The map Ξ :

24 We also implicitly assume here that the law is isotropic since c is radially symmetric.

25 A rigorous analysis of the error term can be conducted by looking at the second order derivative in δ, but for the ease of the reading we only focus on the first term.

. (1.1.67) We now bound the right-hand side of (1.1.67) as follows. We recall that we have shown in Section 1.1.4 that

) 2 ), therefore by using in addition the bound 1 2 + ˆηR ( y c )|u(T, y)| 2 dy

Quand on débute une thèse, on ne sait pas trop par où commencer ou comment s'y prendre, l'exercice est tellement différent de ce qu'on a pu faire pendant le cycle licence-master. Mais quand on est bien entouré et guidé, on peut finalement apprendre et apprécier l'expérience de la recherche. Mes premiers mots vont naturellement à mes deux directeurs de thèse qui ont amplement rempli cette tâche et qui ont su me guider et me permettre de passer du statut d'étudiant que j'étais à celui de jeune chercheur en mathématiques appliqué que je suis aujourd'hui. Je remercie tout d'abord Antoine Gloria, avec qui j'ai pour la première fois découvert l'homogénéisation stochastique, pour m'avoir fait confiance pour cette thèse. Il a été d'une grande disponibilité et a su m'apporter l'aide et les conseils dont j'avais besoin pour mener à bien cette aventure. J'ai également appris beaucoup de notre collaboration, que ce soit à propos de ta méthode de travail ou ta manière d'aborder les problèmes. J'ai aussi eu la chance, grâce à toi, de passer une année au Max Planck Institut für Mathematik in den Naturwissenschaften à Leipzig sous la supervision de Felix Otto. Je le remercie pour son accueil si chaleureux. Le cadre de travail que propose l'institut est vraiment exceptionnel, favorisant les rencontres et les échanges. J'ai beaucoup appris au contact de Felix durant cette collaboration, d'une part pour sa manière d'appréhender les problèmes et d'autre part pour son recul et sa façon de comprendre les choses, qui sont une grande source d'inspiration pour moi. Vielen danke! J'ai vraiment hâte de collaborer davantage ! Mon séjour à Leipzig se caractérise aussi par de nombreuses rencontres professionnelles et amicales. Je remercie mes deux autres collaborateurs Marc Josien et Qiang Xu qui m'ont énormément aidé et soutenu pour ce projet et j'espère à l'avenir collaborer davantage avec vous! Je remercie tous les doctorants et post-doctorants que j'ai rencontré là-bas pour la bonne ambiance et les séances de Padel, de ping-pong et de musculation qui m'ont permis de me vider l'esprit (et de me maintenir en forme).

Le reste du temps, je l'ai passé au laboratoire Jacques-Louis Lions, dans le 5ième arrondissement de Paris. Je remercie tous les doctorants avec qui j'ai commencé cette thèse ainsi que tous les nouveaux qui sont arrivé les deux années qui ont suivi. Merci pour toutes ces pauses café, ces séances d'ultimate et ses soirées qu'on a passées ensemble à décompresser. Ça va me manquer ! Enfin, mes remerciements vont à ma famille. Tout d'abord à ma mère pour son soutient quotidien et pour avoir toujours cru en moi tout au long de la thèse. Ensuite à mon père pour tous ces déjeuners que tu as passé avec moi à m'écouter parler de mes maths pour lesquelles tu as toujours montré un intérêt. Et enfin à mon frère, pour ces rendez-vous à la pizzeria rue du Faubourg Saint-Antoine qu'on a passé ensemble à discuter et à se soutenir concernant notre quotidien.

Foreword

L'étude approfondie de la nature est la source la plus féconde des découvertes mathématiques.

Joseph Fourier -Discours préliminaire à la théorie analytique de la chaleur This thesis was carried out starting on September 2018 during a three-year PhD program in Laboratoire Jacques-Louis Lions at Sorbonne Université in Paris and at the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig, under the co-supervision of Prof. Antoine Gloria and Prof. Felix Otto. The present manuscript contains different results in stochastic homogenization theory of elliptic equations and systems, aiming at a better understanding of the quantitative theory. Starting from quantitative estimates for linear elliptic systems with strongly correlated coefficient fields, we then develop a quantitative homogenization theory for nonlinear elliptic equations and systems with strongly monotone coefficients. In the last part, we analyse the so-called representative volume element method to compute the homogenized coefficient and we show optimal rate of convergence with respect to the size of the box. The Chapters 2 and 3 can be read independently of each others. The Chapter 5 make use of the results proved in Chapter 4.

Introduction to the homogenization theory

General introduction

The homogenization theory aims at deriving the effective properties of heterogeneous systems, when heterogeneities are small compared to the characteristic size of the system. Such systems are, at first sight, very complicated and the general question arising from the homogenization theory is : "Can we replace the complicated heterogeneous system by a simple equivalent homogeneous model (meaning that it is efficiently computable) keeping the relevant physical properties of the heterogeneous system ?".

Figure 1.1: Left panel: A schematic of a two-phase material with properties a 0 and a 1 . L and ε represent the macroscopic and the microscopic length-scales, respectively. Right panel: When L ε, the heterogeneous system can be replaced as a homogeneous one with effective property a depending on a 0 , a 1 and the distribution of the two phases.

The challenge of the homogenization theory is to find a suitable formula for the effective property a, as a function of a 0 , a 1 and of the distribution of the two phases. The understanding and the determination of effective properties of heterogeneous materials, such as transport, electromagnetic, heat conduction or mechanical, has a long history. We can cite, for instance, four pioneering works in that direction, back to the last two centuries. First, Clausius and Mossotti in [START_REF] Mossotti | Sur les forces qui régissent la constitution intérieure des corps aperçu pour servir à la détermination de la cause et des lois de l'action moléculaire[END_REF]126,[START_REF] Clausius | Die mechanische behandlung der electricität[END_REF] investigated the effective dielectric constant for isotropic spherical inclusions in a isotropic reference medium. Later, James Clerk Maxwell, in his Treatise on Electricity and Magnetism [START_REF] Clerk | A treatise on electricity and magnetism[END_REF] popularized the works of Clausius and Mossotti and worked on the same equations. He investigated the effective thermal conductivity of composite materials, model as a continuous matrix of constant conductivity a 0 containing multiple spherical inclusions with identical radii of constant conductivity a 1 . He derived an expression for the effective thermal conductivity, via far-field perturbations to solutions of the steady heat equation, that is exact for dilute sphere concentrations. John William Strutt Rayleigh in [START_REF] Rayleigh | A treatise on electricity and magnetism[END_REF] developed a formalism to compute the effective conductivity of regular arrays of spheres that is still used to this day. Finally, Einstein, in his PhD thesis [START_REF] Einstein | Eine neue bestimmung der molekuüldimensionen[END_REF], determined the effective viscosity of a dilute suspension of spheres, that is, assuming that the particles are scarce. It leads to the so-called Einstein's formula, which reads1 µ = (1 + 5 2 Φ)µ 0 , where µ represents the effective viscosity of the composite system, µ 0 the viscosity of the pure liquid and Φ the volume concentration of the particles. This formula played an important role in the physic community at that time and, in particular, it served as a basis for Perrin's experiment to estimate the Avogadro number. More recently heterogeneous media have been extensively used in physics and engineering for their many interesting properties. Examples of manufactured heterogeneous materials include particulate composites, block copolymers or aligned and chopped fiber composites whereas heterogeneous materials in nature include polycrystals, Earth's crust or cell aggregates and tumors.

Figure 1.2: Example of polycrystals (see [START_REF] Sevostianov | On the effective properties of polycrystals with intergranular cracks[END_REF]): (a) aluminium titanate (b) mixed of titanium, molybdenum, vanadium, iron and aluminium (c) quartz (d) silesian granite.

The rich variety of examples encouraged the development of methods to characterise the effective medium and averaged fields, which took the form of the homogenization theory in the mathematical community. Important mechanical and physical properties include:

• The effective elasticity tensor C, where pioneer works started from the celebrated papers [START_REF] Douglas | The elastic field outside an ellipsoidal inclusion[END_REF] and [START_REF] Douglas | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] about a single ellipsoidal inhomogeneity. The quantity C corresponds to the proportionality constant between the average stress and average strain, commonly referred to averaged Hooke's law for the composite material. More general nonlinear models describing the energy of deformation arise naturally in nonlinear elasticity, see for instance [START_REF] Ciarlet | Mathematical elasticity: Volume i: three-dimensional elasticity[END_REF]. The elastic nonlinear energy model takes the form of an integral ´Ω W (∇u(x))dx where ∇u denotes the deformation gradient and W the effective stored elastic energy function. Computing C and W for heterogeneous material finds, for instance, its interest in biological material such as bone or tendon or also in various synthetic materials.

• The fluid permeability tensor κ, describing slow viscous flow through porous media. The quantity κ corresponds to the proportionality tensor between the average fluid velocity V and the applied pressure gradient in the porous medium ∇P , commonly referred to Darcy's law for the porous medium. In turbulent regimes, a nonlinear model of Darcy's law is often used and takes the form V = -κ(∇P ) = -κ|∇P | p-2 ∇P , for some parameter p > 1 (see for instance [START_REF] Li | Nonlinear diffusion equations[END_REF]). Computing κ for porous media finds, for instance, its interest in extraction of oil or gas from porous rocks, chromatography or filtration.

• The effective thermal or conductivity tensor σ. The quantity σ corresponds to the proportionality tensor between the average of the local electric current (resp. heat flux j h ) and the average of the local electric field (resp. temperature gradient ∇T ) in the composite material, commonly referred to as averaged Ohm's law or Fourier's law in the electrical or thermal problems, respectively. For certain temperature range, we rather apply to nonlinear laws for heat transfers of the form j h = -σ(T, ∇T ) = -σT q |∇T | p-2 ∇T , for some parameters q > 0 and p ≥ 2 (see for instance [START_REF] Pascal | Nonlinear model of heat conduction[END_REF][START_REF] Barth | Lecture notes in computational science and engineering[END_REF]). Computing σ for heterogeneous material finds, for instance, its interest in electrical components and oil drilling operations, where electric conductivity measurements of the brine-saturated rock are used to infer information about the permeability of the pore space. Thermal applications range from composites used for insulation, heat exchangers, and heat sinks for electronic cooling to geophysical problems (determination of the geothermal temperature gradient).

On large scales, microstructures are expected to average out due to a kind of a law of large numbers, in such a way that the heterogeneous physical properties can be replaced by homogeneous ones.

This procedure is what we call the homogenization process. The homogenization theory can be applied to many situations and the most simple examples are obtained when the heterogeneities are periodically or quasi-periodically distributed through the material. However, in many instances, the microstructures can be characterized only statistically, and therefore are referred to as random heterogeneous materials, the main focus of this thesis. There is a vast family of random microstructures that are possible, both natural or synthetic, ranging from porous media, multilayer-structures (heterogeneous in one dimension) to complex connected multiphase media or perforated materials. Physical or mechanical phenomenas are usually described via partial differential equations or energy functionals, depending on the microstructure via a random variable (corresponding to the distribution of the heterogeneities) and the scale separation parameter 0 < ε 1. For a long time, homogenization problems for partial differential equations were being mostly considered by specialists in physics and mechanics. The general mathematical theory only emerged in the 1970's with the pioneer works of Murat and Tartar in [START_REF] Tartar | Homogénéisation compacité par compensation[END_REF][START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF][START_REF] Murat | Calculus of variations and homogenization[END_REF] (in the periodic setting), Kozlov in [START_REF] Mikhailovich | Averaging of random operators[END_REF], Papanicolaou and Varadhan in [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] (in the random setting) and later in the 1980's by Braides [START_REF] Braides | Homogenization of some almost periodic coercive functional[END_REF] and Müller in [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF] (in the periodic setting) as well as Dal Maso in [START_REF] Maso | Nonlinear stochastic homogenization[END_REF] (in the random setting) in the general framework of hyperelasticity, mixing tools from probability, calculus of variation and partial differential equation theory. Since then, the homogenization theory is a very active domain of research and a lot of references will be given in the course of this introduction.

In mathematical models of random heterogeneous media, constitutive properties are usually described by functions of the form a( x ε ), where x denotes the space variable and ε > 0 the characteristic size of the heterogeneities small compared to the characteristic size L of the material-see Figure 1.1.1. In this thesis, we are interested in heterogeneous random media, so that we always assume that a = a(x) is a realization of a random distribution. The determination of the effective properties of the random heterogeneous material is a difficult task since the coefficients of the corresponding differential problem rapidly oscillate as ε ↓ 0. However, by assuming stationarity and ergodicity of the random distribution (in form of a law of large number, a precise statement will be given in section 1.1.2), meaning that the distribution of the heterogeneities is statistically invariant by translation and decorrelates on large scales, we expect an averaging procedure in the limit ε ↓ 0. In general, the averaging that occurs is more subtle than the simple average a of the coefficients2 . We illustrate this by the following simple example. We consider the case of a one-dimensional heterogeneous material on the segment [0, 1] with a series of rectangular inclusions of size ε 1 (that we can see as resistors in series): We are interested in the effective conductivity of the material as ε ↓ 0, where for simplicity we impose that 1 ε ∈ N. The conductivity of the first phase (that is without the heterogeneities) is assumed to be constant equal to a 0 whereas for the second phase (corresponding to the heterogeneities) the conductivity is assumed to be constant equal to a 1 . Furthermore, we assume that the resistors are independently and identically distributed via a sequence of Bernouilli's random variables (b i ) i∈N ⊂ {0, 1} N , with parameter q > 0. More precisely, the conductivity a ε of the heterogeneous material is given by

a ε (x) := a( x ε ) = a 0 + (a 1 -a 0 ) +∞ i=1 b i 1 [i-1,i] ( x ε ). (1.1.1)
We impose a difference of potential (denoted by u) on this configuration of resistors, say ´1 0 ∂ x u = 1, and we assume that there is no charge. Thus, the electrical flux is conserved, which reads:

∂ x (a( x ε )∂ x u) = 0.
By integrating once, we learn that the flux F ε = a( x ε )∂ x u is constant. By averaging ∂ x u = aε a( x ε ) on [0, 1], we obtain that the flux satisfy F ε = ( ´1 0 1 a( x ε ) dx) -1 , which turns into the following formula using (1.1.1),

F ε = ε 1 ε i=1 1 a 0 + (a 1 -a 0 )b i -1
.

(1.1.2)

Since the potential difference is 1, we have F ε = a ε ×1, where a ε represents the effective conductivity of the system. The formula (1.1.2) converges as ε ↓ 0 by the law of large numbers, and the limit provides the formula for the effective conductivity a = a(a 0 , a 1 ) of the homogenized material:

a := lim ε↓0 a ε = E 1 a 0 + (a 1 -a 0 )b 1 -1 = q a 1 + 1 -q a 0 -1 . (1.1.3)
In particular, a = a , which shows that the effective conductivity (1.1.3) is obtained by a more subtle averaging procedure than arithmetic averaging. Actually, the formula (1.1.3) is not surprising: the effective behaviour of a series of resistors is given by adding the resistivity of each resistor, Chapter 1. Introduction expressed by 1 a 1 and 1 a 0 , with a probability to contribute equal to q and 1 -q respectively (which is equivalent to (1.1.19) since the resistivity is the inverse of the conductivity).

The rest of the introduction is organized as follows. Section 1.1.2 is devoted to qualitative stochastic homogenization of elliptic systems in divergence form. In Section 1.1.3 we argue by a formal two-scale expansion on a simple nonlinear model, which allows us to introduce the classical objects of stochastic homogenization, such as the correctors and the homogenized operator. We then rigorously treat in Section 1.1.4 a nonlinear one-dimensional example and relate this result to the formal computations of Section 1.1.3. We also compute the homogenized operator as well as the two-scale expansion in a nonlinear two-dimensional example of a layered material. We then slowly turn to the contents of this thesis: we present in Section 1.1.5 quantitative assumptions and important tools that we use in this thesis. We show in particular how those tools can be applied on the simple one-dimensional example of Section 1.1.4 to get optimal estimates. Finally, in Section 1.2.1 and then in Section 1.2, we describe the questions and the contributions of this thesis.

Qualitative assumptions for a qualitative homogenization theory

We denote by d ≥ 1 the dimension and we consider an open subset Ω ⊂ R d . We fix two exponents m ≥ 1 and p ≥ 2 as well as f ∈ L p (Ω, R m×d ) and we consider the following system of equations with solution

u ε ∈ W 1,p (Ω, R m ) in Ω: -∇ • a( x ε , ∇u ε ) = ∇ • f, (1.1.4)
with either Neumann or Dirichlet boundary conditions and where ε 1 denotes the microscopic scale 3 . Systems of type (1.1.4) model physical phenomena such as (nonlinear) conductivity (of thermal or electrical type) obtained by combining:

• A conservation law: ∇ • (q + f ) = 0, where q denotes either the heat flux or the local electric current and f some exterior forcing.

• A constitutive law (such as Fourier's or Ohm's law): q = a( x ε , ∇u ε ), where u ε is either the temperature or the electric potential. The spatial dependence in the constitutive law in form of x ε comes from the composite structure of the material, made of different materials with different conductivities. 4The map a : R d × R m×d → R m×d is assumed to be strongly monotone, that is: there exist two constants λ ≥ 1 and s ≥ 0 such that for all x ∈ R d and ξ 1 , ξ 2 ∈ R m×d , we have

(a(x, ξ 1 ) -a(x, ξ 2 )) • (ξ 1 -ξ 2 ) ≥ 1 λ (s + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 , (1.1.5) and |a(x, ξ 1 ) -a(x, ξ 2 )| ≤ λ(s + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 |. (1.1.6)
The well-posedness of (1.1.4), under (1.1.5) and (1.1.6), is ensured by the Hartman-Stampacchia's theorem (see for instance [START_REF] Hartman | On some non-linear elliptic differential-functional equations[END_REF]). We are interested in random distributed microstructures and therefore the operators a = a(x, ξ) are realizations of a probability distribution. We use the following framework: we define the space of all admissible operators M := {a : R d × R m×d → R m×d measurable |(1.1.5) and (1.1.6) are satisfied}.

(1.1.7)

We endow M with the σ-algebra F defined as the smallest σ-algebra that makes the evaluations5 a → ´a(x, ξ)ψ(x)dx measurable for all ξ ∈ R m×d and ψ ∈ C ∞ c (R d ) (which is countably generated since C ∞ c (R d ) is separable). We then equip the measurable space (M, F) with a probability measure P. Two additional important conditions have to be added on the probability measure P for the homogenization process of the equation (1.1.4) to hold, that we list hereafter.

• The action group (R d , +) acts on M and associates for all z ∈ R d and a ∈ M the translated field T z a : (x, ξ) ∈ R d × R m×d → a(x + z, ξ). We assume that P is stationary with respect to R d -translations, that is: P • T z = P for every z ∈ R d .

(1.1.8)

• P is ergodic by the action of (R d , +) in the sense that: for all F ∈ L 1 loc (R d , L 1 (dP)) 6 with the shift-invariant property for all a ∈ M and z ∈ R (1.1.9)

The stationarity assumption (1.1.8) imposes that the statistics of a(x) does not depend on x. The ergodic assumption (1.1.9) encodes decorrelation of a at large distances r ↑ +∞, such that statistical averages can be recovered by the spacial averages almost surely. This can be seen as a law of large numbers (extending the case of discrete i.i.d random variables) and we have already seen its importance in the homogenization process in the small example of resistors in series in Section 1.1.1.

The simplest way to construct examples satisfying the assumptions (1.1.5), (1.1.6), (1.1.8) and (1.1.9) is from a "random checkerboard" structure: we pave the space by unit-sized cubes and color each cube either white or black independently at random, where the origin O ∈ [0, 1] d is randomly distributed according to the uniform distribution on [0, 1] d . 8 Each color is then associated with a particular value of (for instance) a diffusivity matrix. More precisely, consider a sequence of independent and identically distributed Bernouilli's random variables (b(z)) z∈Z d ⊂ {0, 1} Z d with parameter q ∈ [0, 1], namely for every z ∈ Z d , P({b(z) = 0}) = 1 -P({b(z) = 1}) = q.

We consider also a random variable O ∈ [0, 1] d distributed according to the uniform distribution on [0, 1] d . We fix two matrices a 0 and a 1 belonging to the set Another important class of examples are constructed using point processes. A simple case to illustrate this is given by Poisson point processes. We recall that a Poisson point process on a measurable space (X, A) with (non-atomic, σ-finite) intensity measure µ is a random subset Π ⊂ X such that the following properties hold:

{ã ∈ R d×d | for all h ∈ R d , 1 λ |h| 2 ≤ ãh • h ≤ λ|h| 2 }. ( 1
• For every A ∈ A, the number of points in Π ∩ A, which is denoted by N (A), follows a Poission law of mean µ (A), that is: for all k ∈ N P({N (A) = k}) = (µ(A)) k k! e -µ (A) .

• For every pairwise disjoint measurable sets (A 1 , ..., A k ) ∈ A k , the random variables N (A 1 ), ..., N (A k ) are independent.

Let Π be a Poisson point process on R d with intensity measure µ = γdx, for some γ > 0. We fix a 0 , a 1 in the set (1.1.10) and we define (x, ξ) → a(x, ξ) by setting for all x ∈ R d a(x, ξ) = (s + |ξ| p-2 )ξ × a 0 if dist(x, Π) ≤ 1 2 , a 1 otherwise.

(1.1.12) Figure 1.5: A sample of the coefficient field defined in (1.1.12) by the Poisson point cloud. The conductivity matrix is equal to a 1 in the black region and to a 0 in the white region.

A third important class of examples is obtained by defining the map (x, ξ) → a(x, ξ) as a local function of a Gaussian field. We recall that a scalar white noise9 on R d of variance σ 2 ∈ (0, +∞) is a family of real random variables (W(f ), f ∈ L 2 (R d )) such that:

• W is a linear form almost surely.

• For all f ∈ L 2 (R d ), W(f ) ∼10 N (0, σ 2 f 2 L 2 (R d ) ). For a rigorous construction of a scalar white noise, we refer to [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF]Proposition 5.14]. Now, for a given scalar white noise W of variance 1, we consider a kernel m satisfying for some C > 0, and β > 0, |m(x)| ≤ C(1 + |x|) -1 2 (d+β) as well as a smooth function A from R d into the set (1.1.10) and we define a(x, ξ) = A(m W(x))(s + |ξ| p-2 )ξ.

(1.1.13)

The convolution g := m W corresponds to an infinite dimensional Gaussian field, with covariance function c(x) = E[g(x)g(0)] = m m(x). Thus, we call (1.1.13) a Gaussian model of randomness with correlation c. Note that the bound on m implies that11 |c(x)| m,C (1 + |x|) -β . This class includes for instance log-normal random coefficients of the form (g-γ) , where b, c > 0 and κ, γ ∈ R.

A = b + e -κ(g-γ) c + e -κ
The qualitative homogenization of the problem (1.1.4) aims at characterizing the asymptotic behaviour of (u ε , ∇u ε ) as ε ↓ 0. More precisely, we ask the following questions: • If (u, ∇u) = lim ε↓0 (u ε , ∇u ε ) exists, how can we characterize u ?

• Does u solve a problem of the type (1.1.4)? If yes, what is the structure of the operator? At first sight, the convergence of (u ε , ∇u ε ) (at least along a subsequence) is immediate: the monotonicity conditions (1.1.5) and (1.1.6) yield the energy estimate ∇u ε L p (Ω) λ,s f L p (Ω) , which in turn implies (together with Rellich-Kondrachov's theorem) the existence of u ∈ W 1,p (Ω, R m ) for which (up to an extraction) u ε → ε↓0 u in L p (Ω, R m ) and ∇u ε ε↓0 ∇u in L p (Ω, R m×d ). The characterization of u, however, is more subtle. Rigorous mathematical studies started in the 1970's, with contributions of the French, the Italian and the Russian schools, in the linear elliptic setting, that is for the map a(x, ξ) = a(x)ξ with a periodic or realization of a random distribution. As references, we cite in particular the works of Murat and Tartar in [START_REF] Murat | Calculus of variations and homogenization[END_REF][START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF], Bakhvalov in [START_REF] Sergeevich | Averaged characteristics of bodies with periodic structure[END_REF][START_REF] Sergeevich | Averaging of partial differential equations with rapidly oscillating coefficients[END_REF], De Giorgi and Spagnolo in [START_REF] De | Convergence problems for functionals and operators[END_REF], Tartar in [START_REF] Tartar | Homogénéisation compacité par compensation[END_REF], Kozlov in [START_REF] Mikhailovich | Averaging of random operators[END_REF] and Papanicolaou and Varadhan in [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] as well as Allaire in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. In the general nonlinear setting (1.1.4), we cite in particular the works of Dal Maso and Modica in [START_REF] Maso | Nonlinear stochastic homogenization[END_REF][START_REF] Maso | Nonlinear stochastic homogenization and ergodic theory[END_REF] (in the variational framework a = ∇ ξ V , with V convex), Braides in [START_REF] Braides | Homogenization of some almost periodic coercive functional[END_REF], Braides and Defranceschi in [START_REF] Braides | Homogenization of multiple integrals[END_REF] as well as Müller in [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF] (in the variational and general framework of integral functionals with p-growth and periodic integrands), Chiadò Piat, Dal Maso and Defranceschi in [START_REF] Chiadò Piat | G-convergence of monotone operators[END_REF], Dal Maso and Defrancesch in [START_REF] Maso | Correctors for the homogenization of monotone operators[END_REF]. For a nice summary of those results, we refer to the textbooks [START_REF] Shen | Periodic homogenization of elliptic systems[END_REF] of Shen and [100] of Jikov, Kozlov and Oleȋnik. In our context, the homogenization result can be summarized as follows:

Theorem 1 (Qualitative homogenization of (1.1.4)). The sequence (u ε ) ε>0 converges (strongly in L p (Ω, R m ) and weakly in W 1,p (Ω, R m )) to some u ∈ W 1,p (Ω, R m ) almost surely. In addition, there exists a deterministic map a : R m×d → R m×d such that we have the weak convergence of the flux a( • ε , ∇u ε ) ε↓0 a(∇u) in L p (Ω, R m ) and u solves -∇ • a(∇u) = ∇ • f, (1. 1.14) with the same boundary conditions than the one applied to (1.1.4). Furthermore, the map a is monotone and satisfies, a priori, a weaker 12 version of (1.1.5), namely there exists a constant C > 0 depending on λ and s such that for all ξ 1 , ξ 2 ∈ R m×d (a(ξ 1 ) -a(ξ 2 ))

• (ξ 1 -ξ 2 ) ≥ 1 C (s + |ξ 1 -ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 , (1.1.15
)

and |a(ξ 1 ) -a(ξ 2 )| ≤ C(s + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 |. (1.1.16)
The convergence of ∇u ε in Theorem 1 is only weak in L p (Ω, R m×d ) and, in fact, cannot be expected to hold strongly in L p (Ω, R m×d ) due to the oscillations at scale ε of the variation of u ε . It tells us that a correction of ∇u ε is needed to take into account the oscillations at the microscopic scale. We explain in the next section how we can formally reconstruct a posteriori the oscillations of ∇u ε given ∇u and some intrinsic quantities, in such a way that it is close to ∇u ε in the strong topology of L p (Ω, R m×d ). A refinement of Theorem 1 will then be given.

The formal two-scale expansion

In this section, we develop formal arguments to reconstruct the oscillations of ∇u ε as ε ↓ 0, given ∇u. For the ease of the reading, we investigate the simple nonlinear model given by a p-Laplacian (for m = 1) regularized at 0: a(x, ξ) := A(x)(1 + |ξ| p-2 )ξ, and we will use the short-hand notation a 0 (x, ξ) = A(x)|ξ| p-2 ξ for the p-Laplacian part. In addition, we use the general notation • to denote either the average on the periodic cell (for periodic operator) or the expectation in the random setting.

