On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

DOI:10.1051/m2an/2013111 www.esaim-m2an.org

ON A VARIANT OF RANDOM HOMOGENIZATION THEORY:

CONVERGENCE OF THE RESIDUAL PROCESS AND APPROXIMATION OF THE HOMOGENIZED COEFFICIENTS

Fr´ ed´ eric Legoll

and Florian Thomines

Abstract. We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C.

Le Bris and P.-L. Lions,C. R. Acad. Sci. S´erie I343(2006) 717–724.; X. Blanc, C. Le Bris and P.-L.

Lions, J. Math. Pures Appl.88(2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions,C. R. Acad. Sci. S´erie I343(2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.

Mathematics Subject Classification. 35R60, 35B27, 60H, 60F05.

Revised June 19, 2013. Accepted September 9, 2013 Published online February 7, 2014.

1. Introduction

Homogenization theory for linear second-order elliptic equations with highly oscillatory coefficients is a well developed topic. In the periodic case, the homogenized problem is known, and convergence rates of the oscillatory solution (denotedu^ε) towards the homogenized solutionu have been obtained.

The situation is less clear in the random (say stationary ergodic) setting. The convergence ofu^ε(·, ω) to some deterministicu is a classical result. However, rates of convergence are much more difficult to obtain. A central difficulty in stochastic homogenization is that the corrector problem, which needs to be solved to next compute the homogenized matrix, is set on the entire space (in contrast with the periodic case, where it is set on the periodic cell). This induces many theoretical and practical difficulties.

Keywords and phrases.Stochastic homogenization, random stationary diffeomorphisms, central limit result, approximation of homogenized coefficients.

1 Laboratoire Navier, ´Ecole Nationale des Ponts et Chauss´ees, Universit´e Paris-Est, 6 et 8 Avenue Blaise Pascal, 77455 Marne- La-Vall´ee Cedex 2, France.[email protected]

2 INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

In what follows, we are interested in the problem

−div A

x ε, ω

∇u^ε(x, ω)

=f(x) inD, u^ε(·, ω) = 0 on∂D, (1.1) where the random matrix A satisfies standard coercivity and boundedness properties (and some structure assumptions that we detail below),Dis an open bounded set ofR^d andf ∈L²(D).

The analysis of the residual, which we define as the difference between the oscillatory solution u^ε and the homogenized solution u, was first taken up in [9], and next complemented in [3]. Both studies consider the equation −d

dx

a x

ε, ω du^ε

dx

= f(x) in the one-dimensional setting, where a(x, ω) is a random stationary process. The behavior, when ε → 0, of the residualu^ε(x, ω)−u(x) turns out to depend on the asymptotic behavior of the correlation function η(x) :=Cov(a(0,·), a(x,·)) of the conductivity coefficient. In [9], the case of small correlation lengths is studied, which amounts to assuming that η(x) ∼_x→∞ x^−α with α > 1. The correlation function is thus integrable. In that case, whenε→0, the random process u^ε(x, ω)−u(x)

√ε converges in distribution to a Gaussian random process. The case of long correlation lengths, namely whenη(x)∼_x→∞x^−α for some 0< α < 1, is studied in [3], where it is shown that the random process u^ε(x, ω)−u(x)

ε^α/2 converges in distribution to a Gaussian random process (defined using a fractional Brownian motion). This result shows that the rate of convergence ofu^ε toucan be as slow as ε^α/2for any α >0, if no further assumptions on the stationary processaare made.

Of course, in both works, the one-dimensional setting allows to get some analytical expression for the residual.

In turn, the analysis of the asymptotic behavior of the residual performed in [3,9] relies on this analytical expression. In higher dimensions, the case of the equation−Δu^ε+q

x ε, ω

u^ε=f(x) has been studied in [2].

Our first aim here is to study a similar question for a variant of the classical stochastic homogenization theory.

We consider in the sequel the following problem, which has been introduced in [7] and further studied in [8]:

−div A_per

φ⁻¹

x ε, ω

∇u^ε(x, ω)

=f(x) in D, u^ε(·, ω) = 0 on∂D, (1.2) where φ is almost surely a diffeomorphism from R^d to R^d with some stationary properties andA_per is a Z^d- periodic matrix, which satisfies the classical coercivity and boundedness properties (see precise assumptions in Sect.2.1below). This model is appropriate to represent a periodic, ideal material, that is randomly deformed (think of fibers in a composite material that are placed at a random position, rather than on a perfect, periodic lattice). In [7], it is shown that the solutionu^ε(·, ω) to the above problem converges asεgoes to 0 tou, solution to some homogenized problem (see Sect.2 below). In the sequel, we aim at obtaining the rate of convergence ofu^εtou, in the one dimensional setting. We make below an assumption on the random diffeomorphism which implies that our setting is close to the one studied in [9] (rather than that studied in [3]). Under this assumption, we show that the random process u^ε(x, ω)−u(x)

√ε converges in distribution to a Gaussian random process that we completely characterize (see Sect.3.1, Thm.3.2).

