongs to SC. Therefore, any sequence of laws of random functions F¯ε weakly converges to the Dirac measureδ_F0. Then, by classical probabilistic arguments, any sequenceF¯_εcon- verges in probability to the constant F₀ inSC, i.e.,P{˜ ω˜ ∈Ω :˜ d_Γ( ¯F_ε(˜ω), F₀)>

η} →0asε→0 for all η >0.

Corollary 2. The random geodesic metric d_ε converges in probability to the deterministic metric d₀ in the following sense: for all η > 0 and all points (a, b) in R^N,

P{ω :|d_ε(ω)(a, b)−d₀(a, b)|> η} →0 asε→0.

Proof. It is equivalent to establish thatP{˜ ω˜ :|d_ε(˜ω)(a, b)−d0(a, b)|> η}

tends to 0asε→0, which is a straightforward consequence of P{˜ ω˜ ∈Ω :˜ dΓ( ¯Fε(˜ω), F0)> η} →0

and the continuity of the real valued mapG7→minGdened in(SC, d_Γ).

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