The reconstruction of the behaviour of u ε is based on a "two-scale expansion", that we formally establish hereafter. Based on Figure 1.1.2, we observe that u ε varies at a macroscopic and a microscopic scale (at scale ε, revealed by the oscillations). We translate this observation by formally Chapter 1. Introduction assuming that u ε depends on two variables: a slow variable x (macroscopic variable) and a fastvariable y = x ε (microscopic variable), that we treat as two independent variables. This dependence is made explicit through a formal expansion of u ε into a power series in ε, where each term of the sum is a function depending on x and y. The expansion takes the form of the following Ansatz:

u ε (x) ∼ +∞ n=0 ε n u n (x, x ε ), (1.1.17)
where we do not worry about summability at this stage. The functions u n = u n (x, y) are the unknowns that we have to determine. For simplicity, we restrict ourself to identify the first order expansion in (1.1.17) which, in other words, means that we only identify u 0 and u 1 . For the formal derivation of higher expansion, we refer to [START_REF] Papanicolau | Asymptotic analysis for periodic structures[END_REF].

First of all, we plug the Ansatz (1.1.17) into the equation (1.1.4). To do so, we observe that the chain rule, for all map F = F (x, y), give us

∇[F (•, • ε )](x) = ∇ x F (x, x ε ) + 1 ε ∇ y F (x, x ε ). (1.1.18)
By considering the second-order expansion in (1.1.17) and by computing the gradient of (1.1.17) using the rule (1.1.18), that in turn we plug into (1.1.32), we obtain

ε -1 ∇ y • a( x ε , 1 ε ∇ y u 0 (x, x ε ) + ∇ x u 0 (x, x ε ) + ∇ y u 1 (x, x ε ) + ε∇ x u 1 (x, x ε ) + ε∇ y u 2 (x, x ε ) + ε 2 ∇ x u 2 (x, x ε )) + ∇ x • a( x ε , 1 ε ∇ y u 0 (x, x ε ) + ∇ x u 0 (x, x ε ) + ∇ y u 1 (x, x ε ) + ε∇ x u 1 (x, x ε ) + ε∇ y u 2 (x, x ε ) + ε 2 ∇ x u 2 (x, x ε )) = ∇ • f. (1.1.19)
Now, the idea is to expand the left-hand side of (1.1.19) in powers of ε and then do a formal identification. By noticing that ξ → a 0 (•, ξ) is smooth, our main tool is Taylor expansions. To begin with, using that a 0 , Da 0 and D 2 a 0 are respectively p -1, p -2 and p -3 homogeneous 13 , one has for all X 1 , X 2 ∈ R m×d a 0 (y, 1 ε X 1 + X 2 ) = ε -p+1 a 0 (y, X 1 + εX 2 ) = ε -p+1 a 0 (y, X 1 ) + ε -p+2 Da 0 (y, X 1 )X 2 + O(ε -p+3 ).

(1. 1.20) By applying (1.1.20) with X 1 = ∇ y u 0 (x, y) and X 2 = ∇ x u 0 (x, y) + ∇ y u 1 (x, y) + ε∇ x u 1 (x, y) + ε∇ y u 2 (x, y) + ε 2 ∇ x u 2 (x, y), we observe that the term which appears with a factor ε -p in (1. 1.19) has to vanish, namely -∇ y • a 0 (y, ∇ y u 0 ) = 0, so that the monotonicity of a 0 implies ∇ y u 0 = 0 and therefore u 0 does not depend on the fast-variable y. Next, we perform a Taylor expansion in form of, for all X 1 , X 2 , X 3 ∈ R m×d a 0 (y, X 1 +εX 2 +ε 2 X 3 ) = a 0 (y, X 1 )+εDa 0 (y, X 1 )X 2 +ε 2 ( 1 2 D 2 a 0 (y, X 1 )X 2 •X 2 +Da 0 (y, X 1 )X 3 )+O(ε 3 ), that we use in (1.1.19) for X 1 = ∇ x u 0 (x, x ε ) + ∇ y u 1 (x, x ε ), X 2 = ∇ x u 1 (x, x ε ) + ∇ y u 2 (x, x ε ) and X 3 = ∇ x u 2 (x, y), to deduce by a formal identification:

• At order ε -1 :

-∇ y • a(y, ∇ x u 0 + ∇ y u 1 ) = 0, (1.1.21)

• At order ε 0 :

-∇ x • a(y, ∇ x u 0 + ∇ y u 1 ) -∇ y • Da(∇ x u 0 + ∇ y u 1 )(∇ x u 1 + ∇ y u 2 ) = ∇ • f.

(1. 1.22) In the equation (1.1.21), x is a parameter so that the equation (1.1.21) identifies u 1 and the latter can be seen as a function of ∇ x u 0 . Indeed, let us slightly reinterpret (1.1.21). We define a new function φ ξ , depending on the direction ξ ∈ R m×d , which solves the equation

-∇ • a(•, ξ + ∇φ ξ ) = 0. (1.1.23)
The function φ ξ is called the corrector in the direction ξ ∈ R m×d . From (1.1.21), one can see that u 1 can be generated with help of the corrector in the direction ∇ x u 0 , namely u 1 (x, y) = φ ∇xu 0 (x) (y).

(1.1.24)

The complexity of the well-posedness of the equation (1.1.23) depends on the context. In the Tperiodic setting, it is easy to give a sense to (1.1.23), since the equation is naturally posed on the torus T T := R d /T Z d . This way, using the monotonicity of a, the equation (1.1.23) has a unique periodic solution with vanishing mean. In the random setting, the story is different and (1.1.23) has to be set in R d . Therefore the well-posedness is more subtle and the stationarity assumption (1.1.8) on the random distribution plays an important role. Indeed, morally speaking, the stationarity assumption allows us to set the equation (1.1.23) on the probability space and construct, from the monotonicity of a, a solution such that ∇φ ξ is stationary 14 , of vanishing expectation, anchored by ´B1 φ ξ = 0 and satisfying

E[|∇φ ξ | 2 + |∇φ ξ | p ] λ,s |ξ| 2 + |ξ| p . (1.1.25)
Hence, in the random setting, (1.1.23) has a distributional sense in R d (φ ξ belongs to W 1,p loc (R d )) and a weak sense in probability. For more precision on the rigorous proof of the well-posedness of (1.1.23), we refer to [START_REF] Pankov | G-convergence and homogenization of nonlinear partial differential operators[END_REF]Chapter 15] and [80, Lemma 1]. Finally, the corrector φ ξ is sublinear at infinity 15 , as a consequence of the fact that ∇φ ξ is stationary, E[∇φ ξ ] = 0 and (1.1.25) holds as well as the ergodicity assumption of the random distribution (1.1.9). For more details, we refer to [80, Proof of Corollary 1]. The sub-linear property of the correctors plays an important role in the homogenization theory, since it controls the error between u ε and u 0 at first order. Indeed, the first order expansion in (1.1.17) together (1.1.24) gives u ε (x) -u 0 (x) ≈ εφ ∇u 0 (x) ( x ε ) which vanishes as ε ↓ 0 by sub-linearity.

It remains to identify u 0 and this is done via the equation (1.1.22). To do so, we average the equation (1. 1.22) with respect to the fast-variable y. By noticing that ∇ y • Da 0 (∇ x u 0 + ∇ y u 1 )(∇ x u 1 + ∇ y u 2 ) = 0 (which comes from an integration by parts in the periodic setting or the stationarity in the random setting 16 ), averaging (1.1.22) has the effect of:

-∇ x • a(•, ∇ x u 0 + ∇ y u 1 ) = ∇ • f.
We slightly rewrite the latter equation, using (1.1.24), by making the homogenized operator a(ξ) = a(•, ξ + ∇φ ξ ) appear:

-∇ • a(∇u 0 ) = ∇ • f.

(1.1.26) 14 We say that F = F (a, x) is stationary if it has the shift-invariance property: for all z ∈ R d , F (a, • + z) = F (a(• + z), •). 15 Meaning that ε 2 ffl

B 1 ε (x) |φ ξ | 2 → ε↓0 0.
16 Formally: if F = F (a, y) is stationary, E[∂y i F ] = E[lim h↓0 F (a,he i )-F (a,0) h ] = lim h↓0 E[ F (a,he i )-F (a,0) h ] = 0, which is for instance rigorous if F is bounded.

The equation (1.1.26) (that we call the homogenized equation) combined with suitable boundary conditions uniquely defines u 0 , that we justify by showing that the homogenized operator a is monotone (in the sense of (1.1.15) and (1.1.16)). We fix ξ 1 , ξ 2 ∈ R m×d . For the lower bound (1.1.15), we use the corrector equation (1.1.23), the strong monotonicity of a (1.1.5) and the vectorial inequality 1.27) in form of (a(ξ 1 ) -a(ξ 2 ))

|ξ 1 -ξ 2 | 2 (1 + |ξ 1 | p-2 + |ξ 2 | p-2 ) p |ξ 1 -ξ 2 | 2 (1 + |ξ 1 -ξ 2 | p-2 ), (1.
• (ξ 1 -ξ 2 ) = (a(•, ξ + ∇φ ξ 1 ) -a(•, ξ 2 + ∇φ ξ 2 )) • (ξ 1 -ξ 2 ) (1.1.23) = (a(•, ξ + ∇φ ξ 1 ) -a(•, ξ 2 + ∇φ ξ 2 )) • (ξ 1 + ∇φ ξ 1 -(∇φ ξ 2 + ξ 2 ))
(1.1.5),(1.1.27)

p 1 λ (1 + |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| p-2 )|ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| 2 ,
which turns into, by applying Jensen's inequality and using the fact that ∇φ ξ = 0 for all ξ ∈ R m×d :

(a(ξ 1 ) -a(ξ 2 )) • (ξ 1 -ξ 2 ) p 1 λ (1 + |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| p-2 )|ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| 2 = 1 λ (1 + |ξ 1 -ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 .
For the upper bound (1.1.16), we use the following energy estimate

|ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| 2 (1 + |ξ 1 + ∇φ ξ 1 | p-2 + |ξ 2 + ∇φ ξ 2 | p-2 ) p,λ,s (s + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 , (1.1
C |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )|1 + |ξ 1 + ∇φ ξ 1 | p-2 + |ξ 2 + ∇φ ξ 2 | p-2 ) ≤ |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )|(1 + |ξ 1 + ∇φ ξ 1 | p-2 + |ξ 2 + ∇φ ξ 2 | p-2 ) 1 2 × 1 + |ξ 1 + ∇φ ξ 1 | p-2 + |ξ 2 + ∇φ ξ 2 | p-2 1 2
(1.1.28), (1.1.25) p,λ,s

(1 + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 .
We briefly argue in favour of (1.1.28). Using the corrector equation (1.1.23), we have -∇ • (a(•, ξ 1 + ∇φ ξ 1 ) -a(•, ξ 2 + ∇φ ξ 2 )) = 0, so that by testing the equation with φ ξ 1 -φ ξ 2 , we arrive at

(a(•, ξ+∇φ ξ 1 )-a(•, ξ 2 +∇φ ξ 2 ))•(ξ 1 +∇φ ξ 1 -(ξ 2 +∇φ ξ 2 )) = a(•, ξ+∇φ ξ 1 )-a(•, ξ 2 +∇φ ξ 2 ))•(ξ 1 -ξ 2 ) ,
which, using (1.1.5) and (1.1.6), turns into

1 λ |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| 2 (1 + |ξ 1 + ∇φ ξ 1 | p-2 + |ξ 2 + ∇φ ξ 2 | p-2 ) ≤ λ|ξ 1 -ξ 2 | |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )|(1 + |ξ 1 + ∇φ ξ 1 | p-2 + |ξ 2 + ∇φ ξ 2 | p-2 ) ,
which gives (1.1.28) by Young's inequality in form of |ξ 1 - (1.1.25). Note that, a priori, except for the particular case of p = 2, the homogenized operator satisfies a weaker monotonicity condition than the original ones (1.1.5) and (1.1.6), which is enough for existence and uniqueness (under suitable boundary conditions). It does also imply a regularity theory for the operator -∇ • a(∇u) in the case of scalar equations (that is for m = 1) for certain range of p (see for more precisions Section 3.1.1 of Chapter 3). The question of showing that the homogenized operator satisfies also the strong monotonicity conditions (1.1.5) and (1.1.6) is actually a challenging question, and we refer to Theorem 13 and Theorem [START_REF] Armstrong | Quantitative analysis of boundary layers in periodic homogenization[END_REF] for a partial answer to this question and to Chapter 3 for a short discussion about this issue.

ξ 2 ||ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| ≤ λ 2 2 |ξ 1 - ξ 2 | 2 + 1 2λ 2 |ξ 1 + ∇φ ξ 1 -(ξ 2 + ∇φ ξ 2 )| 2 and
In conclusion, we have totally characterised the first order expansion in (1.1.17), called the (firstorder) two-scale expansion of u ε , which reads u ε (x) ≈ ε↓0 u 0 (x) + εφ ∇u 0 (x) ( x ε ).

(1.1.29)

Formula (1.1.29) give a drastic reduction of complexity since u is the solution of a constant-coefficient equation and φ ξ does not depend on f . We have the following refinement of Theorem 1.

Theorem 2 (Qualitative homogenization of (1.1.4)-Refinement). For any ξ ∈ R m×d , there exists a unique almost sure distributional solution φ ξ ∈ W 1,p loc (R d ) of the corrector problem (1.1.23) posed in R d , anchored by ´B1 φ ξ = 0 and its gradient ∇φ ξ is stationary, has vanishing expectation E[∇φ ξ ] = 0, and satisfies (1.1.25).

The homogenized operator a has an explicit formula which can be expressed with help of the correctors: for all ξ ∈ R m×d a(ξ) = E[a(0, ξ + ∇φ ξ )].

(1.1.30)

In addition, in the linear setting, we define the two-scale expansion 17 :

u 2sc ε := u + εφ ∇u ( • ε ), (1.1.31)
where u is given in (1. 1.14), and we have that (∇u 2sc ε -∇u ε ) ε>0 converges, almost-surely, to 0 in L 2 (Ω, R m×d ).

We now illustrate the asymptotic (1.1.29) by a complete one-dimensional example (d = m = 1) and we compute the homogenized operator (1.1.30) as well as the two-scale expansion (1.1.31) in the two-dimensional example (d = 2 and m = 1) of a layered material.

Two examples of explicit homogenization

Example 1: A one-dimensional example

We illustrate the asymptotics (1. 1.29) with the model a(x, ξ) = ξ + A(x)|ξ|ξ for d = m = 1. More precisely, we consider A ∈ L ∞ ([0, 1]) such that 0 < c 1 ≤ A ≤ c 2 (for some fixed constants c 1 , c 2 > 0) seen as a realization of a random distribution P satisfying (1.1.8) and (1.1.9). Let f ∈ C 0 ([0, 1]) such that f ≤ f (0) and we consider the sequence (u ε ) ε>0 ⊂ W 1,3 ([0, 1]) ⊂ C 0 ([0, 1]) where for all ε > 0, u ε is the weak solution of the nonlinear equation

-u ε -(A( • ε )|u ε |u ε ) = f in (0, 1], u ε (0) = u ε (0) = 0.
(1.1.32)

The equation (1.1.32) can be solved explicitly. Indeed, a first integration and u ε (0) = 0 give us -u ε -A( • ε )|u ε |u ε = f -f (0). Therefore, since f ≤ f (0), u ε is non-negative, which yields

u ε + A( • ε )(u ε ) 2 = f (0) -f
. By solving this equation and by keeping only the relevant solution (that is the one compatible with u ε ≥ 0) we obtain: for all x ∈ [0, 1]

u ε (x) = - 1 2 ˆx 0 1 A( t ε ) dt + ˆx 0 f (0) -f (t) A( t ε ) + 1 4A 2 ( t ε )
dt.

(1. 1.33) From the explicit formula (1.1.33), the asymptotics of (u ε ) ε>0 is clear: using the ergodicity assumption (1.1.9) on the distribution of A 18 , one has

u ε (x) → ε↓0 u(x), with u(x) := - 1 2 E 1 A x + ˆx 0 E f (0) -f (t) A + 1 4A 2 dt.
(1. 1.34) We now identify the nonlinear equation solved by u. Let us slightly rewrite u with help of the function Φ :

ξ ∈ R + → -1 2 E[ 1 A ] + E ξ A + 1 4A 2
, namely u = Φ(f (0) -f ). Now, observe that Φ is invertible (since Φ > 0), such a way that Φ -1 (u ) = f (0) -f and thus -(a(u )) = f in (0, 1], u(0) = 0 and u (0) = 0, (

for a := Φ -1 . The structure of the equation (1.1.35) is the same as (1.1.32), namely a : R + → R + is strongly monotone in the sense of (1.1.5) and (1.1.6) (for p = 3), that we prove hereafter. Since a ∈ C 1 (R + ), we first reduce the proof to an estimate on a : the monotonicity will be inferred by showing that for all ξ ∈ R + , a (ξ) = (Φ -1 ) (ξ) ∼ 19 1 + 2ξ.

(1.1.36) Indeed, knowing (1.1.36) implies the strong monotonicity via the fundamental theorem of calculus in form of: for all ξ 1 , ξ 2 ∈ R + ,

(a(ξ 1 ) -a(ξ 2 ))(ξ 1 -ξ 2 ) = (ξ 1 -ξ 2 ) ˆξ1 ξ 2 a (t)dt (1.1.36) ∼ (ξ 1 -ξ 2 ) ˆξ1 ξ 2 1 + 2t dt = (1 + ξ 1 + ξ 2 )(ξ 1 -ξ 2 ) 2 .
Also, since (Φ -1 ) = 1 Φ •Φ -1 , (1.1.36) will finally be justified by showing that Φ (Φ -1 (ξ)) ∼ (1+2ξ) -1 .

For α = (E[ 1 A ]E[ 1 √ A ] -1 ) 2 , one has Φ(ξ 2 + α) = - 1 2 E 1 A + E ξ 2 + α A + 1 4A 2 ≥ 1 2 ξE 1 √ A + 1 2 √ αE 1 √ A -E 1 A ξ,
and similarly, Φ(ξ 2 + α) ξ. Therefore, since Φ -1 is non-decreasing (because Φ is), one has Φ -1 (ξ) ∼ ξ 2 + α. Finally, since

Φ (ξ) = 1 2 E A ξ A + 1 4A 2 -1 ∼ (ξ + 1) -1 2 ,
we indeed have Φ (Φ -1 (ξ)) ∼ (1 + 2ξ) -1 , where the constant in ∼ depends on α, c 1 and c 2 .

Thus, (1.1.36) holds and a : R + → R + is strongly monotone in the sense of (1.1.5) and (1.1.6) (for p = 3). Note that this proof is very specific to the one-dimensional case, and in general we cannot ensure such a property for d ≥ 2 (see Chapter 3 for a discussion about this issue).

We now investigate the asymptotic behaviour of the derivative u ε . Differentiating Formula (1.1.33) yields: for all x ∈ [0, 1]

u ε (x) = - 1 2A( x ε ) + f (0) -f (t) A( x ε ) + 1 4A 2 ( x ε ) , (1.1.37)
and thus by the ergodic assumption (1.1.9),

u ε u in L 3 ([0, 1]), (1.1.38)
and the convergence is not strong in L 3 ([0, 1]). The structure of the oscillations of u ε can be made more precise with help of u and an explicit corrector. Indeed, observe that from (1.1.34), one has f (0) -f = Φ -1 (u ) so that we can rewrite u ε given by (1.1.37) as

u ε (x) = - 1 2A( x ε ) + Φ -1 (u ) A( x ε ) + 1 4A 2 ( x ε )
, which can be itself rewritten as:

u ε (x) = u (x) + φ u ( x ε ), (1.1.39)
where φ ξ denotes the corrector 20 , defined in the direction ξ ∈ R + by

φ ξ (x) := - 1 2 ˆx 0 1 A(t) dt + ˆx 0 Φ -1 (ξ) A(t) + 1 4A 2 (t) dt -ξx. (1.1.40)
Observe that φ ξ is indeed the corrector (anchored at the origin via φ ξ (0) = 0) in the sense of (1.1.23), since a direct computation shows that for all ξ ∈ R + :

-φ ξ -(A|ξ + φ ξ |(ξ + φ ξ )) = 0 in R,
with, by definition of Φ, E[φ ξ ] = 0. The equation (1.1.39) describes the oscillations of u ε , and this is a good hint for reconstructing the oscillations of u ε via the two scale expansion:

u 2sc ε := u + εφ u ( • ε ). (1.1.41)
We now verify that (1.1.41) indeed reconstructs the oscillations of u ε , in the sense that the sequence

(u 2sc ε -u ε ) ε>0 tends to 0 in W 1,3 ([0, 1]).
First, the definition (1.1.40) of the corrector combined with the ergodic assumption (1.1.9) and the definition of Φ show that φ ξ is sub-linear at infinity in the form of: for all x ∈ [0, 1] and almost all A

εφ ξ ( x ε ) → ε↓0 ˆx 0 Φ(Φ -1 (ξ))dt -ξx = 0, ( 1
(u 2sc ε -u ε ) = u + φ u ( • ε ) -u ε + ε ∂ ∂ξ φ ξ ( • ε )| ξ=u u (1.1.39) = ε ∂ ∂ξ φ ξ ( • ε )| ξ=u u .
The quantity ∂ ∂ξ φ ξ , that we call the linearized corrector, can be computed explicitly via (1.1.40): for all x ∈ [0, 1]

∂ ∂ξ φ ξ (x) = (Φ -1 ) (ξ) ˆx 0 1 2A(t) Φ -1 (ξ) A(t) + 1 4A 2 (t) dt -x = 1 Φ (Φ -1 (ξ)) ˆx 0 1 2A(t) Φ -1 (ξ) A(t) + 1 4A 2 (t) dt -x = E A Φ -1 (ξ) A + 1 4A 2 -1 -1 ˆx 0 1 A(t) Φ -1 (ξ) A(t) + 1 4A 2 (t)
dt -x, so that, from the ergodic assumption (1.1.9), the linearized corrector is sub-linear at infinity in form of: for all x ∈ [0, 1] and almost all A

ε ∂ ∂ξ φ ξ ( x ε ) → ε↓0 E A Φ -1 (ξ) A + 1 4A 2 -1 -1 ˆx 0 E A Φ -1 (ξ) A + 1 4A 2 -1 dx -x = 0, (1.1.43)
and therefore (u 2sc ε (x) -u ε (x)) → ε↓0 0, which in turn can be upgraded to

(u 2sc ε -u ε ) → ε↓0 0 in L 3 ([0, 1])
by dominated convergence. To conclude, we have shown that

u 2sc ε -u ε W 1,3 ([0,1]) → ε↓0 0 almost surely. (1.1.44)
Example 2: A two-dimensional example of a layered material

We study the effective conductivity of a material represented by the square Ω = [0, 1] 2 , for the nonlinear conductivity model

a((x 1 , x 2 ), ξ) = (1 + |ξ| 2 ) A 1 (x 1 )ξ 1 A 2 (x 1 )ξ 2 ,
corresponding to the case of a layered material.

Figure 1.8: A layered material with conductivity matrix a 0 0 0 ã0 in the white region and a 1 0 0 ã1 in the black region, given conductivities a 0 , a 1 , ã0 , ã1 > 0.

We start by computing the effective nonlinear conductivity model given by the formula (1.1.30). To do so, we solve the corrector equation (1.1.23) which, in our case, reads

∂ x 1 (A 1 (x 1 )(ξ 1 + ∂ x 1 φ ξ )(1 + |ξ + ∇φ ξ | 2 )) + A 2 (x 1 )∂ x 2 ((ξ 2 + ∂ x 2 φ ξ )(1 + |ξ + ∇φ ξ | 2 )) = 0. (1.1.45)
Since the material is heterogeneous in one direction, it is natural to seek a correction which depends only on x 1 , that is φ ξ = φ ξ (x 1 ). Therefore, (1.1.45) turns into

∂ x 1 (A 1 (x 1 )(ξ 1 + ∂ x 1 φ ξ )(1 + (ξ 1 + ∂ x 1 φ ξ ) 2 + ξ 2 2 )) = 0,
which give us by integrating once,

A 1 (x 1 )(ξ 1 + ∂ x 1 φ ξ )(1 + (ξ 1 + ∂ x 1 φ ξ ) 2 + ξ 2 2 ) = a 1 (ξ), (1.1.46)
with a 1 (ξ) that we have to determine. To do so, we define

Ψ ξ 2 : X ∈ R → X(1 + X 2 + ξ 2 2 ). Observe that Ψ ξ 2 is invertible (since Ψ ξ 2 > 0) and Ψ -1
ξ 2 is given by Cardan's formula:

Ψ -1 ξ 2 (X) = X 2 + 1 2 X 2 + 4(1 + ξ 2 2 ) 3 27 1 3 + X 2 - 1 2 X 2 + 4(1 + ξ 2 2 ) 3 27 1 3
.

(1.1.47) Therefore, by dividing the equation (1.1.46) by A 1 (x 1 ), applying Ψ -1 ξ 2 and using that E[∂ x 1 φ ξ ] = 0, we deduce that a 1 (ξ) is characterized by the nonlinear equation 21

ξ 1 = E[Ψ -1 ξ 2 ( a 1 (ξ) A 1 (0) )].
To conclude, using in addition that from (1.1.46) we have

ξ 1 + ∂ x 1 φ ξ = Ψ -1 ξ 2 ( a 1 (ξ) A 1 (x 1 )
), the effective conductivity model is given by

a(ξ) = E A 1 (0)(ξ 1 + ∂ x 1 φ ξ )(1 + (ξ 1 + ∂ x 1 φ ξ ) 2 + ξ 2 2 ) A 2 (0)ξ 2 (1 + (ξ 1 + ∂ x 1 φ ξ ) 2 + ξ 2 2 ) = a 1 (ξ) a 2 (ξ) , (1.1.48) with a 2 (ξ) = E[A 2 (0)ξ 2 (1 + (Ψ -1 ξ 2 ( a 1 (ξ) A 1 (0) )) 2 + ξ 2 2 )]. 30
Chapter 1. Introduction

Finally, from (1.1.46), the corrector (anchored at the origin via φ ξ (0) = 0) can be expressed with help of a 1 (ξ):

φ ξ (x 1 ) = ˆx1 0 Ψ -1 ξ 2 ( a 1 (ξ) A 1 (y) )dy -ξ 1 x 1 ,
so that the two-scale expansion (1.1.31) is explicitly given by

u 2sc ε (x 1 , x 2 ) = u(x 1 , x 2 ) + ε ˆx1 ε 0 Ψ -1 ξ 2 ( a 1 (∇u(x 1 ,x 2 )) A 1 (y) )dy -ξ 1 x 1 ε .
We see with this example that we have reduced the complexity of the problem, since now we only have to compute u which solves a nonlinear elliptic equation in divergence form with a monotone operator given by (1.1.48).

In the next section, we turn to the quantitative theory of stochastic homogenization of (1.1.4) (the main focus of this works), and we present and explain the assumptions which lead to such a theory.

Quantitative ergodic assumptions for a quantitative homogenization theory

We present in this section the quantitative assumptions that allow to develop a quantitative theory of homogenization. First, we illustrate the quantitative theory for the model a(x, ξ) = A(x)(1+|ξ| p-2 )ξ in the setting of the small contrast regime, in the sense that A is of the form A = (1 + f (δ Ã))Id with a smooth function f such that f (0) = 0 and for à stationary and ergodic (that we will specify to the Gaussian setting for the ease of the reading) and δ 1. Second, we present the general framework of quantitative homogenization.

The small contrast regime

We consider the particular model of the p-laplacian regularized at 0 (for

d ≥ 2) 22 a(x, ξ) = A(x)(1 + |ξ| p-2 )ξ, (1.1.49)
where A takes the specific form23

A(x) = (1 + 1 π arctan(πδg(x)))Id,
where g is a real valued Gaussian random variable with E[|g(0)| 2 ] = 1 and δ 1. This perturbative regime allows us to linearise the problem with repsect to δ in the corrector equation (1.1.23). It leads to a linear elliptic equation with constant coefficient where all the randomness is put in the right-hand side.