We next turn to a question of a different nature. As pointed out above, the homogenized matrixAassociated to (1.2) depends on the solution of the corrector problem (see (2.9) below), which is set on the entire space.

Computing an approximation of A is thus, in practice, a challenging question. A standard strategy is to consider the corrector problem on a bounded (large) domainQ_N, supplemented with (say periodic) boundary conditions. An approximation of theexactcorrector is thus computed, from which an approximate homogenized matrix A_N(ω) is inferred. As a by-product of working on a bounded domain, the approximate homogenized matrix is random. In the classical random homogenization setting (that is (1.1) whereAis a stationary matrix), the convergence (and its rate) of A_N(ω) to A has been studied in [10], using some previous approximation

results [20]. It is shown there thatA_N(ω) almost surely converges toA, and thatE

|A_N−A|²

converges to 0 as N^−α, for some α >0 which implicitly depends on the mixing properties of the random coefficientA of the equation (1.1). It is expected that, depending on the properties of that random coefficient,αcan be arbitrary small.

In this work, we consider the above variant (1.2) of the classical random homogenization setting. We describe a strategy (originally introduced in [12]) to approximateAwhich is based, as in the classical setting, on solving the corrector problems on bounded domainsQ_N. We prove here the convergence of this approach (see Sect.3.2, Thm.3.4).

Our article is articulated as follows. In Section 2, we present in details the variant of the classical random homogenization introduced in [7,8]. We next present in Section 3 our two main results, first on the residual process in dimension one (see Sect.3.1and Thm.3.2), second on a practical approximation of the homogenized matrix in dimensiond≥2 (see Sect.3.2and Thm.3.4). The subsequent two sections are devoted to the proof of Theorem3.2. The actual proof is performed in Section4, and needs some technical results which are proved in Section5. Our final section, Section 6, collects the proof of Theorem3.4.

2. A variant of the classical random homogenization

To begin with, we introduce the basic setting of stochastic homogenization we employ. We refer to [13] for a general, numerically oriented presentation, and to [5,11,15] for classical textbooks. We also refer to [7,8] for a presentation of our particular setting (see also [1]). Throughout this article, (Ω,F,P) is a probability space and we denote byE(X) =

Ω

X(ω)dP(ω) the expectation value of any random variableX∈L¹(Ω, dP). For any fixed d∈N (the ambient physical dimension), we assume that the group (Z^d,+) acts onΩ. We denote by (τ_k)_k∈Zd

this action, and assume that it preserves the measureP, that is, for allk∈Z^d and allB ∈ F,P(τkB) =P(B).

We assume that the actionτ isergodic, that is, ifB ∈ F is such that τ_kB=B for any k∈Z^d, thenP(B) = 0 or 1. In addition, we define the following notion of (discrete) stationarity (see [7,8]): anyF ∈L¹_loc R^d, L¹(Ω) is said to bestationaryif

∀k∈Z^d, F(x+k, ω) =F(x, τ_kω) almost everywhere and almost surely. (2.1) In this setting, the ergodic theorem [16,18,19] can be stated as follows:LetF ∈L^∞ R^d, L¹(Ω)

be a stationary random variable in the above sense. Fork= (k₁, k₂, . . . , k_d)∈Z^d, we set|k|∞= sup

1≤i≤d|ki|. Then 1

(2N+ 1)^d

|k|∞≤N

F(x, τ_kω) −→

N→∞E(F(x,·)) inL^∞(R^d), almost surely.

This implies (denoting byQ= (0,1)^d the unit cube in R^d) that F

x ε, ω

−∗

ε→0E

Q

F(x,·)dx

inL^∞(R^d), almost surely.

2.1. Mathematical setting and homogenization result

As pointed out in the introduction, we consider in this article the following problem, which has been introduced in [7] and further studied in [8]:

−div A_per

φ⁻¹

x ε, ω

∇u^ε(x, ω)

=f(x) in D, u^ε(·, ω) = 0 on∂D, (2.2)

whereDis a bounded open set ofR^d,f ∈L²(D),φis almost surely a diffeomorphism fromR^dtoR^d, andA_peris aZ^d-periodic matrix, that satisfies the classical coercivity and boundedness properties: there existsa⁺≥a₋ >0 such that

∀ξ∈R^d, a₋|ξ|²≤A_per(x)ξ·ξ almost everywhere onR^d, and a⁺=A_perL^∞_(R^d₎<∞. (2.3) In addition, we assume that the mapφ(·, ω) satisfies

EssInf

ω∈Ω, x∈R^d(det(∇φ(x, ω))) =ν >0, (2.4) EssSup

ω∈Ω, x∈R^d|∇φ(x, ω)|=M <+∞, (2.5)

∇φis stationary in the sense of (2.1). (2.6)

Assumptions (2.4) and (2.5) mean thatφis a well-behaved diffeomorphism, uniformly inω. Note thatA_per◦φ⁻¹ is in general not stationary. The above setting is thus not a particular case of the classical stationary setting.

In [7], it is shown that, under the above conditions,u^ε(·, ω) converges toualmost surely (strongly inL²(D) and weakly inH¹(D)) whenεgoes to 0, whereu is the solution to the homogenized problem

−div [A∇u(x)] =f(x) in D, u= 0 on∂D. (2.7) In (2.7), the homogenized matrix coefficientA is equal to

∀1≤i, j ≤d, A_ij= det

E

Q

∇φ(y,·)dy

₋₁

E

φ(Q,·)e^T_iA_per φ⁻¹(y,·)

e_j+∇wej(y,·) dy

, (2.8) whereQ= (0,1)^d and where, for allp∈R^d,w_p solves the following corrector problem:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

−div

A_per φ⁻¹(y, ω)

(p+∇wp(y, ω))

= 0 inR^d,

w_p(y, ω) =w_p(φ⁻¹(y, ω), ω), ∇w_p is stationary in the sense of (2.1), E

φ(Q,·)∇wp(y,·)dy

= 0.

(2.9)

2.2. The one-dimensional case

Our first main result, presented in Section3.1, is a convergence result in the one-dimensional case. In that set- ting, it is possible to write some explicit formulas. ChoosingD= (0,1), the problems (2.2) and (2.7) respectively read

−d dx

a_per

φ⁻¹

x ε, ω

du^ε dx(x, ω)

=f(x) in (0,1), u^ε(0, ω) = 0, u^ε(1, ω) = 0, (2.10) and

−d dx

adu

dx(x)

=f(x) in (0,1), u(0) = 0, u(1) = 0. (2.11) The corrector problem (2.9), which reads

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

−d dy

a_per φ⁻¹(y, ω) 1 +dw dy(y, ω)

= 0 inR, w(y, ω) =w(φ ⁻¹(y, ω), ω), dw

dy is stationary in the sense of (2.1), E

φ(Q,·)

dw dy(y,·)dy

= 0,

(2.12)

can be analytically solved. Its solutionwsatisfies 1 + dw

dy(y, ω) = a

a_per(φ⁻¹(y, ω)), (2.13)

where the homogenized coefficienta is given by (a)⁻¹= 1

E₁

0 φ(y,·)dyE

₁

0

φ(y,·) a_per(y)dy

. (2.14)

As pointed out in [7], we observe on (2.13) that, in the one-dimensional case, the gradient of the corrector has the same structure as the highly oscillatory coefficient in (2.10): it is equal to a periodic function composed withφ⁻¹. This is not the case in dimensionsd≥2, as shown in [7].

3. Main results

In this article, we show the following two main results, Theorems3.2and3.4.

3.1. Residual process in dimension one

Our first aim is to characterize how the residual process u^ε(x, ω)−u(x) converges to zero, where u^ε solves (2.10) and u solves (2.11). To this aim, we make the following assumptions. Let us introduce the 1-periodic function

ψ(x) = 1

a_per(x)− 1

a (3.1)

and the random variables

Y_k(ω) = _k+1

k

ψ(t)φ(t, ω)dt. (3.2)

Asψis periodic andφ is stationary, the random variablesY_kare identically distributed. Due to (2.14), we have E(Y₀) =E

₁

0 ψ(t)φ(t,·)dt

= 0.

We furthermore assume that the random variablesY_k are independent, and hence that

the variablesY_k are i.i.d. (3.3)

Likewise, we consider the random variables

D_k(ω) = _k+1

k

φ(t, ω)dt, (3.4)

which are identically distributed, and make the assumption that

the variablesD_k are i.i.d. (3.5)

Remark 3.1. Suppose that the derivative of the random diffeomorphismφreads φ(y, ω) = 1 +

k∈Z

X_k(ω)G_per(y)1_[k,k+1)(y),

where X_k(ω) are independent and identically distributed random variables and G_per is a 1-periodic bounded function, such that, for some 0< m <1,

|X₀(ω)| ≤malmost surely and G_perL^∞(R)≤m.