We have already seen that the ergodic assumption (1.1.9) is enough for the qualitative homogenization theory, as summarized in Theorem 1 and 2 and illustrated in Section 1.1.4. Naturally, for the quantitative homogenization, we need a quantification of ergodicity, which can be understood by imposing a convergence rate in (1.1.9). In our particular setting, we quantify the ergodicity, or in other words the decorrelation over large distances, by imposing an algebraic decay on the covariance function:

|c(x, y)| = |E[g(x)g(y)]| ∼ (1 + |x -y|) -d-α , (1.1.50)
for some α > 0. 24 Let us now show that, in the regime δ ↓ 0, this quantitative assumption implies a convergence rate on the homogenization error u 2sc ε -u ε , where we recall that the two-scale expansion u 2sc ε is defined in (1.1.31). In view of the analysis conducted in Section 1.1.4, the optimal convergence rate in (1.1.44) will follow from the optimal convergence rate in (1.1.42) and (1.1.43), namely from the quantification of the sub-linearity of φ ξ and ∂ ∂ξ φ ξ . We start by investigating the growth of x → φ ξ (x).

By linearizing around δ = 0, we formally expand φ ξ at first order in δ: φ ξ = δφ ξ + o(δ), 25 where the equation for φ ξ is obtained by differentiating with respect to δ the equation (1.1.23) for the model (1.1.49), which reads

-∇ • Id + (p -2) |ξ| p-4 (1 + |ξ|) p-2 ξ ⊗ ξ ∇φ ξ = ∇ • (gξ).
Since the above equation is a linear elliptic PDE with constant coefficients, we have an explicit formula for φ which can be expressed with help of the Green's function associated to

a ξ := Id + (p - 2) |ξ| p-4 (1+|ξ|) p-2 ξ ⊗ ξ: for all x ∈ R d the Green's function reads G ξ (x) = 1 det(a ξ ) × 1 4π log(x • a -1 ξ x) for d = 2, 1 d(2-d)|B 1 | (x • a -1 ξ x) -d-2 2 for d ≥ 3,
and we have

φ ξ (x) = -ˆ∇G ξ (x -y) • g(y)ξdy. (1.1.51)
We first briefly justify that G ξ (resp. its derivatives) can be dominated by the standard Green's function (resp. its derivatives) of the Laplace operator, uniformly in |ξ|. Indeed, since for all x ∈ R d ,

x • a ξ x = |x| 2 + (p -2) |ξ| p-4 (1+|ξ|) p-2 (ξ • x) 2 ≥ |x| 2 and a ξ ≤ 1 + (p -2) |ξ| p-4 (1+|ξ|) p-2 ξ ⊗ ξ 1, we thus have x • a -1 ξ x ∼ |x| 2 uniformly in |ξ|.
Also, it shows that the eigenvalues of a ξ are contained in [1, +∞), and thus det(a ξ ) ≥ 1 (note that a ξ is symmetric). We conclude that for all x, ξ ∈ R d

|∇G ξ (x)| + |x||∇ 2 G ξ | |x| 1-d . (1.1.52)
Because of the singularity of the Green's function at the origin, we have to treat differently the nearfield contribution and the far-field contribution (denoted by φ n (x) and φ f (x) respectively), that we express with help of a smooth cut

-off χ ∈ C ∞ c (R d ) for B 1 in B 2 : φ n (x) = -ˆχ(x -y)∇G ξ (x -y) • g(y)ξ dy and φ f (x) = -ˆ(1 -χ(x -y))∇G ξ (x -y) • g(y)ξ dy.
For the near-field contribution, we use the triangle inequality combined with the fact that the singularity of ∇G ξ is integrable (see (

E[|φ f (x) -φ f (0)| 2 ] = ˆˆ[(((1 -χ)∇G ξ )(x -y) -((1 -χ)∇G ξ )(y)) • ξ][(((1 -χ)∇G ξ )(x -y ) -((1 -χ)∇G ξ )(y )) • ξ] × E[g(y)g(y )]dy dy (1.1.50) |ξ| 2 ˆˆ|((1 -χ)∇G ξ )(x -y) -((1 -χ)∇G ξ )(y))||((1 -χ)∇G ξ )(x -y ) -((1 -χ)∇G ξ )(y ))| × (1 + |y -y |) -d-α dy dy .
Now, using Cauchy-Schwarz' inequality (in the form of ´´f (y)f (y )h(y-y )dy dy ≤ ´´|f (y)| 2 |h(yy )|dy dy ) and the fact that the covariance function c(x, 0) :

x → (1 + |x|) -d-α is integrable, we arrive at E[|φ f (x) -φ f (0)| 2 ] |ξ| 2 ˆ|((1 -χ)∇G ξ )(x -y) -((1 -χ)∇G ξ )(y)| 2 dy.
Then, the large-scale behaviour |y -x| ≥ 3|x| is treated by the Taylor expansion:

|((1 -χ)∇G ξ )(x - y) -((1 -χ)∇G ξ )(y)| |x| sup t∈[0,1] |∇((1 -χ)∇G ξ )(tx + (1 -t)y)| |x||y -x| -d
and gives the contribution:

ˆRd \B 3|x| (x) |((1 -χ)∇G ξ )(x -y) -((1 -χ)∇G ξ )(y)| 2 dy |x| ˆRd \B 3|x| |y| -2d dy 1.
The near-scale behaviour |y -x| < 3|x| (where here, without loss of generality, we may assume that |x| ≥ 1) is treated by the triangle inequality:

|((1 -χ)∇G ξ )(x -y) -((1 -χ)∇G ξ )(y)| (1 -χ)(x -y)|y -x| -d+1 + (1 -χ(y))|y| -d+1
and gives the leading order contribution:

ˆB3|x| (x) |((1-χ)∇G ξ )(x-y)-((1-χ)∇G ξ )(y)| 2 dy ˆB4|x| \B 1 |y| -2(d-1) dy log(2 + |x|) for d = 2, 1 for d = 3.
To summarize, one has

φ ξ = δφ ξ + o(δ), with E[|φ ξ (x) -φ ξ (0)| 2 ] |ξ| 2 log(2 + |x|) for d = 2, 1 for d > 2. (1.1.53)
For the linearized corrector ∂ ∂ξ φ ξ , similar computations can be done and yield the same conclusion. The growth (1.1.53) quantifies the sub-linearity of the corrector in the sense of εφ ξ (

• ε ) → ε↓0 0 and ε ∂ ∂ξ φ ξ ( • ε ) → ε↓0 0.
Once we have understood the sub-linearity of the correctors, one has to find a suitable equation for the homogenization error z ε = u 2sc ε -u ε . For the ease of the reading, we perform the computations for the linear setting (that is, by taking p = 2 in (1.1.49)), which captures all the ideas. For the details in the nonlinear setting we refer to Section 3.6 in Chapter 3. By scaling, we may assume that ε = 1. We start by computing

∇z 1 = ∇u 1 -(∇u + ∂ x i u∇φ e i + φ e i ∇∂ x i u),
where we use Einstein's convention of implicit summation on repeated indices. Then, applying -∇•a together with equation (1.1.4) and the homogenized equation (1.1.14) has the effect of

-∇ • a∇z 1 (1.1.4) = ∇ • f + ∇ • (a(∇φ e i + e i )∂ x i u + φ e i a∇∂ x i u) (1.1.14) = ∇∂ x i u • (a(∇φ e i + e i ) -ae i ) + ∇ • (φ e i a∇∂ x i u).
(1.1.54)

We now rewrite the first right-hand side term of (1.1.54) with help of a new corrector σ = (σ ijk ) i,j,k (called the flux corrector), skew-symmetric in the last two-variables (that is σ ijk = -σ ikj ) and that satisfies

∇ • σ i := ∂ k σ ijk = a(∇φ e i + e i ) -ae i . (1.1.55)
The flux corrector σ i is a d -1-form and (1.1.49) only defines σ i up to gauge invariance. It is standard to fix the gauge by requiring σ i to satisfy (which indeed yields the skew symmetry)

-∆σ ijk = ∂ x j (a(∇φ e i + e i ) • e k ) -∂ x k (a(∇φ e i + e i ) • e j ).
With help of the flux corrector and the chain rule, the first right-hand side of (1.1.54) can be rewritten as

∇∂ x i u • (a(∇φ e i + e i ) -ae i ) = ∂ j ∂ i u∂ k σ ijk = ∂ k (∂ j ∂ i uσ ijk ) -∂ k ∂ j (∂ i u)σ ijk = -∇ • (σ i ∇∂ i u),
where the last line comes from the fact that from the skew-symmetry of (σ ijk ) j,k and the symmetry of

(∂ k ∂ j (∂ i u)) j,k one has ∂ k ∂ j (∂ i u)σ ijk = 0.
To conclude, one has the following equation for the homogenization error (after the rescaling

x x ε ) -∇ • a( • ε )∇z ε = ε∇ • ((a( • ε )φ e i ( x ε ) -σ i ( • ε ))∇∂ i u).
In the regime of small ellipticity contrast, we get similarly the growth (1.1.53) for σ, where the latter corresponds to the first order approximation of σ in δ ↓ 0.

By the energy estimate and (1.1.53), we deduce that the homogenization error satisfies (in the regime δ ↓ 0)

E ˆ|∇z ε | 2 1 2 εE ˆ(|φ( x ε )| 2 + |σ( x ε )| 2 )|∇ 2 u(x)| 2 dx 1 2 (1 + log 1 2 (•)1 d=2 )∇ 2 u L 2 (R d ) × ε log 1 2 (ε -1 ) for d = 2, ε for d > 2.

The general framework of quantitative homogenization

We now present the tools which lead to a quantitative theory of homogenization. As mentioned in Subsection 1.1.5, the quantitative homogenization theory requires to quantify the ergodicity assumption (1.1.9). There are several approaches to encode quantitatively the decorrelation on large-scales.

To begin with, a fundamental example is given by discrete i.i.d coefficients, which means that the coordinate projections a ∈ M → a(x, •) ∈ M 0 (corresponding to the set (1.1.7) with space independent operator), for x ∈ Z d , are independent and identically distributed. The associated probability distribution is given by P = P ⊗Z d 0 , the Z d -fold product of P 0 , for P 0 a probability measure on M 0 . We can consider weaker assumptions as standard nonlinear mixing conditions that we define hereafter. For more details on linear mixing conditions, we refer to [START_REF] Doukhan | Mixing: properties and examples[END_REF].

Definition 1 (Linear mixing conditions). For all Borel subsets U ⊂ R d , we define F U as the σ-algebra generated by the family {a → ´a(x, ξ)ψ

(x)dx|ψ ∈ C ∞ c (U), ξ ∈ R d }.
We say that the probability measure P satisfies:

• A α-mixing condition if there exist C > 0 and α > 0 such that for all U, V ⊂ R d and (A, B) ∈ F U × F V , we have 27 |P(A ∩ B) -P(A)P(B)| ≤ C(1 + dist(U, V)) -α . • A η-range of dependence (for some distance η > 0), if for all Borel subset U, V ⊂ R d , one has dist(U, V) ≥ η ⇒ F U and F V are independent. (1.1.56)
In other words, a| U and a| V are independent as soon as dist(U, V) ≥ η.

The mixing conditions in Definition 1 are known to imply concentration properties and in particular the finite range of dependence (1.1.56) is satisfied by most examples of Section 1.1.2. More precisely, the discrete models (1.1.11) and (1.1.12) have by construction a unit range of dependence, that is they satisfy (1.1.56) for η = 1. The model (1.1.13) has a η-range of dependence provided we choose the kernel m be compactly supported in B η . The quantitative homogenization theory of (1.1.4) within the above linear mixing conditions, either in the linear setting with symmetric coefficients or in the variational setting with a(x, ξ) = ∇ ξ W (with W strongly convex) and p = 2, has been extensively studied by Armstrong, Smart, kuusi, Ferguson, Mourrat in [START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF]5,[START_REF] Armstrong | the additive structure of elliptic homogenization[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF][START_REF] Armstrong | Lipschitz regularity for elliptic equations with random coefficients[END_REF][START_REF] Armstrong | Mesoscopic higher regularity and subadditivity in elliptic homogenization[END_REF] and by Gloria and Otto in [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]. The strongest results are obtained in the case of finite range of dependence (1.1.56) where central limit theorem scalings are obtained. Linear mixing conditions require an involved analysis as they only allow to unravel local cancellations after iteration (see in particular the renormalization procedure in [START_REF] Armstrong | Mesoscopic higher regularity and subadditivity in elliptic homogenization[END_REF] and the notion of approximate locality in [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]). Importantly, such iterations lead to (nearly) optimal stochastic integrability. However, the quantitative homogenization of (1.1.4) under mixing conditions is so far restricted to the case p = 2 and the theory for the full range p ≥ 2 is currently unknown. See the short discussion about this issue in Chapter 6.

We refer to the textbook [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF] for a summarize of the quantitative homogenization theory under finite range of dependence.

An other point of view, which is the one taken in this thesis, is based on functional inequalities in probability like spectral gap or logarithmic Sobolev inequalities which are powerful tools to prove nonlinear concentration properties and central limit theorem scalings, see for instance [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]. Such inequalities are well known to hold in the simple situation of i.i.d random variables, referred to as Efron-Stein' inequality (see [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]Lemma 7]) or also for the Boltzmann-Gibbs distribution dµ = Z -1 µ e -V dx with V strongly convex (see [START_REF] Herm | On extensions of the brunn-minkowski and prékopaleindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF]). In infinite dimension, standard spectral gap and logarithmic Sobolev inequalities depend on a suitable notion of derivative which applies to random variable F = M → R k (depending on the structure of the randomness). Such standard inequalities take the following form.

Definition 2 (Standard functional inequalities). We assume that there exists a suitable notion of derivative x → ∂ x,1 F , which applies to random variables F : M → R k , depending on the structure of the randomness. Morally speaking, ∂ x,1 F "measures" how sensitive F (a) reacts to changes of the value of a(x, ξ) localized at x, for all ξ ∈ R m×d .

We say that the probability measure P satisfies a logarithmic Sobolev inequality (LSI) if there exists ρ > 0 such that for all random variables F : M → R k (for some k ≥ 1) with Ent(F

) := E[F 2 log(F 2 )] -E[F 2 ] log(E[F 2 ]) < +∞, the following inequality holds Ent(F ) ≤ 1 ρ E ˆ|∂ x,1 F | 2 dx . (1.1.57)
We say that the probability measure P satisfies a spectral gap inequality (SG) if there exists ρ > 0 such that for all random variables F :

M → R k with E[|F | 2 ] < +∞, the following inequality holds E[|F -E[F ]| 2 ] 1 2 ≤ 1 ρ E ˆ|∂ x,1 F | 2 dx 1 2
.

(1.1.58)

For a discrete distribution a = (a(x, •)) x∈Z d , we say that the probability distribution P satisfies a spectral gap inequality if Proposition 1. Assume that the probability distribution P satisfies SG (1.1.58). Then, there exists a constant C > 0 such that for all q < +∞ and for all random variables F :

E[|F -E[F ]| 2 ] 1 2 ≤ 1 ρ E x∈Z d |∂ x F | 2 1 2 . ( 1
M → R k with E[|F | q ] < +∞, the following inequality holds E[|F -E[F ]| q ] 1 q ≤ Cq 2 E ˆ|∂ x,1 F | 2 dx q 2 1 q
, where the multiplicative factor Cq

∂ x F = F -E[F |F x ],
with E[•|F x ] refers the conditional expectation where we condition on the σ-algebra F x generated by the family {a(z) := a b(z) : z = x}. The derivative ∂ x F is precisely the L 2 (Ω)-orthogonal projection of F onto the subspace of random variables in L 2 (M) that do not depend on a(x). In other words,

E[•|F

x ] denotes the conditional expectation, where we condition on the event that we know the value of a(z) for all sites z = x.

The Poisson point process based model (1.1.12) satisfies (1.1.57) where the derivative

∂ x,1 := ∂ osc x,1
denotes the oscillation of F = F (A) (taking values in R d ) with respect to the variations of A on B(x), that is formally28 ,

∂ osc x,1 F = sup A :A | R d \B 1 (x) =A| R d \B 1 (x) F (A ) - inf A :A | R d \B 1 (x) =A| R d \B 1 (x) F (A ). (1.1.60)
The Gaussian based model (1.1.13) satisfies (1.1.62) provided that A is Lipschitz and β > d, where the derivative ∂ x,1 := ∂ fct x,1 denotes the functional derivative, defined as the Gâteaux derivative of F (A) with respect to A, that is (we use the short-hand notation A := A(m ξ))

∂ fct x,1 F = sup δA lim sup h↓0 F (A + hδA) -F (A) h , (1.1.61)
where the supremum runs over coefficient fields δA that are compactly supported in B 1 (x) and bounded by

1 in L ∞ (R d ).
Functional inequalities imply a powerful calculus in probability, which is in particular convenient to unravel probabilistic cancellations. Optimal scaling can then be easily captured, but stochastic integrability often remains suboptimal, since functional analysis cannot be iterated nicely (compared to the first approach by mixing conditions). As pointed out in [START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF], standard functional inequalities of Definition 2 are very restrictive: the randomness essentially has to be either Gaussian with integrable covariance as in the model (1.1.13) (for β > d) or to display a product structure as for the models (1.1.11) and (1.1.12). This rules out most models of interest for heterogeneous materials considered in applied sciences in [START_REF] Torquato | Random heterogeneous materials[END_REF], such as random parking process (see [START_REF] Torquato | Random heterogeneous materials[END_REF]Chapter 3]), Voronoi and Delaunay tessellations of a Poisson point set (see [START_REF] Torquato | Random heterogeneous materials[END_REF]Chapter 8]) or Gaussian fields with long-range correlations. A way to relax Definition 2 is to introduce multiscale weighted generalizations of such standard functional inequalities, as introduced by Gloria and Duerinckx in [START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF].

Definition 3 (Multiscale functional inequalities). We say that the probability measure P satisfies a multiscale logarithmic Sobolev inequality (MLSI) if there exist ρ > 0 and an integrable weight π such that for all random variables F : M → R k (for some k ≥ 1) with Ent(F ) < +∞, the following inequality holds

Ent(F ) ≤ 1 ρ E ˆ+∞ 1 -d π( ) ˆ|∂ x, F | 2 dx d , (1.1.62)
where now x → ∂ x, F depends on the multiscale parameter by looking also on how it depends on the ball of size in where the values of x → a(x, •) are localized.

We say that the probability measure P satisfies a multiscale spectral gap inequality (MSG) if there exists an integrable weight π and a constant ρ > 0 for all random variables F :

M → R k (for some k ≥ 1) with E[|F | 2 ] < +∞, the following inequality holds E[|F -E[F ]| 2 ] 1 2 ≤ 1 ρ E ˆ+∞ 1 -d π( ) ˆ|∂ x, F | 2 dx d 1 2
.

( Proposition 2. Assume that the probability distribution P satisfies MSG (1.1.63). Then, there exists a constant C > 0 such that for all q < +∞ and for all random variables F : M → R k with E[|F | q ] < +∞, the following inequality holds

E[|F -E[F ]| q ] 1 q ≤ Cq 2 E ˆ+∞ 1 -d π( ) ˆ|∂ x, F | 2 dx d q 2 1 q
, where the multiplicative factor Cq

∂ fct x, F = sup δA lim sup h↓0 F (A + hδA) -F (A) h , (1.1.64)
where the supremum runs over coefficient fields δA that are compactly supported in B (x) and bounded by 1 in L ∞ (R d ). 29 Voronoi tessellations of a Poisson point set satisfy MLSI (1.1.62) with weight π( ) = Ce

∂ osc x, F = sup A :A | R d \B (x) =A| R d \B (x) F (A ) - inf A :A | R d \B (x) =A| R d \B (x) F (A ).
Random parking process with hardcore radius R = 1 satisfies MLSI (1.1.62) with weight π( ) = Ce -1 C (for some C > 0) and with the oscillation derivative ∂ x, = ∂ osc x, . For more precisions on those examples and for some generalizations, we refer to [START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF]Section 3]. In particular, all the examples of the textbook [START_REF] Torquato | Random heterogeneous materials[END_REF] satisfy a MLSI or a MSG. We also point out that since certain random coefficient fields (as Gaussians) satisfy only one of the assumptions in Definition 1 (mixing conditions) and in Definition 2 and 3 (functional inequalities), it is important to consider both separately.

We now focus on the application of MLSI/LSI/MSG/SG in stochastic homogenization of systems (1.1.4). We recall that the interest of MLSI/LSI/MSG/SG inequalities and what it makes them very powerful is that they entail fluctuations bounds for nonlinear functionals of a. To picture this, it is instructive to apply (1.1.62) to a simple example given by spatial average of a smooth nonlinear transformation of the scalar Gaussian model

A = A(m ξ) ∈ R, for |m(x)| ≤ (1 + |x|) -1 2 (d+β) : F r := ffl Br f (A(x))dx with r > 0 and f ∈ C 1 (R d ) ∩ W 1,∞ (R d ). The functional derivative (1.1.64) is explicitly given by ∂ fct x, F r = |B 1 | -1 ´B (x) f (A)

inequality

´ ´B

(x) f (A)1 Br 2 d ´|f (A)|1 Br 2 in the regime > r, to obtain Ent(F r ) ≤ 1 ρ E ˆ+∞ 1 -d π( ) ˆ |B 1 | -1 ˆB (x) f (A)r -d 1 Br 2 dx d sup |f | 2 r -d ˆr 1 d-β-1 d + ˆ+∞ r -1-β d d,β sup |f | 2 (r -β 1 β<d + r -d log(r)1 β=d + r -d 1 β>d ), (1.1.65)
which gives a convergence rate of (1.1.9) in this special case. For more general results on spatial averages, we refer to [START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF]Section 1.3]. Such estimates are expected to hold for more complex nonlinear maps of the operator a, as the corrector defined in (1.1.23).

Inequalities of type MLSI/LSI/MSG/SG are very rich of applications in stochastic homogenization of (1.1.4) and in particular lead to optimal convergence rate of the homogenization error. To illustrate this, we prove how MSG (1.1.63) give the optimal convergence rate in the one-dimensional example of Section 1.1.4, in the particular scalar Gaussian case

A = A(m ξ), for |m(x)| ≤ (1 + |x|) -1 2 (d+β)
. In view of the analysis conducted in Section 1.1.4, the optimal convergence rate in (1.1.44) will follow from the optimal convergence rate in (1.1.42) and (1.1.43), namely the sub-linearity of φ ξ and ∂ ∂ξ φ ξ . Let us show that (1.1.63) indeed quantify this two convergences. For simplicity, we only treat (1.1.42), (1.1.43) is treated the same way. In the following, we use the short-hand notation

F ε := εφ ξ ( x ε ) = - ε 2 ˆx ε 0 1 A(t) dt + ε ˆx ε 0 Φ -1 (ξ) A(t) + 1 4A 2 (t) dt -ξx, (1.1.66)
where we recall that

Φ(ξ) = -1 2 E[ 1 A ] + E ξ A + 1 4A 2 .
We first compute the functional derivative of F ε in the sense of (1.1.61) and to do so, we fix y ∈ R d , ≥ 1 and a parturbation δA

∈ L ∞ (R d ) compactly supported in [y -, y + ] with δA L ∞ (R d ) ≤ 1. A direct differentiation give d dt F ε (A + tδA) t=0 = ˆε 2 1 [0, x ε ] 1 A 2 - 2AΦ -1 (ξ) + 1 2A 3 Φ -1 (ξ) A + 1 4A 2 δA,
which yields, by the definition of the functional derivative (1.1.64),

∂ fct y, F ε = ˆy+ y- ε 2 1 [0, x ε ] 1 A 2 - 2AΦ -1 (ξ) + 1 2A 3 Φ -1 (ξ) A + 1 4A 2 . Therefore, since E[F ε ] = 0, applying MSG (1.1.63) yields E[F ε | 2 ] 1 2 ≤ 1 ρ E ˆ+∞ 1 -2-β ˆ ˆy+ y- ε 2 1 [0, x ε ] 1 A 2 - 2AΦ -1 (ξ) + 1 2A 3 Φ -1 (ξ) A + 1 4A 2 2 dy d 1 2
1.1. Introduction to the homogenization theory 39

0 < c 1 < A < c 2 , one has 1 A 2 - 2AΦ -1 (ξ)+1 2A 3 Φ -1 (ξ) A + 1 4A 2 c 1 ,c 2 ,α 1 + ξ.
We then split the integral as ´+∞

1 d = ´1 ε 1 d + ´+∞ 1 ε
d and (1.1.67) turns into (using similarly Jensen's and Minkowski's inequality as in (1.1.65))

E[F ε | 2 ] 1 2 ρ (1 + ξ)ε ˆ1 ε 1 + ˆ+∞ 1 ε -2-β ˆ ˆy+ y- 1 [0, x ε ] 2 dy d 1 2 (1 + ξ) √ ε ˆ1 ε 1 -β d + ˆ+∞ 1 ε -1-β d 1 2 (1 + ξ) ×    ε β 2 for β ∈ (0, 1), √ ε log(ε -1 ) for β = 1, √ ε for β > 1.
Similarly,

E[|ε ∂ ∂ξ φ ξ ( x ε )| 2 ]
1 2 enjoys the same bound. To conclude, the convergence rate of the homogenization error is given by: for all q < +∞ E

[ u ε -u-εφ u ( • ε ) 2 W 1,3 ([0,1]) ] 1 2 ρ,c 1 ,c 2 ,α (1+ u L ∞ ([0,1]) ) u L ∞ ([0,1]) ×    ε β 2 for β ∈ (0, 1), √ ε log(ε -1 ) for β = 1, √ ε for β > 1.
Note that in the case of weak correlations β > 1, we recover the central limit theorem scaling √ ε. In higher dimension, the strategy is the same and the whole game is to quantify the sub-linearity of the correctors (for which we have no explicit formula in general).

The quantitative homogenization theory of (1.1.4) under functional inequalities has been intensively studied by Otto, Gloria, Neukamm, Fischer and their collaborators, inspired by an unpublished work by Naddaf and Spencer [START_REF] Naddaf | Estimates on the variance of some homogenization problems[END_REF]. The linear model a(x, ξ) = A(x)ξ is by now well understood, started from the works of Gloria and Otto in [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF] and Gloria, Neukamm and Otto in [START_REF] Gloria | An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF] for discrete equation under the spectral gap inequality (1.1.59). Then, the continuum case has been studied by Gloria and Otto in [84,[START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] for more general coefficient fields which satisfy MLSI/MSG/LSI/SG. For the general (nonlinear) model (1.1.4), quantitative results were so far restricted to the case p = 2 and has been studied in the recent work of Fischer and Neukamm in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] under LSI. The quantitative homogenization of (1.1.4) under LSI, for the full range p ≥ 2, is studied in Chapter 3, however, the case of MLSI is currently out of reach. We summarize the results for the linear setting in the following theorem: Theorem 3 (Quantitative homogenization of (1.1.4)-Linear case). Assume that a(x, ξ) = A(x)ξ and that the probability measure P satisfies a MLSI (1.1.62) or a MSG (1.1.63) inequality with weight π( ) = -β-1 , for some β > 0.

The corrector φ = (φ e i ) i∈ 1,d admits the following growth. There exist a random variable x → C(x) and a constant C > 0 as well as an exponent α > 0 such that sup

x∈R d E[exp( 1 C C α (x))] < +∞ and ˆB(x) |φ| 2 + |σ| 2 1 2 ≤ ˆB φ + ˆB σ + C(x)µ(|x|), (1.1.68) with µ(|x|) :=          (|x| + 1) 1-β 2 for β < 2, log 1 2 (|x| + 2) for β = 2, d > 2 or β > 2 and d = 2, log(|x| + 2) for β = d = 2, 1 for β > 2 and d > 2.
The two-scale expansion

u 2sc ε := u ε + ε d i=1 φ e i ( • ε )∂ i u ε (where u ε (x) := ffl Bε(x) u
) converges quantitatively with the following optimal rate: for all ω ⊂ Ω, there exist a random variable C ω,ε,f , a constant C > 0 as well as an exponent α > 0 such that E[exp( 1 C C α ω,ε,f (x))] < +∞ (uniformly in ε and f ) and

∇u 2sc ε -∇u ε L 2 (ω) ≤ C ω,ε,f εµ(ε -1 ) ˆµ2 |∇f | 2 1 2
.