Then, the conditions (2.4), (2.5) and (2.6) are satisfied withν = 1−m²>0 andM = 1 +m². By construction, the assumptions (3.3) and (3.5) are also fullfilled.

The first main result of this article is the following theorem, the proof of which is postponed until Section4.2.

Theorem 3.2. Assume that a_per andφsatisfy (2.3),(2.4),(2.5)and (2.6). Assume furthermore the indepen- dence conditions (3.3)and (3.5). We consideru^εsolution to(2.10)andu solution to (2.11). Then the residual process converges in distribution to a Gaussian process,

u^ε(x, ω)−u(x)

√ε

−→L

ε→0G₀(x, ω), where

G₀(x, ω) =

Var(Y₀)

E₁

0 φ

₁

0 K₀(x, t) dW_t, (3.6)

whereW_t denotes the classical Brownian motion and K₀(x, t)is given by K₀(x, t) = 1_[0,x](t)−x

1

0 F(s)ds−F(t)

with F(t) = _t

0 f(s) ds. (3.7)

Remark 3.3. It might be possible to weaken assumptions (3.3) and (3.5), and to only assume that the iden- tically distributed variablesY_k are such that

k∈Z

|Cov(Y₀, Y_k)|<+∞, and likewise forD_k. We however do not pursue in that direction.

3.2. Approximation of the homogenized matrix

In this section, we return to the multidimensional setting. To compute the homogenized matrixA defined by (2.8), we first need to solve the corrector problem (2.9), which is set on the entire space. In practice, approximations are therefore in order.

In the sequel, we describe a strategy introduced in [12], and that mimicks the approach proposed in [10] to approximate standard corrector problems in classical random homogenization. In this article, we analyze this approach and prove its convergence (see Thm.3.4below). This is our second main result. We refer to Section 3.2 of [1] for some illustrative numerical tests.

Convention: Following Lemme 2.1 of [7], we adopt the convention that [∇φ]_ij = ∂φ_i

∂x_j for any 1 ≤i, j ≤d.

Hence, for any scalar-valued function ψ, the gradient of ψ =ψ◦φ is given by∇ψ(z) = (∇φ(z)) ^T∇ψ(φ(z)).

This convention implies that

∇φ(φ⁻¹)

∇(φ⁻¹) = Id.

3.2.1. Presentation of the approximation

The weak formulation of the corrector problem (2.9) reads as follows (see [7]): for all ψ stationary in the sense of (2.1), we have

E

φ(Q,·)(∇ψ(y, ω))^TA_per φ⁻¹(y, ω)

(p+∇w_p(y, ω)) dy

= 0, (3.8)

whereψ=ψ◦φ⁻¹. The above expression can be rewritten, after a change of variables, as E

Q

det(∇φ)

∇ψ _T

(∇φ)⁻¹A_per p+ (∇φ)^−T∇w_p

= 0,

where (∇φ)^−T is defined as

(∇φ)⁻¹_T

. Sinceψ, ∇φ, A_per and∇w_p are stationary in the sense of (2.1), the ergodic theorem yields

Nlim→∞

1

|QN|

QN

det(∇φ)

∇ψ T

(∇φ)⁻¹A_per p+ (∇φ)^−T∇w_p

= 0 a.s.

where Q_N =N Q. For a fixed N, we now define the approximate corrector w^N_p as the Q_N-periodic function satisfying:

for allψ Q _N-periodic,

QN

det(∇φ)

∇ψ _T

(∇φ)⁻¹A_per p+ (∇φ)^−T∇w_p^N

= 0. (3.9)

Note thatw_p^N is uniquely defined up to an additive constant.

In turn, recall thatA is defined by (2.8). After a change of variables, we infer from that equation that, for any 1≤i, j≤d, we have

A_ij= det

E

Q

∇φ(y,·)dy

₋₁

E

Q

det(∇φ(y,·))e^T_i A_per(y) e_j+ (∇φ)^−T∇w_ej(y,·) dy

.

The ergodic theorem yields A_ij = lim

N→∞

det

1

|QN|

QN

∇φ(·, ω) ₋₁

1

|QN|

QN

det(∇φ)e^T_iA_per e_j+ (∇φ)^−T∇w_ej a.s.

It is thus natural to approximateA by the matrixA_N(ω) defined by A_N(ω) = det

1

|QN|

QN

∇φ(·, ω) ₋₁

B_N(ω), (3.10)

where the matrixB_N(ω) is defined by, for any 1≤i, j≤d, [B_N(ω)]_ij = 1

|QN|

QN

det(∇φ)e^T_i A_per

e_j+ (∇φ)^−T∇w^N_ej

= 1

|QN|

φ(QN,ω)e^T_i A_per φ⁻¹(y, ω) e_j+∇w^N_ej(y, ω)

dy, (3.11)

where, for anyp∈R^d,w^N_p is defined by (3.9) and

w_p^N(y, ω) =w^N_p(φ⁻¹(y, ω), ω).

Note that, as is standard in stochastic homogenization, the approximation A_N(ω) is a random matrix, even though the exact homogenized matrix A is deterministic. This is a by-product of working on the truncated domainQ_N rather thanR^d.

3.2.2. Convergence of the approach

We prove in Section6below the following convergence result:

Theorem 3.4. Let φbe a diffeomorphism satisfying (2.4),(2.5)and (2.6), andA_per be a periodic matrix that satisfies the ellipticity condition (2.3). Then the random matrixA_N(ω)defined by(3.10)converges almost surely to the deterministic homogenized matrix A defined by (2.8)whenN → ∞:

N→∞lim A_N(ω) =A almost surely.

4. Asymptotic behavior of the residual

The aim of this Section and of the next one is to prove our first main result, Theorem 3.2 of Section 3.1.

Using the one dimensional setting, we first establish a “representation” formula for the residual (see Sect.4.1, Thm.4.2). Using this formula, we are next in position to study the asymptotic behavior of the residual when ε→0 (see Sect.4.2). Section5collects the proofs of some technical results used in Sections4.1and4.2.

4.1. Representation formulas

The following technical result will be useful in the sequel. Its proof is postponed until Section5.1.

Lemma 4.1. Assume thata_perandφsatisfy(2.3),(2.4),(2.5)and(2.6). Assume furthermore the independence conditions (3.3)and(3.5). For any0≤α≤β≤1, and anyA ∈L^∞(α, β)withA∈L²(α, β), define the random variable

Z_ε(α, β, ω) = 1

√ε _β

α

A(t)ψ

φ⁻¹ t

ε, ω

dt, (4.1)

where the functionψ is defined by (3.1). For any p∈N, there exists a deterministic constantC_p independent of A,ε,αandβ, such that

∀ε >0, E

Z_ε(α, β,·)^2p

≤C_p

(β−α)^p+ε^(p−1)/2 A^2p_L∞(α,β)+ (β−α)^pA^2p_L2(α,β)

.

The above result heuristically implies that the quantity _β

α

A(t)ψ

φ⁻¹ t

ε, ω

dt is of the order of√ ε.

We will show below a convergence result for the random variables Z_ε(α, β, ω) (see Lem. 4.6 below). The boundedness result stated in the above lemma is however sufficient for now. Using it, we indeed prove the following theorem, which is a key ingredient to prove Theorem3.2.

Theorem 4.2. Assume that a_per and φ satisfy (2.3), (2.4), (2.5) and (2.6). Assume furthermore the inde- pendence conditions (3.3) and (3.5). Let u^ε be the solution to (2.10) and u be the solution to (2.11). Then

u^ε(x, ω)−u(x) = ₁

0

K₀(x, t)ψ

φ⁻¹ t

ε, ω

dt+r_ε(x, ω), (4.2)

where K₀ is defined by (3.7),ψis defined by (3.1), and there exists a deterministic constant C independent of εsuch that, for any ε >0,

sup

x∈[0,1]E|rε(x,·)| ≤Cε and E

rε²_L2(0,1)

≤Cε². (4.3)

In addition, for anyp∈N, there exists a deterministic constantC_p independent ofεsuch that

∀ε∈(0,1), ∀(x, y)∈(0,1)², E

|r_ε(x,·)−r_ε(y,·)|^2p

≤C_pε^2p

(x−y)^2p+ε^p−1/2 (4.4) and

∀ε∈(0,1), ∀(x, y)∈(0,1)², E

|rε(x,·)−r_ε(y,·)|^2p

≤C_pε^p(x−y)^2p. (4.5) In view of Lemma4.1, the first term of the right-hand side of (4.2) is of the order of√

ε. The termr_ε, which is of the order ofεin view of (4.3), is hence a higher-order term. The bounds (4.4) and (4.5) will be useful below to show that some random process is tight (see Sect.4.2, Thm.4.5).