(1.1.69)

The need of local averages at scale ε of u in Theorem 3 is due to the fact that we assume no regularity of the coefficients at small-scales which imposes that the growth of the correctors (1.1.68) has a no pointwise form. However, the standard De Giorgi-Nash-Moser theory (for m = 1) or the Schauder theory (in the case of systems with A is Hölder continuous), allow us to upgrade (1.1.68) into a pointwise estimate and therefore there is no need to take local average of u in (1.1.69) in those cases. We prove similar results in Chapter 3 for the nonlinear setting (1.1.4) in the full range p ≥ 2 (in dimension 3) under LSI (1.1.57). The quantitative homogenization of (1.1.4) under MLSI is curretly unknown (see Chapter 6 for a short discussion about this issue). We now present the contributions of this thesis in the next section.

Contribution of the thesis

We present in this section the contributions of this thesis.

Questions arising in this thesis

Problem 1. The first part aims for a better understanding of the quantitative homogenization theory of linear elliptic systems (namely, for the model a(x, ξ) = A(x)ξ in (1.1.4)) under MLSI (1.1.62).

We use a parabolic approach to quantitatively control the correctors defined in (1.1.23) and we deduce useful estimates in stochastic homogenization in that case (we provide also a different proof of the results in [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF]). This approach give also convenient tools for the study of numerical methods. This problem is studied in Chapter 2.

Problem 2. The second part aims for a better understanding of the quantitative homogenization theory of the nonlinear system (1.1.4) in the all range p ≥ 2 under LSI (1.1.10). We prove optimal stochastic estimates on the correctors and linearized correctors in the non-perturbative regime as well as an optimal convergence rate of the homogenization error. This work is the result of a collaboration with Antoine Gloria at the Laboratoire Jacques-Louis lions. This problem is studied in Chapter 3.

Problem 3. The third part aims to study an efficient numerical method to compute the homogenized operator (1.1.30), in the particular linear model a(x, ξ) = A(x)ξ, which is characterized by the homogenized matrix, for all e ∈ R d , ae := E[A(0)(∇φ e + e)]. The method that we analyse is the socalled representative volume element method and we prove optimal convergence rates of the method with respect to the size of the box for two approaches of periodization, in the particular setting of Gaussian field. In particular, we extend the so called Price's formula for infinite dimensional Gaussian measures. In particular, the main message of this work is: rather periodize the probability law than the realizations. This work is the result of a collaboration with Marc Josien, Felix Otto and Qiang Xu at the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig. This problem is studied in Chapter 4 (for Price's formula) and Chapter 5 (for the representative volume element method).

List of Preprints

The contributions of this thesis are contained in the two following Preprints and in a work in preparation. In Chapter 2, we investigate the quantitative homogenization of linear elliptic systems with random coefficients, that is for p = 2 and when (1.1.4) takes the particular form of30

-∇ • a( • ε )∇u ε = ∇ • f in R d . (1.2.1)
We derive optimal estimates by following the ideas of the papers [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF][START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF] and [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF]Sec.9] where the authors considered the semigroup associated with the corrector problem (1.1.23) (which, in the linear case, takes the form of -∇ • a(e + ∇φ e ) = 0 in R d , for a given e ∈ R d ), namely the solution u e of the parabolic system

∂ τ u e -∇ • a∇u e = 0 in (0, +∞) × R d , u e (0) = ∇ • a(•)e. (1.2.2)
The relationship between the semigroup u e and the corrector φ e is the following formal integral formula

φ e = ˆ+∞ 0 u e (t, •)dt. (1.2.3)
Indeed, provided we have lim t→+∞ u e (t, •) = 0, integrating the equation (1.2.2) in time yields

-∇ • ae -∇ • a∇ ˆ+∞ 0 u e (t, •)dt = 0,
and implies that ´+∞ 0 u e (t, •)dt is a solution of the corrector equation (1.1.23) so that (1.2.3) follows by uniqueness. Formula (1.2.3) will be made rigorous by showing the optimal decay in time of stochastic moments of u e . This theoretical relationship allow us to transfer optimal estimates on u e into optimal estimates on the corrector φ e . The semigroup u e is also of numerical interest and can be used as a very convenient tool for the study of numerical method for approximating the correctors and the homogenized matrix a. As examples, we present three situations where optimal estimates of u e are used in the context of numerical methods.

(i) First, the study of the representative volume element method, where the method consists of replacing the corrector equation (1.1.23) by an equation posed in a large box Q

L := [-L 2 , L 2 ) d , for L 1, -∇ • a L (∇φ L
e + e) = 0 in Q L , with periodic boundary conditions, for a good choice of periodic realization a L . We refer to [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF] for an analysis via a semigroup approach.

(ii) Second, the semigroup u e has been used more recently in [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] for approximating φ e via exponential regularization, that is we replace the corrector equation (1.1.23) by

u e (T ) -∇ • a(∇φ e,T,R + e) = 0 in Q R , φ e,T,R ≡ 0 on ∂Q R ,
for R 1 and T 1. Optimal estimates on u e are used to control the bias (or the systematic error).

(iii) Third, we can derive optimal convergence rate in the massive term approximation as in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF],

and we propose a proof in Corollary 5 of Chapter 2.

The first result in the vein of this chapter is the optimal decay estimate in time of u e and of its gradient in the case of discrete elliptic equation satisfying SG (1.1.59) proved by Gloria, Neukamm and Otto in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]. In the case of finite range of dependence (1.1.56), similar results are obtained by Gloria and Otto in [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF] and Armstrong, Kuusi and Mourrat in [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF]. Their analysis strongly relies on the fast decay of correlations, and does not treat coefficients with fat tails. The aim of this Chapter is to provide an extension of those results to more correlated coefficient fields such as Gaussian coefficient fields with fat tails. Our quantitative analysis is based on functional inequalities of type MLSI (1.1.62).

The expected optimal time decay of u e can be easily inferred in the small contrast regime, that is, when the coefficient field is given as a perturbation of the identity: a δ = (1 + 1 π arctan(δa))Id, for a small δ 1 and a scalar (as we did in Section 1.1.5). By linearizing around δ ↓ 0, the first order approximation of u e is given by u e = δu + o(δ) where

∂ τ u -∆u = 0 in (0, +∞) × R d , u(0) = 1 π ∇ • a(•)e, (1.2.4)
for which we have an explicit formula. Indeed, using the heat kernel Γ : (T,

x) ∈ R + × R d → 1 (4πT ) d 2 e -|x| 2 4T , we have for all (T, x) ∈ R + × R d u(T, x) = 1 π ˆ∇Γ(T, x -y) • a(y)edy. (1.2.5)
Using a scalar Gaussian model (that is for g a scalar centered Gaussian) a(x) = g(x) and imposing algebraic decay of the correlations in form of |E[g(x)g(y)]| ∼ (1 + |x -y|) -β (for some β > 0), we may express the variance as

E[|u(T, 0)| 2 ] = 1 π 2 ˆˆ∇Γ(T, x) • e ∇Γ(T, y) • e E[g(x)g(y)]dx dy 1 π 2 ˆˆ∇Γ(T, x) • e ∇Γ(T, y) • e (1 + |x -y|) -β dx dy. (1.2.6)
We then distinguish between the two regimes β > d and β ≤ d.

Contribution of the thesis

• In the regime β > d, the application x → (1 + |x|) -β is integrable, so that we may use the Cauchy-Schwarz inequality and bound the variance as:

E[|u(T, 0)| 2 ] 1 π 2 ˆˆ∇Γ(T, x) • e (1 + |x -y|) -β 2 ∇Γ(T, y) • e (1 + |x -y|) -β 2 dx dy ≤ ˆˆ|∇Γ(T, x)| 2 (1 + |x -y|) -β dx dy ˆ|∇Γ(T, x)| 2 dx T -1-d 2 .
This decay is the best that we can expect, since this is the one that we would obtain in the better situation of i.i.d coefficients (for which E[g(x)g(y)] = δ(x -y)).

• In the regime β ≤ d, the decay becomes worse as soon as β gets smaller, and we show this by a more careful domination. We first estimate, for all x ∈ R d , ´∇Γ(T, y) • e (1 + |x -y|) -β dy, that we split into two contributions:

ˆ∇Γ(T, y) • e (1 + |x -y|) -β dy = ˆRd \B √ T (x) ∇Γ(T, y) • e (1 + |x -y|) -β dy + ˆB√ T (x) ∇Γ(T, y) • e (1 + |x -y|) -β dy. The first contribution is directly of order T -β 2 ´|∇Γ(T, x)|dx T -1 2 -β 2 whereas the second contribution, using sup |∇Γ(T, •)| T -1-d
2 and a polar change of coordinates, is of order of

T -1 2 -d 2 ˆ√T 0 r -β r d-1 dr T -1 2 -β 2 if β < d, log(T )T -1 2 -d 2 if β = d.
To conclude, using that ´|∇Γ(T, x)|dx T -1 2 , we obtain from (1.2.6)

E[|u(T, 0)| 2 ] T -1-β 2 if β < d, log(T )T -1-d 2 if β = d.
Similar computations lead to the same decay for E[|∇u(T, 0)| 2 ] with an additional T -1 factor.

Our goal is to extend the previous bound in the non-perturbative regime. Our approach is more in the vein of the series of work [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF][START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF] than in [START_REF] Armstrong | Mesoscopic higher regularity and subadditivity in elliptic homogenization[END_REF][START_REF] Armstrong | The additive structure of elliptic homogenization[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF][START_REF] Armstrong | Lipschitz regularity for elliptic equations with random coefficients[END_REF] (treating the case of finite range of dependence (1.1.56)). More precisely, as in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF], we use a sensitivity calculus and functional inequalities, albeit in the much weaker form of MLSI (1.1.62) introduced in [START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF], as in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF].

As in [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF] however, our starting point is to prove fluctuation bounds on the time dependent flux, for all t ∈ R + ,

q e (t, •) := a(•) ˆt 0 ∇u e (s, •)ds + e , (1.2.7)
after averaging at scales r ≤ √ t. Yet, since functional inequalities cannot be easily iterated, one cannot rely on the same approach as for coefficients with a finite range of dependence. To this aim, as in [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF], we rely on large-scale regularity, this time in the parabolic setting.

The general idea of large-scale regularity is to make use of the nice regularity theory that enjoy the homogenized operator ∂ τ -∇ • a∇. Indeed, the proximity of the two resolvent of the operators ∂ τ -∇ • a∇ and ∂ τ -∇ • a∇ provided by homogenization allows to infer an improvement of regularity for ∂ τ -∇a∇ on large-scales (say, scale much larger than the correlation length or the period in the periodic setting). In other words, on large-scales, the heterogeneous linear parabolic operator ∂ τ -∇•a∇ "inherits" a suitable version of the regularity theory for the homogenized linear parabolic operator ∂ τ -∇ • a∇. In the seminal work [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF], Avellaneda and Lin observed that such "transfer" of regularity is true for periodic coefficient field a, and proved intrinsic C 1,1--a priori estimates on a-harmonic functions, meaning that the estimates are formulated with help of the so-called harmonic coordinates x → e • x + φ ξ (x). Later, such a regularity theory has been extended to the case of linear elliptic operators -∇ • a∇ with stationary and ergodic coefficients in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF], where the large-scales are characterized by a random minimal scale (called in the first reference the minimal radius), for which stretched exponential moments are established under MLSI/MSG (1.1.62) and (1.1.63) in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] as well as under a finite range of dependence (1.1.56) in [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF]. Finally, the theory has been extended to the parabolic case in [START_REF] Bella | A liouville theorem for stationary and ergodic ensembles of parabolic systems[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization and regularity theory of parabolic equations[END_REF] and for completeness we recall the main results and extend them in Appendix 2.4.2.

We obtain (in the non-perturbative regime) optimal decay estimates in time of the semigroup u e , in terms of scaling, both for mildly and strongly correlated coefficient fields, with good stochastic integrability (stretched exponential moments).

Theorem 4 (Main result of Chapter 2). Let T ≥ 1 and e ∈ R d be a unit vector of R d . For an arbitrary map f and r > 0 we denote by f g r its convolution with respect to the Gaussian kernel

g r := r -d e -|•| 2 r 2 .
• For all 1 ≤ r ≤ √ T , the average of the time dependent flux defined in (1.2.7) and the time dependent corrector φ e (T ) := ´T 0 u e (s, •)ds satisfy:

|((q e ) r (T ), ∇(φ e ) r (T )) -E[((q e ) r (T ), ∇(φ e ) r (T ))]| ≤ C ,d,λ,β (r)r -d 2 µ β (T )(1 + log 2 ( √ T r )), with µ β (T ) :=      T d 4 -β 4 if β < d, log 1 2 (T ) if β = d, 1 if β > d.
• We define the exponential kernel η r = r -d e -

+ √ T ˆηR ( y c )|∇u e (T, y)| 2 dy 1 2 ≤ C ,d,λ,β (T )η β (T ),
with for all T ≥ 1

η β (T ) =      T -1 2 -β 4 if β < d, log 1 2 (T )T -1 2 -d 4 if β = d, T -1 2 -d 4 if β > d.

Contribution of the thesis

The random variable X ∈ R + * → C ,d,λ,β (X) depends on d, λ, β and satisfies: for all α <

sup X>0 E exp( 1 C C α ,d,λ,β (X)) ≤ 2.
As applications, the decay of the semigroup gives an alternative proof of the bounds on the correctors recently obtained in [89, Th.1] and yields other results of interest in stochastic homogenization, extending the results of [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF] for this setting.

1.2.4 Quantitative homogenization theory for nonlinear elliptic equations and systems (Chapter 3)

In Chapter 3, we investigate the quantitative homogenization of monotone elliptic systems with random coefficients of the general type (1.1.4). The aim is to make Theorem 2 quantitative, in the all range p ≥ 2, which is so far restricted to the case p = 2.

Whereas for p = 2, the homogenized operator ā (see (1.1.30)) has a good regularity theory provided it is smooth (which is proved in [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF]5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF]), this is not clear in general as soon as p > 2. Indeed, in general, the homogenized monotone map a satisfies the weaker monotonicity condition (1.1.15) than the original one for the operator a in (1.1.5), and presumably the homogenized operator ∇ • a(∇) might not possess C 1,α -regularity even if the monotone operator ∇ • a(•, ∇) does. Since elliptic regularity plays an important role in quantitative homogenization, this raises interesting questions and will impose restrictions on the operators we can consider. We refer to Section 3.1.1 for more precisions.

Quantitative estimates on the two-scale expansion (1.1.31) are derived by the optimal quantification of the sub-linearity of the corrector φ ξ but also of the linearized corrector ∂φ ξ ∂ξ (as we have already seen in the one-dimensional example in Sections 1.1.4 and 1.1.5). It implies that we have to linearise at some point the nonlinear operator, essentially for the two following reasons:

• In order to define the linearized corrector ∂φ ξ ∂ξ , which is formally given by differentiating the corrector equation (1.1.23):

-∇ • Da(•, ξ + ∇φ ξ )(Id + ∂φ ξ ∂ξ ) = 0, (1.2.8)
where we recall that D denotes the derivative of ξ → a(•, ξ).

• In order to control the sub-linearities εφ ξ (

• ε ) → ε↓0 0 and ε ∂φ ξ ∂ξ ( • ε ) → ε↓0
0 via functional inequalities of type LSI (1.1.57), which essentially imposes to linearize the equation of the corrector with respect to the randomness.

When linearizing the nonlinear problem, one obtains the linear operator -∇ • Da(•, ξ + ∇φ ξ )∇, with coefficients that are heterogeneous and depend on the corrector ξ + ∇φ ξ itself. Can we prove perturbative regularity (typically in form of Meyers' estimates) for this operator? In the specific case p = 2 treated in [5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF][START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF], the coefficients of the linearized operator are bounded from above and below, so that Meyers' estimates are standard (which allowed Fischer and Neukamm to essentially follow the linear proof in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF]). This is not the case for p > 2, and this linear equation is hard to handle for two reasons:

• The coefficients may be degenerate (for the p-Laplacian e.g.), and despite recent strong contributions on degenerate models, this degeneracy is currently out of reach. Large-scale regularity has been established for the Laplacian on the percolation cluster by Armstrong and Dario in [START_REF] Armstrong | Elliptic regularity and quantitative homogenization on percolation clusters[END_REF] (and optimal convergence rates by Dario in [START_REF] Dario | Optimal corrector estimates on percolation clusters[END_REF]) and for linear elliptic systems with degenerate and unbounded coefficients under moment bounds assumptions by Bella, Fehrmann and Otto in [START_REF] Bella | A Liouville theorem for elliptic systems with degenerate ergodic coefficients[END_REF]. The main new difficulty we face here is that, unlike in [START_REF] Armstrong | Elliptic regularity and quantitative homogenization on percolation clusters[END_REF][START_REF] Bella | A Liouville theorem for elliptic systems with degenerate ergodic coefficients[END_REF], the degeneracy is not prescribed a priori: it is given by the critical set of harmonic coordinates (that is the set of x ∈ R d such that ∇φ ξ (x) + ξ = 0, cf. (3.1.7)). Precise information on this critical set is however currently unavailable for d > 2 (even for the p-Laplacian, the unique continuation principle is not known to hold for d > 2 and p > 2, e.g. [START_REF] Lindqvist | Notes on the p-Laplace equation[END_REF]).

• The coefficients field Da(•, ξ + ∇φ ξ ) is unbounded. Our sole a priori control is given by the growth exponent p p-2 (the larger p, the weaker the integrability), which can be seen directly from (1.1.5) and (1.1.6): there exists C depending on λ and p such that for all h ∈ R d ,

1 C (s + |ξ + ∇φ ξ | p-2 )|h| 2 ≤ h • Da(•, ξ + ∇φ ξ )h ≤ C(s + |ξ + ∇φ ξ | p-2 )|h| 2 .
The only a priori bound on the coefficients is deduced from (1.1.25) and implies only

E[|Da(0, ξ + ∇φ ξ )| p p-2 ] 1 + |ξ| p , (1.2.9)
which becomes weaker as p becomes larger.

In order to avoid the degeneracy of the the linearized operator, we impose that s > 0 in (1.1.5) and (1.1.6), and therefore -∇ • Da(•, ξ + ∇φ ξ )∇ becomes uniformly elliptic. It rules out the p-Laplacian, but not the p-Laplacian regularized at 0, and only yields the non-degeneracy of the linearized operator in a perturbative way (it disappears in the regime when the solution has a large gradient). Doing so, the remaining difficulty is on the unboundness of the coefficients. By following the standard strategy for Meyers' estimates, passing by Caccioppoli's inequality (in a refined version obtained by optimizing the cut-off function, following an original idea of Bella and Schäffner in [START_REF] Bella | Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations[END_REF]) and a reverse Hölder's inequality (usually obtained from the Poincaré-Sobolev inequality), we prove Meyers type estimates for the operator -∇ • Da(•, ξ + ∇φ ξ )∇ which holds for the all range p ≥ 2 in dimension d = 3 and for p < 2(d-1) d-3 in higher dimension d ≥ 4 (due the lack of integrability in (1.2.9) for large p). For more details on the strategy, we refer to Section 3.2. Combining the Meyer' estimates with functional calculus in form of LSI (1.1.57), we manage to optimally quantify the sub-linearities of the correctors and the linearized correctors which leads to the optimal estimate of the convergence of the homogenization error.

Theorem 5 (Main result of Chapter 3-1). Let ξ, e ∈ R d and ε > 0. There exists two random variables C x,ξ and C ε,u such that

• For all x ∈ R d , | ∂ ∂ξ σ ξ (x)| + |σ ξ (x)| + | ∂ ∂ξ φ ξ (x)| + |φ ξ (x)| C x,ξ ×    1 + |x| for d = 1, log 1 2 (2 + |x|) for d = 2, 1 for d > 2.
• The two-scale expansion (1.1.31) converges to u with the following convergence rate:

∇u ε -∇u 2sc ε L 2 (R d ) ≤ C ε,u ×    √ ε for d = 1, ε log 1 2 (ε -1 ) for d = 2, ε for d > 2.
The two random variables C x,ξ and C ε,u satisfy for some α > 0 and constants c ξ (depending on ξ) and c u (depending on u), the stretched exponential moment bound

sup x∈R d E[c ξ (C x,ξ ) α ] + sup ε>0 E[c u (C ε,u ) α ] ≤ 4.
We also investigate in Chapter 3 whether one can infer more properties on a if we make more assumptions on a. The answer is more subtle than one would expect and we provide two non trivial cases where it can be inferred, one in the periodic setting and another in the random setting.

Theorem 6 (Main result of Chapter 3-2). Let A be a Q-periodic Lipschitz matrix field. For all ξ ∈ R d , denote by ψ ξ ∈ W 1,p per (Q) the unique weak solution of

-∇ • A(x)|∇ψ ξ + ξ| p-2 (∇ψ ξ + ξ) = 0.
Assume that for all ξ ∈ R d , there exists r > 0 such that the r-tubular neighborhood T r (ξ

) = {x + B r | x ∈ C(ξ)} of the critical set C(ξ) = {x ∈ R d | ξ + ∇ψ ξ (x) = 0} is such that R d \ T r (ξ) is a connected set.
Then there exists c > 0 such that ā satisfies the strong monotonicity conditions (1.1.5) and (1.1.6) with λ = c.

The assumptions of the previous Theorem are quite strong. They are satisfied in dimension d = 2 by [2] (which shows that C(ξ) ∩ Q is indeed a finite union of points). For d > 2 this is a widely open problem (somewhat related to unique continuation). For linear equations, this follows from [START_REF] Cheeger | Critical sets of elliptic equations[END_REF]. The theory that we develop in this chapter unfortunately does not apply to probability measure which satisfies MLSI (1.2.8) and we refer to Chapter 6 for a short discussion about this issue.

Price's formula for infinite dimensional Gaussian measures (Chapter 4)

In Chapter 4, we extend the so-called Price's formula for infinite dimensional Gaussian field. Originally introduced by Price in [START_REF] Price | A useful theorem for nonlinear devices having Gaussian inputs[END_REF] (for finite dimensional Gaussian measures), it states, given a family of Gaussian measures • c with covariance c, a general formula for the derivative of F c with respect to c for any smooth random variable F of the realizations.

In finite dimension, since the work of Price, the formula has been extended by McMahon in [START_REF] Mcmahon | An extension of price's theorem[END_REF] and more recently by Voigtlaender in [START_REF] Voigtlaender | A general version of price's theorem[END_REF]. The formula, in its most simple version, is stated as follows. We consider a family of Gaussian measures • C in R N (with N ∈ N) characterized by its covariance matrix (symmetric definite-positive) C ∈ R N ×N and F ∈ C 2 (R N ) with polynomial growth. There holds for all i, j ∈

1, N ∂ ∂C ij F C = 1 2 ∂ ij F C . (1.2.10)
Formula (1.2.10) follows by a direct computation. Indeed, since F has polynomial growth, one has

∂ ∂C ij F C = 1 (2π) d ˆF (x) ∂ ∂C ij exp(- 1 2 x • C -1 x) √ det C -1 dx. (1.2.11)
Then, for all x ∈ R d , we compute

∂ ∂C ij exp(- 1 2 x • C -1 x) √ det C -1 = 1 2 x • C -1 C C ij C -1 x -tr C -1 ∂C C ij exp(- 1 2 x • C -1 x) √ det C -1 = 1 2 tr (C -1 x ⊗ C -1 x -C -1 ) ∂C ∂C ij exp(- 1 2 x • C -1 x) √ det C -1 = 1 2 ∂ ij exp(- 1 2 x • C -1 x) √ det C -1 .
Therefore, by two integration by parts (which are justified since F is of polynomial growth), (1.2.11) turns into

∂ ∂C ij F C = 1 2 ˆ∂ij F (x) 1 (2π) d exp(- 1 2 x • C -1 x) √ det C -1 dx,
which is exactly (1.2.10).

In the infinite dimensional context, the family of measure • c are now Gaussian measures on a Banach space X. We show that a version of formula (1.2.10) holds if we assume an additional structure on X (which is classical in the theory, see [START_REF] Igorevich | Gaussian measures[END_REF]).

1.2.6 Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations (Chapter 5)

In Chapter 5, we continue the investigation of the quantitative homogenization of linear elliptic systems (1.2.1). We analyse the so-called representative volume element method (RVE) that we use to approximate the homogenized matrix given by, for all e ∈ R d , a hom = E[a(0)(∇φ e (0) + e)].

The RVE method. Suppose that the coefficient field a is L-periodic, meaning that a(x + Lk) = a(x) for all x and k ∈ Z d . Given a Cartesian coordinate direction i = 1, • • • , d and denoting by e i the unit vector in the i-th direction, we consider the corrector (up to additive constants) φ

i as the L-periodic solution of -∇ • a(∇φ [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] i + e i ) = 0, (1.2.12)

to which we associate the corresponding homogenized matrix:

āe i := [0,L) d a(∇φ (1)
i + e i ).

(1.2.13)

Observe that the notation ā without reference to the period L is legitimate since (1.2.13) is equivalent to

āe i = lim R↑∞ ffl [0,R) d a(∇φ (1)
i + e i ). In the periodic case, (1.2.13) in fact coincides with the homogenized coefficient a hom . In the random case, we expect that (1.2.13) provides a good approximation of a hom as L ↑ +∞. An important choice in the method is the strategy of periodization of the coefficients. Two strategies can be employed, whence we quantify the convergence for one of them.

Two strategies of periodizing. In order to apply the RVE method in form of (1.2.13), considered as an approximation for a hom , one needs to produce samples a of L-periodic coefficient fields connected to the given ensemble • . The goal of this chapter is to prove that, to procure such L-periodic samples, the strategy of "periodizing the ensemble" leads to a smaller error than the strategy of "periodizing the realizations". The first strategy of "periodizing the realizations" is a direct periodization procedure: Taking a coefficient field a in R d , we restrict it to the box [0, L) d and then periodically extend it. This defines a map a → a L . We then take a L , cf. (1.2.13), as an approximation for a hom . One unfavorable aspect of this strategy is obvious: The pushforward of • under this map a → a L is no longer stationary -an imagined glance at a typical realization would reveal d families of parallel artificial hypersurfaces. Suboptimal convergence rates for this strategy has been establish in [START_REF] Bourgeat | Approximations of effective coefficients in stochastic homogenization[END_REF] and numerical experiments in [START_REF] Schneider | Representative volume elements for matrixinclusion composites-a computational study on periodizing the ensemble[END_REF] is in favour of a bias of order O(L -1 ).

To the best of our knowledge, no formal argument in favour of such a behaviour are established in the literature and such arguments will be added in the work in preparation [START_REF] Clozeau | Bias in the representative volume element method: periodize the ensemble instead of its realizations[END_REF].

The second strategy of "periodizing the ensemble" is more subtle: Given an ensemble • , one construct a "related" stationary ensemble • L of L-periodic fields samples a from • L and takes ā as an approximation. This second strategy was (probably not for the first time) laid out in [79, Remark 5]; we will give the construction of the map •

• L for a specific but relevant class of • given in Assumption 8. This approach obviously capitalizes on the knowledge of the ensemble • and not just of a single realization (a "snapshot"), in the sense of "known unknowns" as opposed to "unknown unknowns". This is in contrast with the numerical analysis on inferring a hom in [START_REF] Mourrat | Efficient methods for the estimation of homogenized coefficients[END_REF], or on constructing effective boundary conditions in [START_REF] Lu | Optimal artificial boundary condition for random elliptic media[END_REF] from a snapshot.

Fluctuations and bias. In this chapter, we are interested in the second strategy and in the study of its bias (also called systematic error): How much do the expected value ā L deviate from a hom , which by qualitative theory is their common limit for L ↑ ∞. We shall prove that

ā L -a hom = O(L -d ), (1.2.14)
see Theorems 9. Here L should be thought of as the (non-dimensional) ratio between the actual period L and a suitably defined correlation length of • set to unity. The quantification of the convergence in L is clearly of practical interest: After a discretization that resolves the correlation length, the number of unknowns of the linear algebra problem (1.2.12) scales with L d for L 1. Numerical experiments confirm the O(L -d ) scaling, see for instance [103].