Using the same arguments, we show the following result:

Theorem 4.3. Assume that a_per andφsatisfy (2.3),(2.4),(2.5)and (2.6). Assume furthermore the indepen- dence conditions (3.3) and (3.5). Let u^ε be the solution to (2.10),u be the solution to (2.11), and w be the corrector, solution to (2.12)withw(0, ω) = 0a.s. Then

d dx

u^ε(x, ω)−u(x)−εw x

ε, ω du

dx(x)

=a⁻¹_per

φ⁻¹ x

ε, ω

1 0

K₁(t)ψ

φ⁻¹ t

ε, ω

dt +f(x)

_x

0 ψ

φ⁻¹ t

ε, ω

dt+r_ε(x, ω), (4.6) whereψ is defined by (3.1),K₁ is given by

K₁(t) =a

F(t)− ₁

0 F(s)ds

with F(t) = _t

0 f(s) ds, (4.7)

and there exists a deterministic constantC independent of εsuch that, for allε >0, E

x∈[0,1]sup |rε(x,·)|

≤Cε and E

x∈[0,1]sup |rε(x,·)|²

≤Cε². (4.8)

Again, in view of Lemma 4.1, the two first terms of the right-hand side of (4.6) are of the order of √ ε. The termr_ε, which is of the order ofε, is hence a higher-order term.

Remark 4.4. It is easy to deduce from (4.6), using Lemma 4.1 and (4.8), that there exists a deterministic constantC independent ofεsuch that

E d

dx

u^ε(x, ω)−u(x)−εw x

ε, ω du

dx(x) ²

L²(0,1)

≤Cε. (4.9)

Likewise, we deduce from (4.2), using Lemma 4.1and (4.3), that E

u^ε(x, ω)−u(x)²_L2(0,1)

≤Cε. (4.10)

Using the expression (4.28) below, we infer from (4.9) and (4.10) that E

u^ε(x, ω)−u(x)−εw x

ε, ω du

dx(x) ²

H¹(0,1)

≤Cε. (4.11)

We recover (in the one-dimensional situation) a classical result of homogenization: the corrector w allows to obtain a convergence result in theH¹ strong norm. We refer to Theorem 3 of [17] for a corresponding result in classical random homogenization (in the multidimensional setting).

The proof of Theorems 4.2 and 4.3 are direct consequences of Lemma 4.1 and of the analytical expression ofu^εandu.

Proof of Theorem 4.2. IntroduceF(x) = _x

0 f(t)dt. The solution to (2.10) reads u^ε(x, ω) =c^ε(ω)

_x

0

1

a_per φ⁻¹ _ε^t, ωdt− _x

0

F(t)

a_per φ⁻¹ _ε^t, ωdt, (4.12)

where

c^ε(ω) = ₁

0

F(t)

a_per φ⁻¹ _ε^t, ωdt ₁

0

1

a_per φ⁻¹ _ε^t, ωdt

· (4.13)

Likewise, the solutionu of the homogenized problem (2.7) is u(x) =c x

a − _x

0

F(t)

a dt, (4.14)

wherea is given by (2.14) and

c= ₁

0 F(t)dt. (4.15)

Step 1.Representation formula

We compute the residual process using (4.14) and (4.12):

u^ε(x, ω)−u(x) =c^ε(ω) _x

0

1

a_per φ⁻¹ _ε^t, ωdt−c x a −

_x

0 F(t)ψ

φ⁻¹ t

ε, ω

dt

=c^ε(ω) _x

0 ψ

φ⁻¹ t

ε, ω

dt+ (c^ε(ω)−c)x a −

_x

0 F(t)ψ

φ⁻¹ t

ε, ω

dt

= (c^ε(ω)−c) _x

0 ψ

φ⁻¹ t

ε, ω

dt+ (c^ε(ω)−c)x a

+ _x

0 (c−F(t))ψ

φ⁻¹ t

ε, ω

dt, (4.16)

whereψ is defined by (3.1). We also infer from (4.13) that c^ε(ω)−c =

₁

0

1

a_per φ⁻¹ _ε^t, ωdt

_{−1 1}

0 (F(t)−c) 1

a_per φ⁻¹ _ε^t, ωdt

=

₁

0

1

a_per φ⁻¹ _ε^t, ωdt

_{−1 1}

0 (F(t)−c)ψ

φ⁻¹ t

ε, ω

dt (4.17)

where we have used that, in view of (4.15), we have ₁

0 (F(t)−c) 1

adt= 0. Observe now that

₁

0

1

a_per φ^{−1 t}_ε, ωdt ₋₁

=a− a

₁

0

a⁻¹_per

φ⁻¹ t

ε, ω

dt ₁

0 ψ

φ⁻¹ t

ε, ω

dt.