We note that fluctuations (which are at the origin of the random part of the error), as for instance measured in terms of the square root of the variance, are in many situations proven to be of the order (see, e.g., [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]Th. 2])

|ā -ā L | 2 1 2 L = O(L -d 2 ), (1.2.15)
and the same is expected to hold for the other strategy [63, (3.3)]

|a L -a L | 2 1 2 = O(L -d 2 ).
Hence the variance scales like the inverse of the volume L d of the periodic cell [0, L) d , as if we were averaging over [0, L) d some field of unit range of dependence instead of the long-range correlated a(e i + ∇φ i ). In view of this identical fluctuation scaling for both strategies, the different bias scaling is significant in the most relevant dimension d = 3, which we mostly focus on in this paper: For the first strategy (by periodization of the realizations), the bias dominates, so that taking the empirical mean of a L over many realizations a does not substantially reduce the total error. It does so in the second scenario, which suggests to use variance reduction methods [START_REF] Bris | Special quasirandom structures: a selection approach for stochastic homogenization[END_REF][START_REF] Fischer | The choice of representative volumes in the approximation of effective properties of random materials[END_REF].

Theoretical results on the random error in RVE, at least for the second strategy like in (1.2.15), are by now abundant, starting from [82, Theorem 2.1] for a discrete medium with i. i. d. coefficient, over [START_REF] Gloria | Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization[END_REF]Theorem 1] for a class of continuum media based on the Poisson point process, to the leading-order identification of the variance in [55, Theorem 2]. The last result arises from the characterization of leading-order variances in stochastic homogenization in general, starting from [128, Theorem 2.1] for correctors, and is in the spirit of the general approach laid out in [START_REF] Gu | Scaling limit of fluctuations in stochastic homogenization[END_REF].

Theoretical results on the systematic error in RVE, again for the second strategy as in (1.2.14), seem to have been restricted to the case of a discrete medium with i. i. d. coefficients, see [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]Proposition 3], where the construction of • L is obvious. The argument for [88, Proposition 3] is based on a (necessarily non-stationary) coupling of • and • L , and introduces a massive term into the corrector equation in order to screen the resulting boundary layer, which leads to a logarithmically worse estimate than (1.2.14). Our analysis avoids this coupling and suggests that such a logarithmic correction is artificial. (Incidentally, the phenomenon that the bias decays to an order that is twice the order of the fluctuation decay occurs also in the analysis of the homogenization error (-∇ • a∇) -1 f -(-∇ • a hom ∇) -1 f itself: While the variance can be characterized to order O(L -d ), where L 1 now is the ratio between the scale of f and the correlation length, see [56, Theorem 1], the expectation seems to be characterized to order O(L -2d ), see [START_REF] Duerinckx | A remark on a surprising result by bourgain in homogenization[END_REF].) From a practical point of view, the first strategy seems more appealing since it is less intrusive, because it only requires snapshots (which could be directly obtained from measures). However, it has been realized quite early in the numerical literature (see, e.g., [102]) that it is less accurate than the second strategy, because, as explained above, it suffers from a large bias. Based on experiments on periodic coefficient fields, oversampling [START_REF] Hou | A multiscale finite element method for elliptic problems in composite materials and porous media[END_REF] and filtering [START_REF] Blanc | Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings[END_REF] strategies were first proposed in order to achieve a better accuracy. Unfortunately, their efficiency is limited for random media (they cannot perform better than O(L -1 )). This motivated an alternative approach that screens the boundary effects by adding a massive zero-th order term to the equation (1.2.12) defining the correctors [START_REF] Gloria | Reduction of the resonance error-part 1: Approximation of homogenized coefficients[END_REF], which is nonetheless inspired from theoretical considerations [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF]. This was combined with a sort of domain decomposition in [START_REF] Mourrat | Efficient methods for the estimation of homogenized coefficients[END_REF] and proved to achieve (almost) the best possible accuracy for a given computational cost in the case where only snapshots are available [127, Prop. 1.1 & Th. 1.2], with an error O(δ) at the cost of O(δ -2 ) elementary operations (up to logarithms).

Assumptions and formulation of rigorous result

We now introduce the class of stationary ensembles • of λ-uniform coefficient fields a considered in this paper. Loosely speaking, it consists of regular pointwise transformation, see (1.2.17), of a stationary Gaussian field g with integrable covariance function. The ensemble is determined by the (translation invariant) covariance function c and the pointwise transformation A. Assumption 8 collects the two quantitative hypotheses on c and A needed, see (1.2.16) and (1.2.18), respectively. Assumption 8. Let • be a stationary and centered Gaussian ensemble of scalar31 fields g on R d , as determined by the covariance function c(x) := g(x)g(0) . We assume that there exists an α > 0 such that

sup x∈R (1 + |x|) d+n+2α |∇ n c(x)| < ∞ for n ∈ {0, • • • , d + 1}.
(1.2.16)

We identify • with its push forward under the map

g → a := x → A(g(x)) , (1.2

.17)

where A : R → R d×d is such that the coefficient field a is λ-uniformly elliptic. We assume that

sup g∈R |A (g)| + |A (g)| < ∞. (1.2.18)
The purely technical reason for going beyond first-order derivatives in characterizing a decay of order d + α in (1.2.16) is that it implies the following property on the level of the (automatically non-negative) Fourier transform

0 ≤ Fc(k) (1 + |k|) -d-2α for k ∈ R d ,
which allows us to appeal to [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. However, the class of ensembles • from Assumption 8 is mainly chosen because it allows for a straightforward periodization, namely through periodizing the covariance function. Note that since α > 0, (1.2.16) in particular ensures that the sum in (1.2.19) converges absolutely.

We periodize the ensemble • by periodizing the covariance function c (see (1.2.19) below). This periodization is natural because it formally amounts to considering realizations of the ensemble • conditioned on [0, L) d -periodicity. We sketch a formal argument for it. 32 We may indeed represent the Gaussian field g generated from • by g = m * ξ, where ξ is the standard Gaussian white noise and where m := F -1 √ Fc is the model associated to c. Assuming that F(c) > 0 (since c is a covariance function, we already have

F(c) ≥ 0), we construct the [0, L) d -periodic realizations g L = m ξ L , for the [0, L) d -periodic realization of the white noise ξ L = k∈Z d ξ(•-kL)1 [0,L) d (•-kL).
Computing the covariance function for such realizations ξ yields (1.2.19). (

lim sup L↑∞ L d | ā L -a hom | < ∞.
Note that the assumption that A is symmetric in Theorem 9 is only for convenience and allow us to simplify the use of second order correctors. For more precisions, we refer to Section. 5.2.2. We also give in Theorem 30 a refinement of Theorem 9 by characterizing the leading-order error term.

Chapter 2

Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

In this context, we recall that • satisfies a multiscale logarithmic Sobolev inequality (see Section 1.1.5 for precisions), that we restate here. For all square integrable functional F of a (see [START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF]Th.3.1,(ii)]): there exists ρ > 0 such that Ent(F (a) In this contribution, we specialize to algebraic decay and assume that there exists β > 0 such that for all ∈ [1, +∞) γ( ) = ( + 1) -β .

) := F 2 (a) log(F 2 (a)) -F 2 (a) log(F 2 (a)) ≤ 1 ρ ˆ+∞ 1 -d π( ) ˆ|∂ fct x, F (a)| 2 dx d , ( 2
(2.1.6)

General notation. For U ⊂ R d open and p ∈ [1, +∞],
we denote by L p (U) the Lebesgue space on U with exponent p, that is, the set of measurable functions f :

U → R d satisfying f L p (U ) := ˆU |f (x)| p dx 1 p < +∞,
and where for p = +∞

f L ∞ (U ) := inf{C > 0||f (x)| ≤ C for almost all x ∈ U}.
The vector space of functions on R d which belongs to L p (U) whenever U is bounded is denoted by

L p loc (R d ). If |U| < +∞ and f ∈ L 1 (U), then we write U f (x)dx := 1 |U| ˆU f (x)dx.
For all U, we denote by H 1 (U) the space of all measurable functions f :

U → R d in L 2 (U) such that ∇f is in L 2 (U).
We also define H 1 loc (R d ) the space of functions which belongs to H 1 (U) whenever U is bounded. For all p ∈ [1, +∞), we denote by L p • (Ω) the space of random variables X : Ω → R d satisfying

X p 1 p < +∞.
If B is a Banach space, then for all p ∈ [1, +∞), we denote by L p (R d , B) (resp. L p loc (R d , B)) the space of measurable functions f :

R d → B such that f (•) B ∈ L p (R d ) (resp. f (•) B ∈ L p loc (R d )).
For all time interval I := [t 1 , t 2 ) and open subset U ⊂ R d , we define the function space

H 1 par (I × U) := {u ∈ L 2 (I, H 1 (U))|∂ τ u ∈ L 2 (I, H -1 (U)}.
for correlated coefficient fields and we say that u ∈ H 1 par (I × U) is a weak solution of

∂ τ u -∇ • a∇u = ∇ • f in I × U, u(t 1 ) = ∇ • g, for r.h.s f ∈ L 2 (I × U) d and initial data g ∈ L q (U) d (for some q ∈ [1, +∞]) if for all ψ ∈ C ∞ c (I × U) - ˆU ˆt2 t 1 u(t, x)∂ τ ψ(t, x)dt dx + ˆU ˆt2 t 1 ∇u(t, x) • a(x)∇ψ(t, x)dt dx + ˆU g • ∇ψ(t 1 , x)dt dx = - ˆU ˆt2 t 1 f (t, x)∇ψ(t, x)dt dx.
For all R ≥ 1, we define the exponential kernel η R by

η R := 1 R d e -|•| R ,
and the Gaussian kernel g R by

g R := 1 R d e -|•| 2 R 2 .
For all measurable functions f and all r > 0, we denote by f r the convolution with the Gaussian kernel g r , namely

f r := f g r = ˆf (y)g r (• -y)dy.
We say that a random field

X : Ω × R d → R is stationary if we have for all x ∈ R d X(a, • + x) = X(a(• + x), •) almost surely. (2.1.7)
For all R > 0 and (s, x) ∈ R d+1 , we write B R (x) := {y ∈ R d ||x -y| < R} for the ball of radius R centered at x and C R (s, x) := (s -R 2 , s) × B R (x) for the parabolic cylinder centered at (s, x) and of radius R (for (s, x) = (0, 0), we do not write the dependance on (s, x)). We use the short-hand notation α 1 ,...,αn for ≤ C for a constant C which depends only on the parameters (α i ) i∈ 1,n .

We write for all (a, b) ∈ R 2 , a ∨ b = max{a, b} and a ∧ b = min{a, b}.

Homogenization theory. We denote by φ e (•, a) ∈ H 1 loc (R d ) the corrector, in the direction of a unit vector e of R d , as the unique distributional solution of, for almost all realization of a

     -∇ • a(∇φ e + e) = 0 in R d , with lim sup R→+∞ 1 R B R |φ e (x)| 2 dx 1 2 = 0 and ˆB1 φ e (x)dx = 0. (2.1.8)
For the existence of correctors, we refer to [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]Lem. 1].

For all T ≥ 1, we denote by φ e,T the massive corrector, defined as the Lax-Milgram solution in

H 1 uloc (R d ) := {ψ ∈ H 1 loc (R d )| sup x∈R d ´B1 (x) |ψ| 2 + |∇ψ| 2 < +∞}, to 1 T φ e,T -∇ • a(∇φ e,T + e) = 0 in R d . (2.1.9)
For the existence and uniqueness of the massive correctors, we refer to [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF]. Likewise, we denote by φ * e and φ * e,T the solutions of (2.1.8) and (2.1.9) with a replaced by a * , the transposed field of a. We denote by u e ∈ H 1,par uloc the semigroup associated with the corrector problem (2.1.8) defined as the weak solution of

∂ τ u e -∇ • a∇u e = 0 in (0, +∞) × R d , u e (0) = ∇ • a(•)e, (2.1.10)
with

H 1,par uloc := u ∈ C 0 (R + * , H 1 loc (R d )) sup T >0 sup R≥ √ T B R |(T ∇u(T, x), √ T u(T, x))| 2 dx + ˆT 0 (∇u(s, x), 1 √ T u(s, x))ds 2 dx < +∞ .
For existence and uniqueness of u e , we refer to [92, Lemma 1].

We also introduce the associated fluxes q e := a(∇φ e + e), (2.1.11) for all T ≥ 1 q e,T := a(∇φ e,T + e),

and for all t ≥ 0 q e (t, We introduce the flux corrector σ = (σ i,j,k ) (i,j,k)∈ 1,d 3 as the unique distributional solution of, for almost all realization of a

•) := a(•) ˆt 0 ∇u e (s, •)ds + e , (2.1
         ∇ • σ i = q e i , -∆σ i,j,k = ∂ j (e k • q e i ) -∂ k (e j • q e i ) in R d , with lim sup R→+∞ 1 R B R |σ i (x)| 2 dx 1 2 = 0 and ˆB1 σ i (x)dx = 0, (2.1.15)
where

(∇ • σ i ) j = d k=0 ∂ k σ i,j
,k and ∂ i denotes the partial derivative with respect to the single coordinate x i . For existence and uniqueness, we refer to [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]Lem. 1].

Finally, for all T ≥ 1, we denote by σ T = (σ T,i,j,k ) (i,j,k)∈ 1,d 3 the massive flux corrector, defined as the Lax-Milgram solution in H

1 uloc (R d ) to 1 T σ T,i,j,k -∆σ T,i,j,k = ∂ j (e k • q e i ,T ) -∂ k (e j • q e i ,T ). (2.1.16)
For the existence and uniqueness of the massive flux corrector, we refer to [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF].

The quantities u e , ∇φ e , ∇σ, q e , φ e,T and q e,T are stationary in the sense of (2.1.7), which implies that the distribution of their convolution with some smooth function f , under the stationary ensemble • , does not depend on the space variable. In the following, we do not distinguish between F f (0) and F f in our notation, for all stationary random fields F .

Quantitative results

Our first main result is split in two quantitative estimates on averages of the time dependent flux (2.1.13). First, we show that the fluctuations of (q e ) r (T ) on scale r ∈ [1, √ T ] decays as the central limit theorem scaling r -d 2 times some growth in time which depends on the correlation (in particular, in the case β > d, we get exactly the central limit theorem scaling). Second, we show that the fluctuations of particular averages q e (r 2 ) f r , for all r ≥ 1 and f r which behaves like ´r2 1 ∇g √ s ds, has some growth in r depending on the correlation. The first result is a key estimate to obtain the optimal decay in time of the semigroup u e , whereas the second is needed to get the optimal growth of the correctors state in Corollary 3. We prove those estimates for stretched exponential moments.

Theorem 10 (Fluctuations of averages of the time dependent flux). Let T ≥ 1 and e ∈ R d be a unit vector of R d .

• For all 1 ≤ r ≤ √ T , we have

|((q e ) r (T ), ∇(φ e ) r (T )) -((q e ) r (T ), ∇(φ e ) r (T )) | ≤ C ,d,λ,β (r)r -d 2 µ β (T )(1 + log 2 ( √ T r )), (2.1.17) with µ β (T ) :=      T d 4 -β 4 if β < d, log 1 2 (T ) if β = d, 1 if β > d. (2.1.18)
• For all r ≥ 1 and function

f r ∈ C 1 b (R d ) which satisfies for all x ∈ R d |f r (x)| |x| ˆr2 1 s -1 g √ s (x)ds and |∇f r (x)| |x| 2 ˆr2 1 s -2 g √ s (x)ds, (2.1.19)
we have

|(q e (r 2 ), ∇(φ e )(r 2 )) f r -(q e (r 2 ), ∇(φ e )(r 2 )) f r | ≤ C ,d,λ,β (r)χ d,β (r), (2.1.20)
with Theorem 10 implies the following optimal decay in time of the semigroup u e (defined in (2.1.10)) and of its gradient. This result is in the spirit of [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]Th.1] established in the discrete setting and extends [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]Cor.4] and [8, Th.9.1] established in the case where the coefficients are randomly distributed according to a stationary ensemble of finite range of dependence to the Gaussian setting. with for all T ≥ 1

χ d,β (r) :=          (r + 1)
η β (T ) =      T -1 2 -β 4 if β < d, log 1 2 (T )T -1 2 -d 4 if β = d, T -1 2 -d 4 if β > d,
T >0 exp( 1 C C α ,d,λ,β (T )) ≤ 2.
In particular for all 1. Let us first comment on the scalings. The time decay η β (T ) of Corollary 1 is optimal, is the sense that it is the optimal scaling in the case of the small contrast regime (see the computations done in Section 1.2.3). We now comment the scalings in Theorem 10. First, the decay r -d 2 µ d (T ) in (2.1.17) is also optimal in the sense that it is satisfied in the case of the small contrast regime (meaning that we consider a scalar model a ∈ R with a δ = (1 + 1 π arctan(πδa))Id, for δ 1, using the notations of Section 1.2). Indeed, it may be verified by an explicit computation. We fix 1 ≤ r ≤ √ T . By linearizing around δ ↓ 0, the first order approximation of the time depend flux is given by q e ≈ δq where for all T ∈ R + q(T,

x ∈ R d |∇u e (T, x)| 2 1 2 d,λ T -1 2 η β (T ). ( 2
•) := ˆT 0 ∇u(s, •)ds + e, (2.1.26)
and where we recall that u is defined in (1.2.4). Using the fact that there exists a constant C depending on d such that Γ(T, •) = Cg √ T and the semigroup property, for all s ∈ [0, T ), g √ s g r = g √ s+r 2 as well as (1.2.5) and (2.1.26), we deduce

q r (T, •) (1.2.5),(2.1.26) = C ˆT 0 ∇(∇g √ s a(•)e g r )ds + e g r = C ˆT 0 ∇(∇g √ s+r 2 a(•)e)ds + e g r (1.2.5) = ˆT 0 ∇u(s + r 2 , •)ds + e g r .
(2.1.27)

Consequently, using the moment bounds shown in Section 1.2.3 of ∇u and the fact that r ≤ √ T , we get (noticing that ∇u = 0)

|q r (T ) -q r (T ) | 2 1 2 = ˆT 0 ∇u(s + r 2 , 0)ds 2 1 2 ≤ ˆT 0 |∇u(s + r 2 , 0)| 2 1 2 ds d,β ˆT 0 (s + r 2 ) -1 2 η β (s + r 2 )ds d,β r -d 2 µ β (T ),
Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields where µ β (T ) is defined in (2.1.18).

Second, the scaling χ d,β (r) in (2.1.20) is optimal, except for β < 2 and d = 2. Indeed, it may be verified by an explicit computation as previously assuming for simplicity the more particular form f r = ´r2 1 ∇g √ τ dτ (which satisfies (2.1.19)). We may argue as in (2.1.27) to get q(r 2 ) f r -q(r 2 ) f r = ´r2

1 ´r2 0 ∇ 2 u(s + τ, •)ds dτ .
We then argue differently between the regimes of β and d. and thus the logarithmic contribution in (2.1.20) in this case is not optimal.

In the regimes β > 2, d > 2 or β < 2, d ≥ 2, we directly estimate |q(r 2 ) f r -q(r 2 ) f r | 2 1 2 = ˆr2 1 ˆr2 0 ∇ 2 u(s + τ, 0)ds dτ 2 1 2 ≤ ˆr2 1 ˆr2 0 ∇ 2 u(s + τ, 0)|
In the regimes β > 2, d = 2, β = 2, d > 2 and β = d = 2, this is more subtle and we have to expand the variance in order to capture the right behaviour in r. Assuming that a = g, c(x -y) := g(x)g(y) = (1 + |x -y|) -β and applying ∇ 2 to (1.2.5), we may express the variance as (we implicitly sum on repeated indices)

|q(r 2 ) f r -q(r 2 ) f r | 2 = ˆˆˆr2 1 ˆr2 0 ∂ ijk Γ(s + τ,
(|x| + 1) -1 (1 + 1 d=2 log( |x| 2 )) + r(|x| + 1) -2 , (2.1.30)
such a way that the far-field contribution is of order of, after a polar change of coordinates and using (2.1.29)

ˆRd \Br γ k (x) ˆγm (y)(1 + |x -y|) -2 dy dx r ˆ+∞ r (1 + ρ) -d-1 (1 + 1 d=2 log(ρ))ρ d-1 dρ ∼ 1 + 1 d=2 log(r + 2).
For the near-field contribution, we proceed the same way as in (2.1.30), expect for the contribution in ´Rd \B |x| 2 (x) that we estimate by a dyadic decomposition

ˆRd \B |x| 2 (x) γ m (y)(1 + |x -y|) -2 dy ≤ +∞ n= log( |x| 2 ) 2 -2n ˆB2 n+1 (x) \B 2 n (x) |γ m | ≤ +∞ n= log( x 2 ) 2 -2n ˆB2 n+1 +|x| |γ m | (2.1.29) +∞ n= log( x 2 )
2 -2n ˆB2 n+1 +|x| (|y| + 1) -d+1 dy

(|x| + 1) -1 ,
so that the near-field contribution is of order of, after a polar change of coordinates and using (2.1.29)

ˆBr γ k (x) ˆγm (y)(1 + |x -y|) -2 dy dx ˆr 0 (ρ + 1) -d (1 + 1 d=2 log(ρ))ρ d-1 dρ ∼ log(r + 2)(1 + 1 d=2 log(r + 2)).
2. Due to the computations done above, the logarithm correction in (2.1.17) is not optimal. In fact, this correction is here for technical reasons and mostly a consequence of the logarithm contribution in (2.2.39). However, in practice, it has no consequences in the proof of the optimal decay in time of ∇u e and also in the proof to obtain the optimal growth of the correctors and their gradients (see Corollary 2 and 3) for which only the regime T ∼ r 2 is needed.

Theorem 10 and Corollary 1 imply the following four results that are of interest in stochastic homogenization. The first one yields bounds on the gradient and flux of the extended corrector (φ e , σ), as well as the massive corrector (φ e,T , σ T ), which gives an alternative proof of [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF]Th.1]

sup r>0 exp( 1 C C α ,d,λ,β (r)) ≤ 2,
and for all r ≥ 1

π (r) =    r β if β < d, r d log -1 (r) if β = d, r d if β > d.
In particular, for β > d, the quantities decay as the central limit theorem scaling r -d 2 .

Corollary 2 combined with Theorem 10 implies the following growth on the extended corrector (φ e , σ).

Corollary 3. We have for all unit vector e ∈ R d and From Corollary 3, we obtain the following quantitative convergence of the two-scale expansion.

x ∈ R d (|(φ e , σ) -(φ e , σ) 1 (0)| 2 ) 1 2 1 (x) ≤ C ,d,λ,β (x)ξ d,β (|x|), (2.1.33) with for all x ∈ R d ξ d,β (|x|) :=          (|x| + 1) 1-β 2 for β < 2, log 1 2 (|x| + 2) for β = 2, d > 2 or β > 2, d = 2, log(|x| + 2) for β = d = 2, 1 for β > 2, d > 2, ( 2
Corollary 4 (Convergence rate of the two-scale expansion).

Let g ∈ H 1 (R d ) such that ξ d,β (| • |)∇g ∈ L 2 (R d )
, and for all ε > 0 let v ε and v hom be the Lax-Milgram solutions, in

Ḣ1 (R d ) := {v ∈ H 1 loc (R d )|∇v ∈ L 2 (R d )}/R, of -∇ • a( • ε )∇v ε = ∇ • g and -∇ • a hom ∇v = ∇ • g,
with, for all e ∈ R d a hom = a(∇φ e + e) .

(2.1.35)

Consider the two-scale expansion error

z ε := v ε -(v hom,ε + ε d i=1 φ e i ( • ε )∂ i v hom,ε ),
where v hom,ε is a simple moving average of v hom at scale ε, that is v hom,ε = (v hom ) ε (0). Then

ˆ|∇z ε (x)| 2 dx 1 2 ≤ C ,d,λ,β,g (ε)εξ d,β (ε -1 ) ˆξ2 d,β (|x|)|∇g(x)| 2 dx 1 2
, where ξ d,β is defined in (2.1.34) and for some random variable ε ∈ R + * → C ,d,λ,β,g (ε) which depends on d, λ, β, g and satisfies: for all α < , there exists some constant C depending on d, λ, β, g and α such that

sup ε∈R + * exp( 1 C C α ,d,λ,β,g (ε)) ≤ 2.
Remark 3. The need for local averages at scale ε of v hom is due to the fact that the corrector estimate (2.1.34) only holds for average of (φ e , σ) under minimal regularity assumption on a. However, from De Giorgi-Nash-Moser theory in the case of equations and from classical Schauder theory in the case of systems with Hölder continuous coefficient field a, we may improve the estimate (2.1.34) into a pointwise estimate. Therefore, in both cases, there is no need to consider local averages of v hom at scales ε.

For a proof of Corollary 4 based on the results of Corollary 3, we refer the reader to [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]. The second consequence of Corollary 1 is a new optimal control of the sub-systematic error, extending the bound obtained in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]Lem.8] in the case of discrete elliptic equations and the one in [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]Th.3] for a finite range of dependence. This corollary is of numerical interest for approximating the homogenized matrix a hom defined in (2.1.35).

Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields Corollary 5 (Sub-systematic error). Let (φ e i ,T ) i∈ 1,d be defined in (2.1.9). For all (i, n) ∈ 1, d ×N, we define the Richardson extrapolation of φ e i ,T with respect to T by

φ n+1 e i ,T = 1 2 n -1 (2 n φ n e i ,2T -φ n e i ,T
) for all n ≥ 1, φ 1 e i ,T = φ e i ,T , and likewise for φ * e i ,T . We define the approximation (a n T ) n∈N of the homogenized coefficients a hom by: for all (i, j, n) ∈ 1, d 2 × N e j • a n T e i = (∇φ * ,n e j ,T + e j ) • a(∇φ n e i ,T + e i ) .

(2.1.36)

We have the following estimates of the sub-systematic errors: for all d ≥ 2 and n > β∨d

4 |∇φ n e i ,T -∇φ e i | 2 1 2 T 1 2 η β (T ), (2.1.37)
and

|a n T -a hom | T η 2 β (T ), (2.1.38)
where η β is as in (2.1.24)

Finally, Corollary 5 implies the following bound on the bottom of the spectrum of -∇ • a∇ projected on ∇•a(0)e and extends [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]Corollary 5], [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]Corollary 1] to correlated fields. Let us recall that stationarity allows us to define a differential calculus in probability through the correspondence for stationary fields: for all stationary fields ψ : Ω × R d → R we define for all i ∈ 1, d :

D i ψ(0) = lim h→0 ψ(a(• + he i ), 0) -ψ(a, 0) h = lim h→0 ψ(a, he i ) -ψ(a, 0) h = ∇ i ψ(a, 0),
and we set Dψ := (D i ψ(0)) i∈ 1,d . We define the Hilbert space

H 1 := {ψ ∈ L 2 • (Ω)| |Dψ| 2 < +∞}.
In the case when the coefficients a are symmetric, the operator L := -D • a(0)D defines a quadratic form on H 1 . We denote by L its Friedrichs extension on L 2

• (Ω). Since L is a self-adjoint nonnegative operator, by the spectral theorem it admits a spectral resolution: for all Θ ∈ L 2

• (Ω), there exists a unique measure ν Θ such that for all g ∈ L ∞ (R + )

g(L)Θ, Θ = ˆ+∞ 0 g(λ)dν Θ (λ).
(2.1.39)

Corollary 6 (Spectral resolution). Let assume that the map A defined in (2.1.1) takes values in the set of symmetric matrices and assume that Θ := D • a(0)e is in L 2 • (Ω) for some unit vector e ∈ R d . We denote by ν Θ the spectral measure, defined in (2.1.39), of the operator -D • a(0) • D associated to the vector Θ. We have for all 0 < µ ≤ 1

ˆµ 0 dν Θ (ζ) η 2 β (µ -1 ),
where η β is as in (2.1.24).

2.2. Structure of the proof 65

Extension to other model of coefficient field

The approach we develop here is not limited to the Gaussian setting. For coefficient field a for which the law satisfies multiscale functional inequalities with oscillation, similar result to the ones presented in this paper hold. More precisely, assume that there exists ρ > 0 such that for all square integrable functional F of a, we have

Ent(F (a)) ≤ 1 ρ ˆ+∞ 1 d π( ) ˆ|∂ osc x, F (a)| 2 dx d , (2.1.40)
with, for some C > 0 and

β > 0 π( ) = e -1 C β , (2.1.41)
and for all

(x, ) ∈ R d × [1, +∞) ∂ osc x, F (a) := sup{F (a ) -F (a )|a = a = a on R d \B (x)}. (2.1.42)
Then, with the notations

µ β (T ) = 1, η β (T ) = T -1 2 -d 4 , π (r) = r d and ξ d,β (|x|) = log 1 2 (|x| + 2) if d = 2 and ξ d,β (|x|) = 1 if d ≥ 3
, the results of section 2.1.2 hold with a random variable C (possibly depending on d, λ, β, g, x, r and T ) with stretched exponential moments for some exponent α (depending on d and β) uniform in x, r and T when it depends on this parameters.

Multiscale logarithm Sobolev inequality of type (2.1.40) are satisfied, for instance, by random inclusions with random radii and random tessellations of Poisson points or the random parking measure. For more precise details, we refer to [START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF] and Section 1.1.5. For completeness and to see the differences compared to the Gaussian setting, we provide in Appendix 2.4.4, a proof of Theorem 10 under the assumption (2.1.40), when u e is real valued and a satisfy a regularity assumption. The proofs of the general case may be extended by following the arguments of Appendix 2.4.4 and Section 2.3.

Structure of the proof

Let us now describe the strategy of the proof of Theorem 10, together with a flow of auxiliary results. In the rest of the paper, for notational convenience, we do not write the dependance of all quantities on the unit vector e, fixed once for all.

Strategy of the proof

General strategy of the proof. The proof uses on two important quantities:

for all t ∈ R + * ∀r ≥ √ t, Q r 1 := √ t ˆη√ 2r (y)|∇u(t, y)| 2 dy 1 2 , (2.2.1) and ∀r ≤ √ t, Q r 2 (t) : y ∈ R d → q r (t, y) -q r (t, y) , (2.2.2)
and their relationship. On the one hand, using the estimate from [92, Lemma 1], we have a deterministic relationship between (2.2.1) and averages in space and in r of (2.2.2), recalled in Lemma 3.

On the other hand, using sensitivity estimates (see Lemma 7 and Proposition 3) and the multiscale logarithmic Sobolev inequality (2.1.4), we can control moments of (2.2.2) by moments of (2.2.

1). for correlated coefficient fields

The main difficulty is that the estimates are coupled in an intricate way, which does not allow to buckle easily. We handle this difficulty by, first deriving nearly-optimal estimates in scaling in r, t of moments of (2.2.2) from a sub-optimal deterministic bound in t of (2.2.1), which is itself based on deterministic energy estimates (see Lemma 1). Second, from the nearly-optimal moment bounds of (2.2.2), we deduce a better decay in t of moments of (2.2.1) than the one provided by energy estimates, which allow us to deduce the optimal scaling in r, t of (2.2.2), which leads to Theorem 10. We then finally obtain the optimal decay in time of moments of (2.2.1), which leads to Corollary 1. Our main effort is to derive the sensitivity estimates and the control of moments of (2.2.2). We focus in the following on the main ideas of the proof of (2.1.17). For (2.1.20), the ideas are very closed and we only say few words on the differences at the end of this section Sensitivity estimates. The proof of the sensitivity estimates combined two different types of arguments.

1. Deterministic arguments. There are two main ingredients. The first ingredient is the classical L 2 theory of parabolic systems in form of localized energy type estimates, see Lemmas 1 and 2. The second ingredient is the large-scale regularity theory for parabolic systems developed in [START_REF] Bella | A liouville theorem for stationary and ergodic ensembles of parabolic systems[END_REF] that we recall and extend in Appendix 2.4.2. It provides, in particular, a large-scale C 0,1 estimate: for all x ∈ R d , there exists a stationary random variable r * (x) ≥ 1 such that for all t ∈ R and weak solution v of, for

R ≥ r * (x), ∂ τ v -∇ • a∇v = ∇ • g in C R (t, x), we have C r * (x) (t,x) |∇v(s, y)| 2 ds dy d,λ C R (t,x) |∇v(s, y)| 2 ds dy + sup ρ∈[r * (x),R] R ρ 2α Cρ g(s, y) - Cρ(t,x)
g(s , z)ds dz 2 ds dy.

(2.2.3)

This properties can be used provided r * has good moment bounds, which have already been established in [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] in our context.

2. Stochastic arguments. Moment bounds on q r (T ) -q r (T ) will be obtained from the multiscale logarithmic Sobolev inequality (2.1.4), and more precisely in the version of Proposition 4 in order to control high moments: for all p ∈ [1, +∞), T ≥ 1 and r ∈ (0,

√ T ], |q r (T ) -q r (T ) | p 1 p √ p ˆ+∞ 1 -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p . ( 2

.2.4)

To use (2.2.4), we have to estimate the sensitivity of q r (T ) with respect to the coefficient field a, namely the quantity

(x, ) ∈ R d × [1, +∞) → ´|∂ fct x, q r (T )| 2 dx d .
The method used here is inspired by the series of articles [START_REF] Fischer | Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations[END_REF][START_REF] Gloria | An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] which treats the case of elliptic systems and proceeds by duality. The results are summarized in Lemma 7 for the computation of the functional derivative and in Proposition 3 for the sensitivity calculus and the control of moments.

The localized energy estimates of the deterministic part are classical and rely only on L 2 theory for parabolic systems. The contribution of this paper is more on the stochastic part. Let us now describe the main ideas of the proof of (2.1.17) in Theorem 10. For simplicity, we do this in a simpler case with two additional assumptions:

(i) We assume that u is real-valued and a is symmetric. In that case, we get the uniform bound of ∇u:

for all t > 0 ∇u(t, •) L ∞ (R d ) d,λ t -1 , (2.2.5)
cf Remark 4 below for a proof. The bound (2.2.5) is our starting point to prove the sensitivity estimate.

(ii) We assume that

∇ • ae ∈ L ∞ (R d ) and ∇ • ae L ∞ (R d ) ≤ 1. (2.2.6)
This assumption allow us to avoid the singularity at t = 0 and to use the localized energy estimate in the form: for all

T ≥ 1, R ≥ √ T and z ∈ R d ˆT 0 B R (z) |∇u(t, x)| 2 dx dt d,λ 1, (2.2.7)
see for instance [92, Lemma 3] for a proof.

We proceed in three steps.

I) The first step identifies the functional derivative of each components

k ∈ 1, d , (x, ) ∈ R d × [1, +∞) → ∂ fct x, q r (T ) • e k , defined in (2.1.5) for T ≥ 1 and r ≤ √ T . Formally, we have for all (x, ) ∈ R d × [1, +∞), ∂ fct x, q r (T ) • e k = ˆB (x) ∂ ∂a(y) q r (T ) • e k dy,
with, for all y ∈ B (x), using that ∂ ∂a(y) a = δ y and the chain rule

∂ ∂a(y) q r (T )•e k = g r (y)e k •e+g r (y)e k • ˆT 0 ∇u(t, y)dt+ ˆgr (z)e k •a(z) ˆT 0 ∇ ∂ ∂a(y) u(t, z)dt dz, (2.2.8) with from (2.1.10) ∂ τ ∂ ∂a(y) u -∇ • a∇ ∂ ∂a(y) u = ∇ • δ y ∇u in (0, +∞) × R d , ∂ ∂a(y) u(0) = ∇ • δ y e.
(2.2.9)

The first two r.h.s terms of (2.2.8) are directly controlled via (2.2.5) and (2.2.7), whereas the control of the last term is more technical. The idea is to rewrite this term by duality, in form of, for all k ∈ 1, d

ˆgr (z)e k • a(z) ˆT 0 ∇ ∂ ∂a(y) u(t, z)dt dz = ∇v T k (0, y) ⊗ e + ˆT 0 ∇u(t, y) ⊗ ∇v T k (t, y)dt, (2.2
.10) where v T = (v T k ) k∈ 1,d solves the corresponding dual problem of (2.1.10) with final time T , that is the backward parabolic system

∂ τ v T k + ∇ • a * ∇v T k = ∇ • ag r e k on (-∞, T ) × R d , v T k (T ) = 0.
(2.2.11)

Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

Consequently, the crucial terms of ∂ fct x, q r (T ) • e k are M k (T, x, ) := ˆB (x) ∇v T k (0, y) ⊗ e dy + ˆB (x) ˆT 0 ∇u(t, y) ⊗ ∇v T k (t, y)dt dy. (2.2.12)
This decomposition is summarized in Lemma 7.

II) In the second step, we deduce the control of moments of q r (T ) -q r (T ) from the formula of

(x, ) ∈ R d × [1, +∞) → ∂ fct x, q r (T ), for T ≥ 1 and 1 ≤ r ≤ √
T , and the estimate (2.2.4). More precisely, at this stage, we are only able to reach a sub-optimal bound, with a log(T ) correction in (2.1.17). This additional contribution is due to the, purely deterministic, sub-optimal uniform bound (2.2.5), as we will clearly see in the computations below. We provide the idea of the control of the main term (2.2.12), for all ≥ 1 and k = 1. In the following, we write M and v T for M 1 and v T 1 respectively. As in Remark 1, we have to distinguish between the two regimes < √ T and ≥ √ T .

Regime < √ T . We start with the first r.h.s term of (2.2.12). We use the plain energy estimate: for all t ≤ T ˆ|∇v

T (t, y)| 2 dy d,λ ˆg2 r (y)dy r -d , (2.2.13)
applied for t = 0 to get, using in addition the Jensen inequality and the estimate

´´B (x) dx d d ´, ˆ ˆB (x) |∇v T k (0, y) ⊗ e|dy 2 dx d,λ 2d r -d . (2.2.14)
We now turn to the estimate of the second r.h.s term of (2.2.12). We start by splitting the time integral into the two contributions in (0, 1) and [1, T ]. For the contribution in (0, 1), we make use of the Cauchy-Schwarz inequality, the energy estimate (2.2.7), the estimate ´´B (x) dx d d ´and the plain energy estimate (2.2.13) to obtain ˆ ˆB

(x) ˆ1 0 ∇u(t, y) ⊗ ∇v T (t, y)dt dy 2 dx ≤ ˆˆ1 0 ˆB (x) |∇u(t, y)| 2 dy dt (2.2.15) × ˆ1 0 ˆB (x) |∇v T (t, y)| 2 dy dt dx (2.2.7) d,λ d ˆˆ1 0 ˆB (x) |∇v T (t, y)| 2 dy dt dx (2.2.13) d,λ 2d r -d . (2.2.16)
For the contribution in [1, T ], we make use of the Jensen inequality, the estimate ´´B (x) dx d d ´, the deterministic uniform bound (2.2.5) and the Minkowski inequality in L 2 (R d ) as well as the plain energy estimate (2.2.13) to obtain for all ≥ 1 ˆ ˆB

(x) ˆT 1 ∇u(t, y) ⊗ ∇v T (t, y)dt dy 2 dx (2.2.5) d,λ 2d ˆ ˆT 1 t -1 |∇v T (t, x)|dt 2 dx ≤ ˆT 1 t -1 ˆ|∇v T (t, x)| 2 dx 1 2 dt 2 d,λ log 2 (T )r -d . ( 2
(1 + log 2 (T ))r -d ˆ√T 1 d-1-β d β (1 + log 2 (T ))r -d µ 2 β (T ), where µ β (T ) is defined in (2.1.18). Regime ≥ √ T .
Here, the bound (2.2.18) is of no use since → -d π( ) ´M2 (T, x, )dx needs to be integrable at infinity. This is why we treat this regime a different way. We start with the first r.h.s term of (2.2.12). This term is more subtle to control in this regime, even with the two additional assumptions (2.2.5) and (2.2.6). We present here the argument in the homogeneous case a = Id. In that case, we may express ∇v T in terms of the Duhamel formula: for all

(t, x) ∈ (-∞, T ) × R d ∇v T (t, x) = - ˆT -t 0 ∇ 2 Γ(s, •) g r (x)ds,
where we recall that Γ denotes the heat kernel. Then, using the Minkowski inequality in L 2 (R d ) and noticing that for all s ∈ R + * , Γ(s, •) = Cg √ s for some C > 0 and using the semigroup property g √ s g r = g √ s+r 2

ˆ ˆB

(x) |∇v T (0, y) ⊗ e|dy 2 dx d d ˆ|∇v T (0, y)|dy 2 = C d ˆ ˆT 0 ∇ 2 g √ s+r 2 (y)ds dy 2 d d ˆT 0 (s + r 2 ) -1 ds 2 d d log 2 (1 + T r 2 ).
In the heterogeneous case, we replace the use of the Green's function by appealing to largescale regularity, in form of estimate (2.2.3), to get a pointwise bound of local average of ∇v T , see Lemma 4, and we get, for all t ∈ (-∞, T ] ˆ ˆB

(x) |∇v T (t, y)| dy 2 dx ≤ C d log 2 (1 + r ), (2.2.19)
with for all p ≥ 1, C p 1 p d p d+2 β∧d . We refer to the estimate of the first l.h.s term of (2.3.79) for more precisions. We now turn to the estimate of the second r.h.s term of (2.2.12). As before, we split the time integral into the two contributions (0, 1) and [1, T ]. For the contribution in (0, 1), we make use of the Minkowski inequality in L

C d log 2 (T ) log 2 (1 + r ). (2.2.22)
The combination of (2.2.19) applied with t = 0, (2.2.21) and (2.2.22) yields, using that r ≤ √ T in the last line

ˆ+∞ √ T -d π( ) ˆM2 (T, x, )dx d log 2 (T ) ˆ+∞ √ T -1-β (1 + C log 2 ( r ))d β log 2 (T )T -β 2 (1 + C log 2 ( √ T r )) β log 2 (T )r -d µ 2 β (T )(1 + C log 2 ( √ T r )).
Let us now talk about the main difficulties and changes which occur in the general case, that is when we do not assume (2.2.5) and (2.2.6).

(i) When the assumption (2.2.6) is not satisfied, u is now singular at t = 0 and thus the second r.h.s term of (2.2.12) is not well defined in the Lebesgue sense. In order to handle this singular part, we have to treat a different way the contribution in (0, 1) of the time integral of the third r.h.s term of (2.2.8). This is done by using the localized energy estimates directly on the equation (2.2.9). As a consequence, we do not obtain an explicit formula for

(x, ) ∈ R d × [1, +∞) → ∂ fct
x, q r (T ) but rather a bound, see Lemmas 7 and 8. Note that the bound get much simpler in the case of fast decay of correlations, namely for β > d, where the case < √ T has to be considered (and so the contribution G r, does not appear). (ii) When u is vector-valued, (2.2.5) fails and has to be replaced by: for all R ≥ √ t

B R |∇u(t, y)| 2 dy t -2 . (2.2.23)
This estimate is however not sufficient for our propose since we see in (2.2.12) that we need to bound the average of ∇u over all balls B , for ∈ [1, +∞). We have to appeal to large-scale regularity theory in form of estimate (2.2.3) to obtain the improvement

B (x) |∇u(t, y)| 2 dy r * (x) ∨ 1 d 1 < √ t + 1 ≥ √ t t -2 . (2.2.24)
Equipped with (2.2.24), we may control the second r.h.s term of (2.2.12) as in the scalar case. The only main change is that we cannot use the plain energy estimate (2.2.13) for the defining equation (2.2.11) as we did in (2.2.17). Instead, we prove a new lemma which states a pointwise bound (depending on the form of the r.h.s of (2.2.11)) on ffl

B r * (x) |∇v T (t, y)| 2 dy for all x ∈ R d and √ T -t ≥ r * (x), see Lemma 4.
The sub-optimal estimate of moments of q r (T ) -q r (T ) is summarized in Proposition 3. III) In the final step, we remove the log(T ) contribution which appears in the previous step. To this aim, we need a little more decay in time of the averages of ∇u than the one obtained in (2.2.24), since the log(T ) contribution clearly comes from this sub-optimal bound. The idea is to use the L 2 -L 1 estimate of Lemma 3 which essentially says, by stationarity, that for all

R ≥ √ t, x ∈ R d and p ∈ [1, +∞) B R (x) |∇u(t, y)| 2 dy p 2 1 p d,λ t -1-d 4 t 2 t 4 √ s 0 r d 2 |q r (s) -q r (s) | p 1 p dr ds.
It gives, using the sub-optimal moment bounds of q r (T ) -q r (T ) of the previous step that, for all R ≥ √ t

B R (x) |∇u(t, y)| 2 dy log 2 (t)t -1 η 2 β (t)D (x), (2.2.25)
with η β defined in (2.1.24) and where D (x) is a random variable with stretched exponential moment. By interpolating (2.2.24) and (2.2.25) we deduce for all ε > 0 and ∈ [1, +∞)

B (x) |∇u(t, y)| 2 dy r * (x) ∨ 1 d 1 < √ t + 1 ≥ √ t t -2(1-ε) η 2ε β (t)D 2ε (x). (2.2.26)
This improved decay allow us to prove the optimal estimates of Theorem 10. The price to pay in this step is a little loss of stochastic integrability due to the random variable D 2ε . Note that, the exponent α that we get in (2.1.22), is neither optimal for β > d (since [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF] indicates that we expect nearly-Gaussian moments), nor for 0 < β 1 (since by [START_REF] Fischer | Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations[END_REF] we can obtain nearly Gaussian moments).

IV) The proof of (2.1.20) follows the same ideas and is even easier since we do not need to begin with a sub-optimal estimate as we need for (2.1.17). We use Lemma 8 for the estimate of We finally mention that in the case of fast decay correlation, that is for β > d, the proof is much simpler and only the regime < √ T has to be considered.

(x, ) ∈ R d × [1, +∞) → ∂ fct x, q(r 2 ) f r ,
Remark 4. When u is real-valued and a is symmetric, we can prove (2.2.5) by using the Nash-Aronson bounds on the parabolic Green function. Indeed, let ψ

: (0, +∞) × R d × R d → R be Chapter 2.
Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields the Green function of the operator ∂ τ -∇ • a∇. Recall that the parabolic Green function ψ is a nonnegative function which solves in the distributional sense, for all (x, y) ∈ R d × R d , the equations

∂ τ ψ(•, •, y) -∇ • a∇ψ(•, •, y) = 0 in (0, +∞) × R d , ψ(0, •, y) = δ y , (2.2.27)
and

∂ τ ψ(•, x, •) -∇ • a∇ψ(•, x, •) = 0 in (0, +∞) × R d , ψ(0, x, •) = δ x .
We have the following estimate (for a reference, see [8, Prop E1]): there exists a constant C > 0 which depends only on d and λ such that for all (x, y) ∈ R d × R d and t > 0

C 3 2 √ t (t,x) |∇ z ψ(s, y, z)| 2 ds dz 1 2 d,λ T -1 2 -d 2 exp - |x -y| 2 Ct . (2.2.28) By noticing that from (2.2.27), ζ : y ∈ R d → ∇ y ψ(•, •, y) solves ∂ τ ζ -∇ • a∇ζ = 0 in (0, +∞) × R d , we deduce from the Caccioppoli estimate (see Lemma 12) that for all (x, y) ∈ R d × R d C √ t (t,x) |∇ z ∇ y ψ(s, z, y)| 2 ds dz 1 2 d,λ t -1 2 C 3 2 √ t (t,x) |∇ y ψ(s, z, y)| 2 ds dz 1 2 . (2.2.29)
Since a is symmetric, we have ψ(s, z, y) = ψ(s, y, z) and so ∇ y ψ(s, z, y) = ∇ z ψ(s, y, z). Consequently, the combination of (2.2.28), (2.2.29) and Lemma 2 yields for all

(x, y) ∈ R d × R d B √ t (y) |∇ x ∇ z ψ(t, x, z)| 2 ds dz 1 2 d,λ t -1-d 2 exp - |x -y| 2 Ct . ( 2

.2.30)

This bound implies, using the explicit formula ∇u(t, x) = ˆ∇x ∇ y ψ(t, x, y) • a(y)e dy, that for all

(t, x) ∈ R + * × R d |∇u(t, x)| ≤ ˆ|∇ x ∇ y ψ(t, x, y)|dy ≤ ˆ B √ t (y) |∇ x ∇ z ψ(t, x, z)| 2 dz 1 2 dy (2.2.30) d,λ t -1 ,
and thus for all

t > 0 ∇u(t, •) L ∞ (R d ) d,λ t -1 . ( 2

.2.31)

We now state the lemmas needed in the proof of Theorem 10. The first section lists the deterministic PDE ingredients, the second section the results derived from the large-scale regularity theory, and finally the third section the sub-optimal control of the fluctuations of the time dependent flux q r (•, •).

Deterministic results

This section displays the deterministic PDE ingredients needed in the proof of Theorem 10 and Corollary 1. We start with two classical results from standard L 2 regularity theory of parabolic system. The first one is the localized energy type estimates for parabolic systems.

Lemma 1 (Localized energy estimates). Let v be the weak solution of the parabolic system

∂ τ v -∇ • a∇v = ∇ • f in (0, +∞) × R d , v(0) = ∇ • q, with q ∈ L 2 loc (R d ) and f : R + × R d → R d such that T ∈ R + → ˆT 0 f (s, •)ds is in L 2 loc (R d )
and for all

(x, ) ∈ R d × [1, +∞), ˆB (x) |f (•, y)| 2 dy 1 2 is in L 1 loc (R + ).
There exists a universal constant c ∈ (0, +∞) such that the three following estimates hold:

(i) Assume that f ≡ 0. We have for all T > 0, R ≥ √ T and x ∈ R d T ˆηR ( y -x c )|∇v(T, y)| 2 dy 1 2 + ˆηR ( y -x c ) ˆT 0 ∇v(s, y)ds 2 dy 1 2 d,λ ˆηR ( y -x c )|q(y)| 2 dy 1 2 . ( 2.2.32)
(ii) Assume that q ≡ 0. We have for all R ≥ 1 and

x ∈ R d ˆηR ( y -x c ) ˆ1 0 ∇v(s, y)ds 2 dy 1 2 d,λ ˆηR ( y -x c ) ˆ1 0 f (s, y)ds 2 dy 1 2 + ˆ1 0 1 1 -t ˆ1 t ˆηR ( y -x c )|f (s, y)| 2 dy 1 2
ds dt.

(2.2.33)

(iii) Assume that q and f are supported in B (x) for some x ∈ R d and ∈ [1, +∞). Then we have

ˆRd \B (x) e |x-z| 2c
B (z) ˆ1 0 ∇v(s, y)ds 2 dy dz 1 2 d,λ ˆB (x) |q(y)| 2 dy 1 2 + ˆB (x) ˆ1 0 f (s, y)ds 2 dy 1 2 + ˆ1 0 1 1 -t ˆ1 t ˆB (x) |f (s, y)| 2 dy 1 2 ds dt. (2.2.34)
We then state a technical lemma needed in order to obtain pointwise estimates in time. for correlated coefficient fields

Lemma 2. Fix r > 0, (s, x) ∈ R d+1 and g ∈ L 2 (R d ) d . Assume that v is a weak solution of ∂ τ v -∇ • a∇v = ∇ • g in (s -4r 2 , s) × B 2r (x), then we have sup t∈(s-r 2 ,s) Br(x) |∇v(t, y)| 2 dt dy d,λ s s-4r 2 B 2r (x) |∇v(s , y)| 2 ds dy + B 2r (x) |g(y)| 2 dy. (2.2.35)
The same holds for the operator ∂ τ + ∇ • a∇ on (s, s + 4r2 ) × B 2r (x) by a time reflexion t → -t.

We conclude this section by the relationship between spatial averages of u(T, •) and averages of q r -q r over scales r ≤ √ T . This lemma allow us to deduce Corollary 1 from Theorem 10. We refer the reader to [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]Lem. 6] for the original proof of this result.

Lemma 3 (L 2 -L 1 estimate). Let u defined in (2.1.10). There exists a universal constant c > 0 such that for all T > 0 and R ≥ √ T √ T ˆηR ( y c )|∇u(T, y)| 2 dy T 2 T 4 √ t 0 r √ t d 2 ˆη√ 2R ( y c )|q r (t, y) -q r (t, y) |dy dr dt, where η R := R -d e -|•| R and q(•, •) is defined in (2.1.13).

Large-scale regularity results

We state in this section two estimates, needed in the proof of Theorem 10, which are obtained from the large-scale regularity theory recalled in the appendix 2.4.2. We start with a lemma which gives a pointwise bound on a local average of the solution of the dual problem (2.2.11), depending on the behavior of the r.h.s. It constitutes the parabolic version of Lemmas 2, 3 and 4 of [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] established for elliptic systems.

Lemma 4 (Pointwise estimates on the dual problem). Let

f 1 ∈ C 1 b (R d
), e a unit vector of R d and v r satisfies, in the weak sense for some r ≥ 1, the parabolic backward system

∂ τ v r + ∇ • a∇v r = ∇ • af r e in (-∞, 0) × R d , v r (0) = 0, (2.2.36)
with f r which satisfies one of the two following assumptions:

• f r := r -d f 1 ( • r ) such that for all x ∈ R d |f 1 (x)| 1 (|x| + 1) d and |∇f 1 (x)| 1 (|x| + 1) d+1 .
(2.2.37)

• For all x ∈ R d |f r (x)| r (|x| + 1) d ∧ 1 (|x| + 1) d-1 and |∇f r (x)| r (|x| + 1) d+1 ∧ 1 (|x| + 1) d .
(2.2.38)

We have

1. If (2.2.37) holds, for all x ∈ R d and √ -t ≥ 2r * (x) B r * (x) (x) |∇v r (t, y)| 2 dy dt 1 2 r * (0) r ∨ 1 d 2 log(1 + |x| r ) (|x| + r) d .
(2.2.39)

2. If (2.2.38) holds, for all x ∈ R d and √ -t ≥ 2r * (x) B r * (x) (x) |∇v r (t, y)| 2 dy 1 2 r d 2 * (0) r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 .
(2.2.40)

Let us briefly comment on Lemma 4.

1. The bound (2.2.39) is needed to replace the plain energy estimate for the solution v T r of (2.2.11), as we did in the heuristic argument (2.2.18). In the homogeneous case, i.e a = Id, and in the case where f 1 = g 1 , the bound (2.2.39) takes the more natural form: for all

(t, x) ∈ R -× R d |∇v r (t, x)| (|x| + r) -d .
(2.2.41) Indeed, (2.2.41) is easy to see using the parabolic Green function Γ of the heat operator ∂ τ -∆:

we have for all

(t, x) ∈ R -× R d ∇v r (t, x) = - ˆ0 t ∇ 2 Γ(s -t, •) g r (x)ds.
Thus, using the fact that, for all s ≥ t, Γ(s -t, •) is proportional to g √ s-t and using the semigroup property of Gaussian kernel g √ s-t g r = g √ r 2 +s-t as well as the estimate, for all

x ∈ R d , e - |x| 2 r 2 +s-t (1 + |x| 2 r 2 +s-t ) -d 2 -2 , we have for all (t, x) ∈ R -× R d |∇v r (t, x)| ˆ0 t |∇ 2 g √ r 2 +s-t (x)|ds |x| 2 ˆ0 t (r 2 + s -t) -d 2 -2 e - |x| 2
r 2 +s-t ds |x| 2 ˆ0 t (|x| 2 + s -t + r 2 ) -d 2 -2 ds (|x| + r) -d .
Also, if we have the more precise structure f r = ´r2 1 ∇g √ s (•)ds (which satisfies the assumption (2.2.38)), using the same ideas as before, the bound (2.2.40) takes the more natural form: for all

(t, x) ∈ R -× R d |∇v r (t, x)| r (|x| + 1) d ∧ 1 (|x| + 1) d-1 .
Therefore, since the bounds (2.2.39) and (2.2.40) are natural in the homogeneous case and we know from homogenization theory that on large-scales the heterogeneous parabolic operator ∂ τ -∇ • a∇ inherits (in form on the C 0,1 estimate (2.4.5)) the regularity theory of the homogenized operator ∂ τ -∇ • a hom ∇, it is natural to expect that the two estimates (2.2.39) and for correlated coefficient fields (2.2.40) hold in the heterogeneous case once we fix the scale (characterized by the minimal radius r * ). Note that the logarithm contributions in (2.2.39) and (2.2.40) are due to the fact that we have less structure on the r.h.s of (2.2.36) than the two we took above. We also point out that the logarithm contribution in (2.2.40) may be removed (see for instance Lemmas 3 and 4 of [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] for elliptic systems). However, we prefer to keep it this way and provide simple arguments for (2.2.40) rather than going trough additional technical difficulties.