We then deduce from (4.17) that

c^ε(ω)−c=a ₁

0

(F(t)−c)ψ

φ⁻¹ t

ε, ω

dt−ρ^ε(ω), (4.18)

where

ρ^ε(ω) =

⎡

⎢⎢

⎢⎣

a ₁

0 a⁻¹_per

φ⁻¹ t

ε, ω

dt ₁

0 ψ

φ⁻¹ t

ε, ω

dt

⎤

⎥⎥

⎥⎦ ₁

0 (F(t)−c)ψ

φ⁻¹ t

ε, ω

dt. (4.19)

Collecting (4.16) and (4.18), we write u^ε(x, ω)−u(x) = (c^ε(ω)−c)

_x

0 ψ

φ⁻¹ t

ε, ω

dt+ _x

0 (c−F(t))ψ

φ⁻¹ t

ε, ω

dt +x

₁

0 (F(t)−c)ψ

φ⁻¹ t

ε, ω

dt− x aρ^ε(ω)

=r_ε(x, ω) + _x

0 (c−F(t))ψ

φ⁻¹ t

ε, ω

dt+ x ₁

0 (F(t)−c)ψ

φ⁻¹ t

ε, ω

dt

=r_ε(x, ω) + ₁

0 K₀(x, t)ψ

φ⁻¹ t

ε, ω

dt with

K₀(x, t) = 1_[0,x](t)−x

(c−F(t)) and

r_ε(x, ω) =−x

aρ^ε(ω) + (c^ε(ω)−c) _x

0 ψ

φ⁻¹ t

ε, ω

dt. (4.20)

In view of (4.15), we recover the expression (3.7) ofK₀. We thus have written the residual in the form (4.2).

Step 2.Proof of the bound (4.3).

We first boundρ^ε(ω). We infer from (4.19) that

|ρ^ε(ω)| ≤a⁺a&&

&& ₁

0 ψ

φ⁻¹ t

ε, ω

dt&&

&& &&

&& ₁

0 (F(t)−c)ψ

φ⁻¹ t

ε, ω

dt&&

&&. (4.21) Using the Cauchy–Schwartz inequality, we deduce that

E(|ρ^ε|)≤εa⁺a '( ()E

&

&&

& 1

√ε ₁

0 ψ

φ⁻¹ t

ε,·

dt&&

&&² '(()E

&

&&

& 1

√ε ₁

0 (F(t)−c)ψ

φ⁻¹ t

ε,·

dt&&

&&²

.

Using Lemma 4.1 with p = 1, α = 0, β = 1, A(t) = 1 and A(t) =F(t)−c, we obtain that there exists a constantC independent ofεsuch that

E(|ρ^ε|)≤Cε. (4.22)

We also deduce from (4.21) that, for anyp∈N, E

|ρ^ε|^2p

≤(a⁺a)^2pε^2p '( ()E

&

&&

& 1

√ε ₁

0 ψ

φ⁻¹ t

ε,·

dt&&

&&^4p '(()E

&

&&

& 1

√ε ₁

0 (F(t)−c)ψ

φ⁻¹ t

ε,·

dt&&

&&^4p

.

Using again Lemma4.1, we obtain that there exists a constantC_p independent ofεsuch that E |ρ^ε|^2p

≤C_pε^2p. (4.23)

Using the obtained bounds onρ^ε, we now estimater_ε. We infer from (4.18), using (4.23) and Lemma4.1, that, for anyp∈N,

E

|c^ε−c|^2p

≤(a)^2pε^pC_pE

&

&&

&√1 ε

₁

0 (F(t)−c)ψ

φ⁻¹ t

ε,·

dt&&

&&^2p

+C_pE

|ρ^ε|^2p

≤(a)^2pC_pε^p+C_pε^2p

≤C_pε^p (4.24)

for a constantC_p independent ofε. In view of (4.20), we thus obtain, using (4.22) and (4.24), that E(|r_ε(x,·)|)≤ x

aE(|ρ^ε|) +E&&

&&(c^ε−c) _x

0 ψ

φ⁻¹ t

ε,·

dt&&

&&

≤Cε+√ ε

E

|(c^ε−c)|²'(()E

&

&&

& 1

√ε _x

0 ψ

φ⁻¹ t

ε,·

dt&&

&&²

≤Cε

for a constantC independent fromεandx∈(0,1). This concludes the proof of the first assertion in (4.3).