2. We may deduce the result of Lemmas 3 and 4 of [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] from (2.2.39) by sending t ↓ -∞. Indeed, one may prove, from the localized energy of Lemma 1 that v r (t, •) → t→-∞ ṽr in L 2 loc (R d ) with ∇ • a∇ṽ r = ∇ • af r e and then pass to the limit in (2.2.39).

The next lemma allows us to control spatial averages of ∇u at scale R < √ T . Combined with the energy estimate (2.2.32), it implies in particular the estimate (2.2.24) needed in the proof of Theorem 10.

Lemma 5 (Control of averages). Let u be defined in (2.1.10). Assume that there exist two monotone functions f and g (increasing and decreasing respectively) such that for all T ≥ 1 and for all x ∈ R d there exists a constant C(x, T ) < +∞ for which

B √ T (x) |∇u(T, y)| 2 dy ≤ C(x, T )f (T )g(T ). (2.2.42)
Then we have for all

T ≥ 1, x ∈ R d and R < √ T B R (x) |∇u(T, y)| 2 dy r * (x) R ∨ 1 d C(x, T )f (T )g T 2 ,
with C(x, T ) := max C(x, T ),

T T 2
C(x, s)ds .

Suboptimal control of fluctuations of the time dependent flux

In this section, we state the suboptimal moment bounds of q r (T )-q r (T ) . We prove that it displays the central limit theorem scaling r -d 2 , a growth in T which depends on the parameter β defined in (2.1.6) and a log(T ) correction (which makes it suboptimal and will be removed later). We first state the following bound on q r (•, •), for r ≤ 1. It is only needed for technical reasons since, in view of the application of Lemma 3, r is allowed to be arbitrary close to 0. Lemma 6. Let q(•, •) be defined in (2.1.13). For all r ∈ (0, 1) and x ∈ R d , there exists a random variable C (r, x) such that for all √ T ≥ 1 2 we have

|q r (T, x)| ≤ (1 + r -d 2 log( √ T r ))C (r, x), (2.2.43) with sup (r,x)∈R + ×R d exp 1 C C 2 β∧d d (r, x)
≤ 2 for some constant C > 0 depending on d and λ.

The next lemmas give an estimate of the functional derivative of averages of the flux q r (T ) and q(T ) f r for T ≥ 1 and r ≤ √ T . This is the starting point for using our assumption (2.1.4).

Lemma 7 (Functional derivative). Let q(•, •) be defined in (2.1.13). There exists a universal constant c ∈ (0, +∞) such that for all T ≥ 1, r > 0, x ∈ R d and ∈ [1, +∞), we have

|∂ fct x, q r (T )| d,λ ˆB (x) g r (y)dy + ˆB (x) ˆT 0 ∇u(s, y)ds g r (y) dy + ˆB (x) |∇v T (1, y)| 1 + ˆ1 0 ∇u(s, y)ds dy + F r, (x)1 < √ T + G r, (x)1 ≥ √ T + ˆB (x) ˆT 1 |∇u(t, y)||∇v T (t, y)|dtdy, (2.2.44)
where v T = (v T k ) k∈ 1,d is a weak solution of the backward parabolic system

∂ τ v T k + ∇ • a * ∇v T k = ∇ • ag r e k on (-∞, T ) × R d , v T k (T ) = 0, (2.2.45) with F r, (x) = d 2 ˆe-|x-z| 2c B (z) |g r (y)| 2 dy dz 1 2 + d 2 ˆe-|x-z| 2c B (z) |∇v T (1, y)| 2 dy dz 1 2 , (2.2.46) and G r, (x) =T x, (η r )(0) + 1 ≥r * (0) ˆRd \B 4 B (y) |∇v T (1, z)| 2 dz 1 2 T x, (η )(y)dy + ˆB7 B r * (y) (y) |∇v T (1, z)| 2 dz 1 2 T x, (η r * (y) )(y)dy + 1 <r * (0) r d 2 * (0) ˆe-|x-z| 2c B (z) |∇v T (1, y)| 2 dy dz 1 2
.

(2.2.47)

as well as for all y ∈ R d and ρ > 0

T x, (η ρ )(y) = ˆB (x) η ρ (z -y) 1 + ˆ1 0 ∇u(t, z)dt 2 dz 1 2 + ˆ1 0 1 1 -t ˆ1 t ˆB (x) η ρ (z -y)|∇u(s, z)| 2 dz 1 2
ds dt.

(2.2.48)

Lemma 8. Let q(•, •) be defined in (2.1.13
) and for all r > 0 we consider

f r ∈ C 1 b (R d ) satisfying, for all y ∈ R d f r (y) = ˆr2 1 fr (s, y)ds with | fr (s, y)| |y|s -1 g √ s (y). ( 2

.2.49)

There exists a universal constant c ∈ (0, +∞) such that for all r ≥ 2, x ∈ R d and ∈ [1, +∞), we have

|∂ fct x, q(r 2 ) f r | ˆB (x) |f r (y)|dy + ˆB (x) ˆr2 0 ∇u(s, y)ds |f r (y)| dy + ˆB (x) |∇v r 2 (1, y)| 1 + ˆ1 0 ∇u(s, y)ds dy + K r, (x) + G r, (x) + ˆB (x) ˆr2 1 |∇u(t, y)||∇v r 2 (t, y)|dtdy, (2.2.50)
Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

where v r 2 = (v r 2 k ) k∈ 1,d is a weak solution of the backward parabolic problem ∂ τ v r 2 k + ∇ • a * ∇v r 2 k = ∇ • af r e k on (-∞, r 2 ) × R d , v r 2 k (r 2 ) = 0, (2.2.51) with K r, (x) :=1 ≥r ˆr2 1 s -1 2 T x, (η √ s )(0)ds + 1 <r T x, (η 1 )(0) + log 2 (3 ) n=0 2 n T x, (η 2 n+1 )(0) + ˆRd \B 2 B (y) |f r (z)| 2 dz 1 2
T x, (η )(y)dy .

(2.2.52)

and G r, as well as T x, are defined in (2.2.47) (for T = r 2 ) and (2.2.48) respectively.

We finally state the main result of this section, which is the sub-optimal moment bound on the time dependent flux q r (T,

•) for 1 2 ≤ r ≤ √ T . Proposition 3 (Sub-optimal fluctuation estimates). Let q(•, •) defined in (2.1.13). For all T ≥ 1, 1 ≤ r ≤ √ T and p ∈ [1, +∞) |q r (T ) -q r (T ) | p 1 p d,λ,β p 1 2 + d+2 β∧d r -d 2 (1 + log(T ) + log 2 ( √ T r ))µ β (T ), (2.2.53)
with

µ β (T ) :=      T d 4 -β 4 if β < d, log 1 2 (T ) if β = d, 1 if β > d.

Proofs

We give in the section the all proofs of the results stated in the sections 2.2.2, 2.2.3 and 2.2.4. For notational convenience, we shall assume that the results of Lemmas 1 and 3 hold for the universal constant c = 1. In the general case, it suffices to change the kernels g r and η r from line to line (by allowing a constant in the exponential). We also drop the dependance on d, λ and β in the inequalities.

Proof of the deterministic results

Proof of Lemma 1: Localized energy estimates

We only provide the arguments for (2.2.33) and (2.2.34), the proof of (2.2.32) can be found in [START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF]Lem.1]. Without loss of generality, we may assume that x = 0.

Step 1. Proof of (2.2.33). We set for all t ∈ [0, 1], w(t, •) := ´t 0 v(s, •)ds and we note that w is a weak solution of

∂ τ w -∇ • a∇w = ∇ • ´τ 0 f (s, •)ds on (0, 1] × R d , w(0) = 0.
The idea of the proof is to use the estimate (2.2.32) by expressing w with help of the Duhamel formula. We denote by S the semigroup associated to the operator -∇ • a∇, namely (S(t)) t∈R + is a family of operators such that for all Schwartz distributions

ζ on R d , z := S(•)ζ is the unique weak solution of ∂ τ z -∇ • a∇z = 0 on (0, 1] × R d , z(0) = ζ.
We express ∇w(1, •) with help of S in form of the Duhamel formula, that is

∇w(1, •) = ˆ1 0 ∇ S(1 -t)∇ • ˆt 0 f (s, •)ds dt.
Thus, we write for all R ≥ 1, by the triangle inequality and ´t 0 f (s,

•)ds = ´1 0 f (s, •)ds -´1 t f (s, •)ds in the last line ˆηR (y)|∇w(1, y)| 2 dy 1 2 = ˆηR (y) ˆ1 0 ∇ S(1 -t)∇ • ˆt 0 f (s, y)ds dt 2 dy 1 2 ≤ ˆηR (y) ˆ1 0 ∇ S(1 -t)∇ • ˆ1 0 f (s, y)ds dt 2 dy 1 2 + ˆηR (y) ˆ1 0 ∇ S(1 -t)∇ • ˆ1 t f (s, y)ds dt 2 dy 1 2 . (2.3.1)
For the first r.h.s term of (2.3.1), we use (2.2.32) for T = 1 in form of ˆηR (y)

ˆ1 0 ∇ S(1 -t)∇ • ˆ1 0 f (s, y)ds dt 2 dy 1 2 ˆηR (y) ˆ1 0 f (s, y)ds 2 dy 1 2
, which gives the first r.h.s term of (2.2.33). For the second r.h.s term of (2.3.1), we use (2.2.32) for T = 1, this time in the pointwise way, combined with the Minkowski inequality in L

2 (R d , η R dx) to get ˆηR (y) ˆ1 0 ∇ S(1 -t)∇ • ˆ1 t f (s, y)ds dt 2 dy 1 2 ≤ ˆ1 0 ˆηR (y) ∇ S(1 -t)∇ • ˆ1 t f (s, y)ds 2 dy 1 2 dt (2.2.32) ˆ1 0 1 1 -t ˆηR (y) ˆ1 t f (s, y)ds 2 1 2 dt ≤ ˆ1 0 1 1 -t ˆ1 t ˆηR (y)|f (s, y)| 2 dy 1 2
ds dt, which gives the second r.h.s term of (2.2.33).

Step 2. Poof of (2.2.34). Since, for all (y, z)

∈ B × R d \B , we have e -|y-z| c ≤ e 1 c - |z| c
, we deduce from (2.2.32), (2.2.33) (applied with R = ) and the fact that f and q are compactly supported in B :

e |z| 2c ˆB (z) ˆ1 0 ∇v(s, y)ds 2 dy e -|z| 2c ˆB |q(y)| 2 dy + ˆB ˆ1 0 f (s, y)ds 2 dy + e -|z| 2c ˆ1 0 1 1 -t ˆ1 t ˆB |f (s, y)| 2 dy 1 2 ds dt 2 ,
which yields (2.2.34) by integrating over R d \B .

Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

Proof of Lemma 2

The arguments are the same as in [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF]Lem 8.2] where a proof is given for g ≡ 0. For reader convenience, we repeat the proof with very small changes in order to consider non-zero r.h.s g. Without loss of generality, we may assume thay (s, x) = (0, 0).

Since g does not depend on time, ∂ τ v is also a weak solution of the equation and therefore the Caccioppoli inequality, see Lemma 12, yields for all ρ ≤ R

Cρ |∇∂ τ v(s, y)| 2 dy ds 1 (R -ρ) 2 C R |∂ τ v(s, y)| 2 dy ds. (2.3.2)
To be entirely rigorous, we need to justify that ∂ τ u belongs to H 1 par (C R ). One may justify it by considering difference quotients in time, obtaining a version of (2.3.2) by the Caccioppoli inequality for these difference quotients, which then allows to pass to weak limits in the difference quotient parameter to obtain that ∂ τ u does indeed belong to H 1 par (C R ) and is a solution of the equation. Next, differentiating gives, for all t ∈ (-r 2 , 0) (2.3.3)

∂ τ Br |∇v(t, y)| 2 dy = 2 Br ∇v(t, y) • ∇∂ τ v(t,
It remains to estimate the second r.h.s term of (2.3.3). For this, fix

ρ < r, η ∈ C ∞ c (B 3 2 ρ ) such that η = 1 in B ρ , 0 ≤ η ≤ 1 and |∇η| 1 ρ .
Testing the equation for v with the function η∂ τ v, we obtain

Cρ |∂ τ v(s, y)| 2 dy 1 ρ C 3 2 ρ |∂ τ v(s, y)||∇v(s, y)|dy ds + C 3 2 ρ |∇∂ τ v(s, y)||∇v(s, y)|dy ds + 1 ρ C 3 2 ρ |g(y)||∂ τ v(s, y)|dy ds + C 3 2 ρ |g(y)||∇∂ τ v(s, y)|dy ds. (2.3.4)
Then, using the Cauchy-Schwarz inequality combined with (2.3.2), we have

C 3 2 ρ |∇∂ τ v(s, y)||∇v(s, y)|dy ds + C 3 2 ρ |g(y)||∇∂ τ v(s, y)|dy ds 1 ρ C 2ρ |∂ τ v(s, y)| 2 dy ds 1 2 B 2ρ |g(y)| 2 dy 1 2 + C 2ρ |∇v(s, y)| 2 dyds 1 2
.

Proofs 81

Therefore, (2.3.4) turns into

Cρ |∂ τ v(s, y)| 2 dy 1 ρ C 2ρ |∂ τ v(s, y)| 2 dy ds 1 2 B 2ρ |g(y)| 2 dy 1 2 + C 2ρ |∇v(s, y)| 2 dyds 1 2
.

Choosing now 1 2 ≤ σ < σ ≤ 1, we obtain by a covering argument (considering the decomposition

C σ 4r ⊂ N i=1 (t i -((σ -σ ) r 2 ) 2 , t i ) × B (σ-σ ) r 2 (x i ) for (t i , x i ) i∈ 1,N ⊂ C σ 4r and N ∼ (σ -σ ) -d-2 ) that C σ 4r |∂ τ v(s, y)| 2 dy 1 (σ -σ ) d+3 r C σ4r |∂ τ v(s, y)| 2 dy ds 1 2 B 4r |g(y)| 2 dy 1 2 + C 4r |∇v(s, y)| 2 dyds 1 2
, which turns to, by a Young's inequality, for some constant C depending on d and λ We conclude the proof by a covering argument by considering r 4 instead of r.

C σ 4r |∂ τ v(s, y)| 2 dy ≤ 1 2 C σ4r |∂ τ v(s, y)| 2 dy ds + C (σ -σ ) 2(d+3) r -2 B 4r |g(y)| 2 dy + C 4r |∇v(s, y)| 2 dyds . Lemma 11 applied with ρ : σ → ffl C σ4r |∂ τ v(s,

Proof of the large-scale regularity results

We provide the proofs of Lemmas 4 and 5. Our main tool here in the large-scale regularity theory for parabolic system recalled in Appendix 2.4.2.

Proof of Lemma 4: Pointwise estimates on the dual problem

We prove Lemma 4 in two steps. The first step is devoted to prove (2.2.39) and we do it in two times. First, we treat the particular case where f r is compactly supported in the ball B r for some r ≥ 1. Second, we treat the general case by decomposing R d into dyadic annuli (B k ) k∈N , defined by B k := B 2 k+1 r \B 2 k r for k ≥ 1 and B 0 := B r , and writing f r = +∞ k=0 f r χ k , where (χ k ) k∈N is a partition of unity according to the decomposition (B k ) k∈N . We then apply the result of the compact supported case for each k ∈ N. The second step is devoted to prove (2.2.40) and this is done by using the results of the first step. This extends Lemmas 2, 3 and 4 of [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] from the elliptic to the parabolic setting. for correlated coefficient fields

Step 1. Proof of (2.2.39). We split the proof into two substeps.

Substep 1.1. We prove that under the assumptions f r is supported in B r and r d sup

x∈R d |f r (x)| + r d+1 sup x∈R d |∇f r (x)| 1, (2.3.5)
we have for all x ∈ R d and

√ -t ≥ 2r * (x) B r * (x) (x) |∇v r (s, y)| 2 dy ds 1 2 r * (0) r ∨ 1 d 2 (|x| + r) d . (2.3.6)
The estimate (2.3.6) will come from the following four relations and estimates:

1. For all (t, x) ∈ R -× R d ∇v r (t, x) = ˆ-t 0 ∇w r (s, x)ds, (2.3.7)
with w r is the weak solution of

∂ τ w r -∇ • a∇w r = 0 in (0, +∞) × R d , w r (0) = ∇ • af r e. (2.3.8)
2. The plain energy estimate: for all t ∈ R - ˆRd |∇v r (t, x)| 2 dy r -d .

(2.3.9)

3. The large-scale regularity estimate: for all

(t, x) ∈ R -× R d such that |x| ≥ 4(r * (x) ∨ r) B r * (x) (x) |∇v r (t, y)| 2 dy r * (0) r ∨ 1 d (|x| + r) 2d . (2.3.10) 4. The large-scale C 0,1 estimates: for all x ∈ R d , √ -t ≥ 2r * (x) and r ≥ r * (x): • For √ -t ≤ r, B r * (x) (x) |∇v r (t, y)| 2 dy 0 t B √ -t (x) |∇v r (s, y)| 2 dy ds + r -2d . (2.3.11) • For √ -t ≥ r, B r * (x) (x) |∇v r (t, y)| 2 dy t+r 2 t Br(x) |∇v r (s, y)| 2 dy ds + r -2d . (2.3.12) Argument for (2.3.7). A direct computation shows that (t, x) ∈ R -× R d →
´-t 0 w r (s, x)ds is the weak solution of (2.2.36). Thus, by uniqueness, for all t ∈ R -, v r (t, •) =

´-t 0 w r (s, •)ds and (2.3.7) follows.

Proofs 83

Argument for (2.3.9). We have, by using the formula (2.3.7) combined with the localized energy estimate (2.2.32) applied to the equation (2.3.8) and the support condition (2.3.5) of f r , for all

(t, x) ∈ R -× R d ˆη√ -t (y -x)|∇v r (t, x)| 2 dy (2.3.7) = ˆη√ -t (y -x) ˆ-t 0 ∇w r (t, x) 2 dy (2.2.32) ˆη√ -t (y -x)|f r (y)| 2 dy (2.3.5) r -2d ˆBr η √ -t (y -x)dy, (2.3.13)
which gives (2.3.9) by integrating the estimate over x ∈ R d .

Argument for (2.3.10). We first prove by a duality argument that for all R ≥ 2(r * (0) ∨ r)

and t ∈ R - ˆRd \B R |∇v r (t, y)| 2 dy r * (0) r ∨ 1 d R -d . (2.3.14) Let h ∈ C ∞ c (R d ) supported in R d \B R . Let t ∈ R -, s ∈ [0,
-t] and k s be the weak solution of the backward parabolic system (corresponding to the dual system of (2.2.36) with final time s), .

∂ τ k s + ∇ • a * ∇k s = 0 on (-∞, s) × R d , k s (s) = ∇ • h. ( 2
(2.3.16)

The r.h.s of (2.3.16) is then dominated as follows. First, we set ṽ(t, •) :=

´-t 0 k s (0, y)ds and by noticing that for all s ∈ (0, -t), k s (0, •) = k 0 (-s, •), we have

∂ τ ṽ + ∇ • a * ∇ṽ = ∇ • h in (-∞, 0) × R d , ṽ(0) = 0.
Second, we denote by v the weak solution of

∂ τ v + ∇ • a * ∇v = ∇ • h1 (-∞,0) in R d+1 , v(0) = 0 in R + × R d . (2.3.17)
v is an extension of ṽ in the sense that ∇ṽ(s, •) = ∇v(s, •) as long as s ≤ 0. Now, since h = 0 in B R , we have by using the estimate ffl

Br ( r * (0) r ∨ 1) d 2 ffl B r * (0)∨r
, Lemma 2 and the large-scale C 0,1 estimate (2.4.6)

Br ˆ-t 0 ∇k s (0, y)ds 2 dy 1 2 = Br |∇v(t, y)| 2 dy 1 2 r * (0) r ∨ 1 d 2 B r * (0)∨r |∇v(t, y)| 2 dy 1 2 r * (0) r ∨ 1 d 2 t+4(r * (0)∨r) 2 t B 2(r * (0)∨r) |∇v r (s, y)| 2 dy ds 1 2 (2.4.6) r * (0) r ∨ 1 d 2 t+R 2 t B R |∇v(s, y)| 2 dy ds 1 2 . (2.3.18)
Now, since v ≡ 0 in R + ×R d , we have using the plain energy estimate (for which the proof is identical as (2.3.9)): for all s ∈ R - ˆ|∇ṽ(s, y)| 2 dy ˆ|h(y)| 2 dy,

(i) for √ -t ≥ R t+R 2 t B R |∇v(s, y)| 2 dy ds = t+R 2 t B R |∇ṽ(s, y)| 2 dy ds R -d ˆ|h(y)| 2 dy. (2.3.19) (ii) for R ≥ √ -t t+R 2 t B R |∇v(s, y)| 2 dy ds = R -2 ˆ0 t B R |∇ṽ(s, y)dy ds √ -t R 2 R -d ˆ|h(y)| 2 dy R -d ˆ|h(y)| 2 dy. (2.3.20)
The combination of (

(0) r ∨ 1 d 2 R -d 2 ˆ|h(y)| 2 dy 1 2
, which gives (2.3.14) by the arbitrariness of h.

We now prove (2.3.10). Let R := 1 2 |x| and assume that R ≥ 2(r * (x) ∨ r). Without loss of generality, we may assume that R ≥ 2r * (0). Indeed, otherwise, we deduce from the 1 8 -Lipschitz property of r * (x) in form of

r * (0) ≤ r * (x) + |x| 8 → r * (x) ≥ |x| 4 ,
and

r * (0) ≥ r * (x) - |x| 8 → 3 2 r * (0) ≥ r * (x),
as well as (2.3.9) that

B r * (x) (x) |∇v r (t, y)| 2 dy r -d * (x) ˆ|∇v r (t, y)| 2 dy r d * (0) (|x| + r) d . We have B R (x) ⊂ R d \B R . Indeed for all y ∈ B R (x), the triangle inequality yields |y| ≥ |y -x| -|x| ≥ 2R -R = R, so that y / ∈ B R .
We then argue once again by extension and we consider v r the weak solution of

∂ τ v r + ∇ • a∇v r = ∇ • af r e1 (-∞,0) in R d+1 , v r = 0 in R + × R d , ( 2
(2.3.5), f r ≡ 0 on B R (x)) that for all t ∈ R - B r * (x) (x) |∇v r (t, y)| 2 dy = B r * (x) (x) |∇v r (t, y)| 2 dy t+4r 2 * (x) t B 2r * (x) (x) |∇v r (s, y)| 2 ds dy (2.4.6) t+R 2 t B R (x) |∇v r (s, y)| 2 ds dy. (2.3.22) Now, since v r ≡ 0 in R + × R d , we have from (2.3.14): (i) For √ -t ≥ R, t+R 2 t B R (x) |∇v r (s, y)| 2 ds dy = t+R 2 t B R (x) |∇v r (s, y)| 2 ds dy ≤ R -d t+R 2 t ˆRd \B R |∇v r (s, y)| 2 ds dy (2.3.14) r * (0) r ∨ 1 d R -2d . (2.3.23)
Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

(ii) For R ≥ √ -t, t+R 2 t B R (x) |∇v r (s, y)| 2 ds dy = R -2 ˆ0 t B R (x) |∇v r (s, y)| 2 ds dy ≤ R -d √ -t R 2 0 t ˆRd \B R |∇v r (s, y)| 2 ds dy (2.3.14) r * (0) r ∨ 1 d R -2d . (2.3.24)
The combination of (

r -2d ˆBr η √ -t (y -x)dy + r -2d ≤ r -2d r d * (0) (|x| + r) d . (ii) If √ -t ≥
r -2d r d * (0) (|x| + r) d .
This concludes the proof of (2.3.6).

Substep 1.2. We prove (2.2.39) without the support condition (2.3.5) on f r . We decompose the r.h.s of (2.2.36) according to a family of dyadic annuli (B k ) k∈N , defined by B k := B 2 k+1 r \B 2 k r for all k ≥ 1 and B 0 := B r . Namely, we set for all k ∈ N, f r,k := f r χ k , where (χ k ) k∈N is a partition of unity according to the decomposition (B k ) k∈N , and we denote by v r,k the weak solution of (2.2.36) with r.h.s ∇ • af r,k e. By uniqueness, we have ∇v r = +∞ k=0 ∇v r,k . Hence, we get by the triangle inequality

B r * (x) (x) |∇v r (t, y)| 2 dy 1 2 ≤ +∞ k=0 B r * (x) (x) |∇v r,k (t, y)| 2 dy 1 2
.

Thanks to (2.2.37), f r,k satisfies (2.3.5) with radius 2 k r, thus, by (2.3.6), we have for all k ≥ 0 and

√ -t ≥ 2r * (x) B r * (x) (x) |∇v r,k (t, y)| 2 dy 1 2
r * (0)

2 k r ∨ 1 d 2 (|x| + 2 k r) d .
Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

We deduce, setting N r := log

2 1 + |x| r B r * (x) (x) |∇v r (t, y)| 2 dy 1 2 +∞ k=0 r * (0) 2 k r ∨ 1 d 2 (|x| + 2 k r) d r * (0) r ∨ 1 d 2 (|x| + r) -d N r + r -d +∞ k=Nr+1 2 -kd ,
which gives (2.2.39).

Step

B r * (x) (x) |∇v r,k (t, y)| 2 dy 1 2 r r * (0) 2 k ∨ 1 d 2 (|x| + 2 k ) d .
We then conclude exactly as in Substep

B r * (x) (x) |∇v r,k (t, y)| 2 dy 1 2 2 k r * (0) 2 k ∨ 1 d 2 (|x| + 2 k ) d .
We then conclude by the same decomposition as in Substep 1.2 : setting N := log 2 (1 + |x|)

B r * (x) (x) |∇v r (t, y)| 2 dy 1 2 +∞ k=0 2 k r * (0) 2 k ∨ 1 d 2 (|x| + 2 k ) d r d 2 * (0) (|x| + 1) -d N k=0 2 k + +∞ k=N+1 2 (1-d)k ,
which concludes the proof since

N k=0 2 k |x| + 1, 2.3. Proofs and +∞ k=N+1 2 (1-d)k 1 (|x| + 1) d-1 .

Proof of Lemma 5: Control of averages

We consider the two regimes: the non-generic case R ≤ r * (x) and the generic case R ≥ r * (x).

1. Let us start with the non-generic case R ≤ r * (x). We distinguish two sub-cases.

(i) In the case where T 2 ≤ 2r * (x), we have, using (2.2.42) and R <

√ T B R (x) |∇u(T, y)| 2 dy √ T R d B √ T (x) |∇u(T, y)| 2 dy (2.2.42) r * (x) R d C(x, T )f (T )g(T ).
(ii) In the case where

T 2 ≥ 2r * (x), we have by noticing that ∂ τ u -∇ • a∇u = 0 on ( T 2 , T ) × B T 2 (x)
and from Lemma 2 combined with the large-scale C 0,1 estimate (2.4.6)

B R (x) |∇u(T, y)| 2 dy r * (x) R d B r * (x) (x) |∇u(T, y)| 2 dy r * (x) R d C 2r * (x) (T,x) |∇u(s, y)| 2 ds dy (2.4.6) r * (x) R d T T 2 B √ T 2 (x) |∇u(s, y)| 2 ds dy r * (x) R d T T 2 B √ s (x) |∇u(s, y)| 2 ds dy (2.2.42) r * (x) R d f (T )g T 2 T T 2
C(x, s)ds.

2.

Let us now consider the generic case R ≥ r * (x) and without loss of generality we may assume that

√ T > 2 √ 2R since otherwise ffl B R ffl B √
T and the conclusion follows from (2.2.42). We

have ∂ τ u -∇ • a∇u = 0 on (0, T ) × B √
T and r * (x) ≤ 2R < T 2 so that from Lemma 2 and the large-scale C 0,1 estimate (2.4.6) we deduce that

B R (x) |∇u(T, y)| 2 dy T T -4R 2 B 2R |∇u(s, y)| 2 dy ds T T 2 B √ T (x) |∇u(s, y)| 2 dy ds C(x, T )f (T )g T 2 .