Similarly, we have

(r_ε(x, ω))²≤ 2

(a)²ρ^ε(ω)²+ 2(c^ε(ω)−c)²&&

&& _x

0

ψ

φ⁻¹ t

ε, ω

dt&&

&&², thus

E

rε²_L2(0,1)

≤ 2

(a)²E

|ρ^ε|² + 2

₁

0 E

(c^ε−c)²&&

&& _x

0 ψ

φ⁻¹ t

ε,·

dt&&

&&²

dx

≤Cε²+ 2 ₁

0

'(

()E[(c^ε−c)⁴]E

&

&&

&

_x

0 ψ

φ⁻¹ t

ε,·

dt&&

&&⁴

dx.

Using (4.24) and Lemma4.1withp= 2, we deduce that E

r_ε²_L2(0,1)

≤Cε²+ 2 ₁

0

√Cε⁴dx≤Cε²

for a constantC independent fromε. This concludes the proof of the second assertion in (4.3).

Step 3.Proof of the bounds (4.4) and (4.5).

We first prove (4.4). In view of (4.20), we have r_ε(x, ω)−r_ε(y, ω) = y−x

a ρ^ε(ω) + (c^ε(ω)−c) _x

y

ψ

φ⁻¹ t

ε, ω

dt, thus

|r_ε(x, ω)−r_ε(y, ω)|^2p≤C_p

y−x a

_2p

(ρ^ε(ω))^2p+C_p(c^ε(ω)−c)^2p&&

&& _x

y

ψ

φ⁻¹ t

ε, ω

dt&&

&&^2p, (4.25) and

E

|r_ε(x,·)−r_ε(y,·)|^2p

≤C_p

y−x a

_2p E

(ρ^ε)^2p

+C_p

E[(c^ε−c)^4p] '( ()E

&

&&

&

_x

y

ψ

φ⁻¹ t

ε,·

dt&&

&&^4p

≤C_pε^2p(y−x)^2p+C_p√ ε^2p

ε^2p (x−y)^2p+ε^(2p−1)/2

≤C_pε^2p

(y−x)^2p+

(x−y)^2p+ε^p−1/2

.

Since|y−x| ≤1, we have (y−x)^2p≤ |y−x|^p≤

(x−y)^2p+ε^p−1/2, and thus E

|r_ε(x,·)−r_ε(y,·)|^2p

≤C_pε^2p

(x−y)^2p+ε^p−1/2. This concludes the proof of (4.4). We now prove (4.5). We infer from (4.25) that

|rε(x, ω)−r_ε(y, ω)|^2p≤C_p

y−x a

_2p

(ρ^ε(ω))^2p+C_p(c^ε(ω)−c)^2p (x−y)^2p ψ^2p_L∞(R), hence, using (4.23) and (4.24),

E

|rε(x,·)−r_ε(y,·)|^2p

≤C_p(x−y)^2p

ε^2p+ε^p .

This concludes the proof of (4.5) and thus that of Theorem4.2.

Proof of Theorem 4.3. Recall that the solution to the corrector problem (2.12) satisfies (2.13). We thus have, using (4.12) and (4.14),

d dx

u^ε(x, ω)−u(x)−εw x

ε, ω du

dx(x)

= du^ε

dx(x, ω)−du dx(x)

1 +w

x ε, ω

−εd²u dx² (x)w

x ε, ω

= c^ε(ω)−c

a_per φ^{−1 x}_ε, ω−εd²u dx²(x)w

x ε, ω

.

Using (4.18), we deduce that d

dx

u^ε(x, ω)−u(x)−εw x

ε, ω du

dx(x)

= a

a_per φ^{−1 x}_ε, ω ₁

0 (F(t)−c)ψ

φ⁻¹ t

ε, ω

dt−εd²u dx² (x)w

x ε, ω

− ρ^ε(ω)

a_per φ^{−1 x}_ε, ω

=a⁻¹_per

φ⁻¹ x

ε, ω

1 0 K₁(t)ψ

φ⁻¹

t ε, ω

dt+εf(x) a w

x ε, ω

+r^ε(x, ω), (4.26)

withK₁ defined by (4.7) and

r^ε(x, ω) =− ρ^ε(ω)

a_per φ^{−1 x}_ε, ω· (4.27)

Observe now that, in view of (2.13) and (3.1), we have w(y, ω) =a

_y

0 ψ φ⁻¹(t, ω) dt,

where we have chosen the integration constant inwsuch thatw(0, ω) = 0 almost surely. Thus w

x ε, ω

=a _x/ε

0 ψ φ⁻¹(t, ω)

dt=a ε

_x

0 ψ

φ⁻¹ t

ε, ω

dt. (4.28)

Collecting this equation with (4.26) yields (4.6). The bound (4.8) follows from (4.27), (4.22) and (4.23). This

concludes the proof of Theorem4.3.