Proof of the suboptimal control of fluctuations of the time dependent flux

We provide in this section the proofs of Lemmas 6, 7, 8 and Proposition 3 of Section 2.2.4. for correlated coefficient fields

Proof of Lemma 6

We prove the lemma in two steps. In the first step we prove a deterministic bound on q r (T, x), using the energy estimates of Lemma 1 and the control of averages of ∇u deduced from Lemma 5. The deterministic bound will depend on a random variable built from an average of r * . In the second step, we prove that the random constant has stretched exponential moments, using the moment bound (2.4.3) of r * .

Step 1. Proof that for all r ≤ min{1,

√ T } and x ∈ R d |q r (T, x)| 1 + r -d 2 log( √ T r ) 1 + r d 2 * (0) + ˆRd \B 1 (x) η 1 (y)r d 2 * (ry + x)dy . (2.3.27)
Without loss of generality, we may assume that x = 0. We have from the definition (2.1.13) of q r (T ) followed by the triangle inequality

´T 0 = ´r2 0 + ´T r 2 , the continuous embedding L 2 (R d , η r dx) → L 1 (R d , η r dx)
B √ T (x) |∇u(T, y)| 2 dy 1 2 T -1 ,
we deduce from the Minkowski inequality in L 2 (B r (x)) and Lemma 5 applied with R = 1, f ≡ 1 and g :

T ∈ R + → T -1 , that, since r ≤ 1 Br(x) ˆT r 2 ∇u(s, y)ds 2 dy 1 2 r -d 2 ˆT r 2 ˆB1 (x) |∇u(s, y)| 2 dy 1 2 ds r -d 2 r d 2 * (x) log( √ T r ).
Consequently dx (r

d 2 * (0) + 1)r d 2 ˆη2 r (y)dy 1 2 r d 2 * (0) + 1.
For the second r.h.s term of (2.3.31), we note that for all x ∈ R d \B r and y ∈ B r (x) we have η r (y) η r (x), so that

ˆRd \Br r d 2 * (x) Br(x) η 2 r (y)dy 1 2 dx ˆRd \Br r d 2 * (x)η r (x)dx = ˆRd \B 1 r d 2 * (rx)η 1 (x)dx.
This concludes the proof of (2.3.30) and the argument for (2.3.27).

Step 2. We prove (2. Therefore for all p ∈ [1, +∞) and

x ∈ R d 1 + r d 2 * (0) + ˆRd \B 1 (x) η 1 (y)r d 2 * (ry + x)dy p 1 p 1 + r dp 2 * (0) 1 p + ˆRd \B 1 (x) η 1 (y) r dp 2 * (ry + x)

Proof of Lemmas 7 and 8: Control of the functional derivatives

We prove Lemma 7 and Lemma 8 independently.

Proof of lemma 7. It is enough to prove (2.2.44) for the quantities q r (T ) • e k , for all r > 0, T ≥ 1 and k ∈ 1, d . In particular, we only treat the case k = 1, since the other contributions are controlled the same way. For notational convenience, we write v T for v T 1 . for correlated coefficient fields

Let x ∈ R d , h ∈ (0, 1), T ≥ 1, r > 0, ∈ [1, +∞)
and δa be compactly supported in B (x) such that sup y∈B (x) |δa(y)| ≤ 1. We compute the finite difference

δ h q r (T ) • e 1 := q r (a + hδa, T ) • e 1 -q r (a, T ) • e 1 h
= ˆgr (y)e where

δ h u(t, •) := u(a+hδa,t,•)-u(a,t,•) h
is the weak solution of We now split the rest of the proof into two steps, and treat the two r.h.s terms of (2.3.35) separately.

∂ τ δ h u -∇ • a∇δ h u = ∇ • δa∇u(a + hδa, •, •) on (0, +∞) × R d , δ h u(0, •) = ∇ • δa(•)e. ( 2
Step 1. First r.h.s term of (2.3.35). We prove that ds dt

lim sup h→0 ˆgr (y)e 1 • a(y) ˆ1 0 ∇δ h u(t, y)dt dy 1 < √ T d 2 ˆe-|x-z| 2c B (z) |g r (y)| 2 dy dz 1 2 + 1 ≥ √ T T x, (η r )(0), (2.3
d 2 ˆ1 0 1 1 -t ˆ1 t s -1 ds dt = d 2 ˆ1 0 -log(t) 1 -t dt d 2 .
(

+ G r, (x)1 ≥ √ T + 1 < √ T d 2 ˆe-|x-z| 2c B (z) |∇v T (1, y)| 2 dy dz 1 2 . ( 2.3.47)
Recall that v T denotes the weak solution of the dual system associated with (2.3.34), which reads

∂ τ v T + ∇ • a * ∇v T = ∇ • ag r e 1 on (-∞, T ) × R d , v T (T ) = 0. ( 2
δ h u(1, •) -∇ • a ˆ1 0 ∇δ h u(t, •)dt = ∇ • δa ˆ1 0 ∇u(a + hδ, t, •)dt + ∇ • δae, which provides by testing with v T (1, •) ˆδh u(1, y)v T (1, y)dy = ˆ∇v T (1, y) • δa(y) ˆ1 0 ∇u(a + hδa, t, y)dt dy -ˆ∇v T (1, y) • a(y) ˆ1 0 ∇δ h u(t,
ˆ∇v T (1, y) • a(y) ˆ1 0 ∇δ h u(t, y)dt dy ≤ ˆ B (y) |∇v T (1, z)| ˆ1 0 ∇δ h u(t, z)dt dz dy = ˆRd \B 4 + ˆB4 B (y) |∇v T (1, z)| ˆ1 0 ∇δ h u(t,
|∇v T (1, z)| ˆ1 0 ∇δ h u(t, z)dt dz dy ˆRd \B 4 B (y) |∇v T (1, z)| 2 dz 1 2
T x, (η )(y)dy.

For the near-field contribution |x| < 4 , we first note that from the assumption > r * (0) we have for all

y ∈ R d B r * (y) (y) ∩ B 5 = ∅ ⇒ y ∈ B 7 . ( 2
ˆB4 B (y) |∇v T (1, z)| ˆ1 0 ∇δ h u(t, z)dt dz dy ≤ ˆB5 |∇v T (1, z)| ˆ1 0 ∇δ h u(t, z)dt dz (2.4.20) ˆ B r * (y) (y) |∇v T (1, z)| ˆ1 0 ∇δ h u(t, z)dt 1 B 5 (z)dz dy (2.3.53) ≤ ˆB7 B r * (y) (y) |∇v T (1, z)| 2 dz 1 2 × B r * (y) (y) ˆ1 0 ∇δ h u(t,
lim sup h→0 ˆB4 B (y) |∇v T (1, z)| ˆ1 0 ∇δ h u(t, z)dt dz dy ˆB7 B r * (y) (y) |∇v T (1, z)| 2 dz 1 2
T x, (η r * (y) )(y)dy.

For the non-generic regime < r * (0), we use the estimate (2. Proof of Lemma 8. We keep the notations of the previous proof and we give only the argument for q(r 2 ) f r • e 1 . First, equality (2.3.33) holds and take the form

δ h q(r 2 ) f r • e 1 =
ˆr2 1 s -1 2 ˆg√ s (y) ˆ1 0 ∇δ h u(t, y)dt dy ˆr2 1 s -1 2 T x, (η √ s )(0)ds.
Regime < r. We first note that from the assumption (2.2.49) we have, for all

y ∈ R d |f r (y)| |y| ˆr2 1 s -1-d 2 (1 + |y| 2 s ) -d 2 -3 2 ∧ (1 + |y| 2 s ) -d 2 -1 ds r (|y| + 1) d ∧ 1 (|y| + 1) d-1 , (2.3.56)
We first make use of the identity ´= ´fflB (y) to get ˆfr (y)e

+ log 2 (3 ) n=0 2 n B 2 n+1 ˆ1 0 ∇δ h u(t, y)dt 2 dy 1 2
which gives the first term in the second r.h.s term of (2.2.52) by passing to the lim sup and using (2.3.45) applied both with R = 1 and R = 2 n+1 as well as (2.3.46). For the far-field contribution, we make use of the Cauchy-Schwarz inequality in form of

ˆRd \B 2 B (y) |f r (z)| ˆ1 0 ∇δ h u(t, z)dt dz dy ˆRd \B 2 B (y) |f r (z)| 2 dz 1 2 × B (y) ˆ1 0 ∇δ h u(t, z)dt 2 dz 1 2
dy, and we get the second term in the second r.h.s term of (2.2.52) by passing to the lim sup and using (2.3.45) applied with R = as well as (2.3.46).

Proof of Proposition 3: Suboptimal fluctuation estimates.

We split the proof into three steps. In the first two steps we control ´|∂ fct x, q r (T )| 2 dx, using the bound of ∂ fct

x, q r (T ) proved in Lemma 7. More precisely, in the first step we treat the regime < √ T and in the second step the regime ≥ √ T . In the last step we deduce the desired moment bound (2.2.53) from the multiscale logarithmic Sobolev inequality, in form of (2.4.1), and the moment bound (2.4.3) on r * . We start with preliminary estimates.

Step 0. Preliminary. First, we will use several times the following sub-optimal deterministic decay in time of averages of ∇u: for all t ∈ (0, T ]

ˆB (x) |∇u(t, y)| 2 dy d r * (x) ∨ 1 d 1 < √ t + 1 ≥ √ t t -2 . (2.3.57)
This estimate is a direct consequence of the combination of the localized energy estimate (2.2.32) applied to (2.1.10) with R = √ t and Lemma 5 applied with f ≡ 1, g : t ∈ R + → t -2 and R = . for correlated coefficient fields

Second, we will use several times the following large-scale regularity result: for all t ∈ (-∞, T ] and

x ∈ R d B r * (x) (x) |∇v T (t, y)| 2 dy r d * (0) log 2 (1 + |x| r ) (|x| + r) 2d + r -d η 4r * (x) (x) + g 2 2r (x), (2.3
( r * (0) r ∨ 1) d ≤ 2r * (0) since r ≥ 1) in form of B r * (x) (x) |∇v T (t, y)| 2 dy = d k=1 B r * (x) (x) |∇ṽ k (t -T, y)| 2 dy r d * (0) log 2 (1 + |x| r ) (|x| + r) d . In the regime 2r * (x) ≥ √ T -t,
η 2r * (x) (x -y)g 2 r (y)dy r -d η 4r * (x) (x) ˆg 1 √ 2 r (y)dy r -d η 4r * (x) (x).
We now turn to the proof of (2.2.53).

Step

1. Regime < √ T . Proof that for all ∈ [1, √ T ) ˆ|∂ fct x, q r (T )| 2 dx 2d r -d (1 + log 2 (T ) + log 2 ( √ T r ))C (r, ), (2.3
ˆB (x) ˆT r 2 ∇u(t, y)dt 2 dy ≤ ˆT r 2 ˆB (x) |∇u(t, y)| 2 dy 1 2 dt 2 (2.3.57) (r * (x) ∨ ) d log 2 ( √ T r ),
so that we finally get, using in the last line the Lipschitz property of r * in form of sup y∈B (x) r * (y) ∨

1 d r * (x) ∨ 1 d combined with the identity ´fflB (x) dx = ´and change of variables x → x r ˆˆB (x) ˆT r 2 ∇u(t, y)dt 2 dy ˆB (x) g 2 r (y)dy dx 1 2 log( √ T r ) ˆ(r * (x) ∨ ) d ˆB (x) g 2 r (y)dy dx 1 2 d r -d 2 log( √ T r ) ˆ r * (rx) ∨ 1 d g 2 1 (x)dx
≤ N -1 j=0 ˆ ˆ2j+1 2 j ˆB (x) |∇u(t, y)||∇v T (T, y)|dy dt 2 dx 1 2 ≤ N -1 j=0 ˆˆ2 j+1 2 j ˆB (x) |∇u(t, y)| 2 dy dt ˆ2j+1 2 j ˆB (x) |∇v T (t, y)| 2 dy dt dx 1 2
(2.3.70)

(2.3.57) N -1 j=0 2 -j 2 ˆ(r * (x) ∨ ) d ˆ2j+1 2 j ˆB (x) |∇v T (t, y)| 2 dy dt 1 2
.

(2.3.71)

In addition, by the 1 8 -Lipschitz property of r * in form of r * (x) ∨ inf y∈B (x) r * (y) ∨ and sup y∈B r * (x) (x) r * (y) ∨ r * (x) ∨ as well as the identity ´fflB (x) dx = ´and the property (2.4.20), we have for all j ∈ 0, N -1

ˆ(r * (x) ∨ ) d ˆ2j+1 2 j ˆB (x) |∇v T (t, y)| 2 dy dt dx 1 2 d 2 ˆˆ2 j+1 2 j (r * (x) ∨ ) d |∇v T (t, x)| 2 dt dx 1 2 (2.4.20) d 2 ˆˆ2 j+1 2 j B r * (x) (x) (r * (y) ∨ ) d |∇v T (t, y)| 2 dy dt dx 1 2 d 2 ˆ(r * (x) ∨ ) d ˆ2j+1 2 j B r * (x) (x)
|∇v T (t, y)| 2 dy dt dx

1 2
.

(2.3.72)

Then, using the large-scale estimate (2.3.58) we have ˆ2j+1

2 j B r * (x) (x) |∇v T (s, y)| 2 dy ds 2 j r d * (0) log 2 (1 + |x| r ) (|x| + r) 2d + r -d η 4r * (x) (x) + g 2 2r (x) . (2.3.73)
Therefore, from (2.3.72), (2.3.73) and the change of variables x → x r , we get as well as r * (x) ∨ 1 r * (x). This concludes the argument for (2.3.60).

ˆ(r * (x) ∨ ) d ˆ2j+1 2 j ˆB (x) |∇v T (t, y)| 2 dy dt dx 1 2 2 j 2 d 2 r -d 2 r d * (0) ˆ(r * (rx) ∨ ) d log 2 (|x| + 1) (|x| + 1) 2d dx + ˆ(r * (x) ∨ ) d η 4r * (x) (x)dx + ˆ(r * (rx) ∨ ) d g 2 2 (x)dx
Step

2. Regime ≥ √ T . Proof that for all ∈ [ √ T , +∞) ˆ|∂ fct x, q r (T )| 2 dx d log 2 (T )(r 2d * (0) + D ,1 (r, ) + D ,2 (r) log 2 ( √ T r )), (2.3
D ,1 (r, ) =r d * (0) d ˆRd \B log 2 (1 + |x| r ) (|x| + r) 2d dx + d ˆRd \B r d * (x)|x| -2d dx + r d * (0) ˆB7 log(1 + |x| r ) (|x| + r) d dx 2 + ˆB7 r d 2 +1 * (x)(|x| + 1) -d-1 dx 2 + r d * (0) ˆrd * (x) (|x| + 1) 2d dx, ( 2
|∇v T (1, y)| 1 + ˆ1 0 ∇u(s, y)ds dy 2 dx + ˆG2 r, (x)dx d D ,1 (r, ), (2.3.79)
where we recall that G r, is defined in (2.2.47).

Argument for (2.3.78). Since for all x ∈ R d , ´B (x) g r (y)dy ≤ ´gr (y)dy 1, we have by the Jensen inequality applied with the measure g r dy and the identity ´fflB

(x) = ´ ˆB (x) ˆT 0 ∇u(t,
ˆ B r * (x) (x) |∇v T (1, y)| 2 1 R d \B 3 (y)dy dx.
(2.3.83)

By the 1 8 -Lipschitz continuity property of r * and the assumption r * (0 . For the near-field contribution |x| < 4 we write, using the Minkowski inequality in L 2 (B 4 ), the Fubini-Tonnelli theorem, the property (2.4.20) and the assumption ≥ r * (0) in form of (2.3.53)

) ≤ one has B r * (x) (x) ∩ R d \B 3 = ∅ → x ∈ R d \B . Indeed, by contradiction, if x ∈ B , then r * (x) ≤ r * (0) + |x| 8 ≤ 15 16 so that B r * (x) (x) ⊂ B + 15 16 ⊂ B 2 and consequently B r * (x) (x) ∩ R d \B 3 = ∅. Hence, ˆ B r * (x) (x) |∇v T (1, y)| 2 1 R d \B 3 (y)dy dx ≤ ˆRd \B B r * (x) (x) |∇v T (1, y)| 2 dy dx, ( 2
ˆB4 ˆB (x) |∇v T (1, y) 1 + ˆ1 0 ∇u(s, y)ds dy 2 dx 1 2 ≤ ˆ ˆB4 |∇v T (1, y)| 2 1 + ˆ1 0 ∇u(s, y)ds 2 1 B (x) (y)dx 1 2 dy d 2 ˆB5 |∇v T (1, y)| 1 + ˆ1 0 ∇u(s, y)ds dy (2.4.20) d 2 ˆ B r * (x) (x) |∇v T (1, y)| 1 + ˆ1 0 ∇u(s, y)ds 1 B 5 (y)dy dx (2.3.53) d 2 ˆB7 B r * (x) (x) |∇v T (1, y)| 1 + ˆ1 0 ∇u(s,
d 2 r d 2 * (0) ˆB7 log(1 + |x| r ) (|x| + r) d dx + ˆB7 η 1 2 4r * (x) (x)dx + ˆB7 g r (y)dy ,
which concludes since ´B7 g r (y)dy 1 and η

1 2 4r * (x) (x) r d 2 +1 * (x)(|x| + 1) -d-1 .
Regime < r * (0). We use the estimate (2.3.63) and we bound one d by r d * (0) and r -d by 1.

We now turn to the second l.h.s term of (2.3.79). The first term in the definition (2.2.47) of G r, is bounded as follows. By definition of T x, (η r )(0), that we recall here

T x, (η r )(0) = ˆB (x) η r (z) 1+ ˆ1 0 ∇u(t, z)dt 2 dz 1 2 + ˆ1 0 1 1 -t ˆ1 t ˆB (x) η r (z)|∇u(s, z)| 2 dz
d 1 + ˆ1 0 -log(t) 1 -t dt 2 d . (2.3.89)
For the other term in (2.2.47), we can use previous estimates. To this aim, we need preliminary inequalities and we distinguish between the two regimes ≥ r * (0) and < r * (0).

Regime ≥ r * (0). For the far-field contribution |x| ≥ 4 , we make use of the Jensen inequality with the measure T x, (η )(y)dy in form of, for all

x ∈ R d ˆRd \B 4 B (y) |∇v T (1, z)| 2 dz 1 2 T x, (η )(y)dy 2 ≤ ˆRd \B 4 T x, (η )(y)dy × ˆRd \B 4 B (y) |∇v T (1, z)| 2 dz T x, (η )(y)
T x, (η )(y) η 1 2 2 (y -x) ˆB (x) 1 + ˆ1 0 ∇u(s, z)ds 2 dz 1 2 + ˆ1 0 1 1 -t ˆ1 t ˆB (x) |∇u(s, z)| 2 dz 1 2 dt d 2 η 1 2 2 (y -x). Consequently, ˆRd \B 2 (x) T x, (η )(y)dy d 2 ˆRd \B 2 (x) η 1 2 2 (y -x)dy d ,
|∇v T (1, z)| 2 dz 1 2
T x, (η r * (y) )(y)dy

2 dx ≤ ˆB7 B r * (y) (y) |∇v T (1, z)| 2 dz 1 2
ˆT 2

x, (η r * (y) )(y)dx

1 2 dy 2 d ˆB7 B r * (y) (y) |∇v T (1, z)| 2 dz 1 2 dy 2 , (2.3.92)
and we then proceed as we did from (2.3.88).

Regime < r * (0). In this regime, we use the estimate (2.3.68).

Substep 2.2. Proof that for ∈ [ √ T , +∞) ˆ ˆT 1 ˆB (x) |∇u(t, y)||∇v T (t, y)|dt dy 2 dx d log 2 (T ) d ˆRd \B log 2 (1 + |x| r ) (|x| + r) 2d dx + d ˆRd \B r d * (x)|x| -2d dx + r d * (0) ˆB7 log(1 + |x| r ) (|x| + r) d dx 2 + ˆB7 r d 2 +1 * (x)(|x| + 1) -d-1 dx 2 + r 2d * (0) + r d * (0) ˆrd * (x) (|x| + 1) 2d dx . (2.3.93)
For the proof of (2.3.93), we argue as previously and we distinguish between the generic case ≥ r * (0) and the non-generic case < r * (0). We mainly make use of previous ideas and estimates.

Regime ≥ r * (0). d 2 by 1.

Step 3. Proof of (2.2.53). We have from the multiscale logarithmic Sobolev inequality in form of (2.4.1): for all p ∈ [1, +∞)

|q r (T ) -q r (T ) | p 1 p p 1 2 ˆ+∞ 1 -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p . (2.3.97)
We split the integral over into two parts.

(i) In the regime < √ T we use (2.3.60):

ˆ√T 1 -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p (2.3.60) r -d 2 (1 + log(T ) + log( √ T r )) ˆ√T 1 d π( )C (r, )d p 2 1 p (2.1.6) ≤ r -d 2 (1 + log(T ) + log( √ T r )) ˆ√T 1 d-1-β C p (r, ) 1 p d 1 2 r -d 2 (1 + log(T ) + log( √ T r ))µ β (T ) sup ≥1 C p (r, ) 1
ˆ+∞ √ T -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p (2.3.75),(2.1.6) log(T ) ˆ+∞ √ T -1-β (r d * (0) + D ,1 (r, ))d p 2 1 p + log( √ T r ) D p 2
,2 (r)

1 p ˆ+∞ √ T -1-β d 1 2 T -β log(T ) ˆ+∞ 1 -1-β (r d * (0) + D ,1 (r, √ T )) p 1 p d 1 2
+ log(

√ T r ) D p 2
,2 (r)

1 p r -d 2 log(T )µ β (T ) ˆ+∞ 1 -1-β (r d * (0) + D ,1 (r, √ T )) p 1 p d 1 2
+ log(

√ T r ) D p 2
,2 (r)

1 p . ( 2
ˆ+∞ 1 -1-β r d * (0)( √ T ) d ˆRd \B √ T log 2 (1 + |x| r ) (|x| + r) 2d dx + ( √ T ) d ˆRd \B √ T r d * (x)|x| -2d dx p 1 p d 1 2
r dp * (0)

1 2p T d 4 ˆ+∞ 1 d-1-β ˆRd \B √ T log 2 (1 + |x| r ) (|x| + r) 2d dx + ˆRd \B √ T |x| -2d dx d 1 2
(2.3.102)

p 1 2 d β∧d ˆ+∞ 1 -1-β (1 + log 2 ( √ T r ))d 1 2 p 1 2 d β∧d (1 + log( √ T r )). ( 2
r d * (0) ˆB7 √ T log(1 + |x| r ) (|x| + r) d dx 2 + ˆB7 √ T r d 2 +1 * (x)(|x| + 1) -d-1 dx 2 + r d * (0) ˆrd * (x) (|x| + 1) 2d dx p 1 p d 1 2 (2.3.102) p d+2 β∧d ˆ+∞ 1 -1-β ˆ7 √ T 0 log(1 + ρ r ) (ρ + r) d ρ d-1 dρ + 1 2 d 1 2 ≤ p d+2 β∧d ˆ+∞ 1 -1-β log 2 (1 + 7 √ T r ) ˆ7 √ T r 0 (ρ + 1) -d ρ d-1 dρ + 1 2 d 1 2 , (2.3.104)
and we estimate the integral in the integrand of the r.h.s of (2.3.104) by, for all ≥ 1

ˆ7 √ T r 0 (ρ + 1) -d ρ d-1 dρ ˆ1 0 (1 + log(1 + T r 2 ρ -2 ))ρ d-1 dρ + ˆ7 √ T r 1 ρ -1 dρ 1 + log(7 √ T r
).

(

Proof of the main results

Proof of Theorem 10: Fluctuations of the time dependent flux

We only gives the argument for the flux q(T ), the computations for φ(T ) are equivalent and are done by a straightforward adaptation of the argument of this proof and the ones to prove Lemmas 7 and 8.

Our first goal is to remove the log(T ) correction in the r.h.s of (2.2.53), which will lead to (2.1.17).

To this aim, we use the L 2 -L 1 type estimate of Lemma 3, which allow us to make the link between the r -d 2 decay of the fluctuations of q r (T, •), for r ≤ √ T , proved in Proposition 3 and the decay in T of moments of averages of ∇u(T, •). It yields a better decay estimate in time T of spatial averages of ∇u(T, •) than the one obtained in (2.3.57) (from which the log(T ) contribution in (2.2.53) comes from). With this new decay in hand, we are able to obtain optimal estimate in scaling. The price to pay in this step is a little loss of stochastic integrability. Our second goal is to prove estimate (2.1.20). It does not require new ideas and this is done by dominating carefully the terms in the derivative (2.2.50) and by using some estimates already done in the proof of (2.1.17).

Proof of (2.1.17). We split the proof into three steps.

Step 1. Improvement of (2.3.57). Proof that for all x ∈ R d , T ≥ 1 and ε ∈ (0, 1)

ˆB (x) |∇u(T, y)| 2 dy ≤ d r d * (x)1 < √ T + 1 ≥ √ T D 2ε (T, , x)η 2 ε,β (T ), (2.3.106)
with

η ε,β (T ) =      (1 + log ε (T ))T -1-ε β 4 if β < d, (1 + log 3 2 ε (T ))T -1-ε d 4 if β = d, (1 + log ε (T ))T -1-ε d 4 if β > d, (2.3.107)
and for some stationary random field D (T, , •) that satisfies for all p ∈ [1, +∞) sup

(T, ,x)∈R + ×[1,+∞)×R d D p (T, , x) 1 p p 1 2 + d+2 β∧d . (2.3.108)
We have from Lemma 3, the Minkowski inequality in L p • (Ω) and the stationarity of q r : for all

p ∈ [1, +∞), T ≥ 4 and R ≥ √ T ˆη√ 2R (y)|∇u(T, y)| 2 dy p 2 1 p 1 T T 2 T 4 √ t 0 r √ t d 2
ˆη2R (y)|q r (t, y) -q r (t, y) |dy dr dt

p 1 p 1 T T 2 T 4 √ t 0 r √ t d 2 |q r (t) -q r (t) | p 1 p dr dt. (2.3.109)
Then, we split the integral over [0, √ t] into the two contributions r ≤ 1 and 1 ≤ r ≤ √ t:

(i) For r ≤ 1 we use (2.2.43) and the change of variable r → √ t r :

1 T T 2 T 4 1 √ t ˆ1 0 r √ t d 2 |q r (t) -q r (t) | p 1 p dr dt (2.2.43) p 1 2 d β∧d T -1-d 4 T 2 T 4 1 √ t ˆ1 0 r d 2 + log( √ t r ) dr dt p 1 2 d β∧d T -1-d 4 ˆ+∞ √ T r -2 (1 + log(r))dr + 1 √ T p 1 2 d β∧d T -3 2 -d 4 log(T ). ( 2
p 1 2 + d+2 β∧d T -1-d 4 µ β (T ) T 2 T 4 1 √ t ˆ√t 1 1 + log(T ) + log 2 ( √ t r
) dr dt

≤ p 1 2 + d+2 β∧d η β (T ) log(T ) + ˆ T 2 1 r -2 (1 + log 2 (r))dr p 1 2 + d+2 β∧d (1 + log(T ))η β (T ), (2.3
|∇u(T, y)| 2 dy D2 (T, R, x)(1 + log 2 (T ))η 2 β (T ), (2.3.112)
where D (T, R,

•) := ffl B R (•) |∇u(T,y)| 2 dy log 2 (T )η 2 β (T )
has the moment bound (2.3.108). It implies, from Lemma 5 applied with f :

t ∈ R + * → 1 + log 2 (t) and g : t ∈ R + * → t -2-d 2 that, for all < √ T ˆB (x) |∇u(T, y)| 2 dy (r * (x) ∨ ) d D2 (T, √ T , x) ∨ T T 2 D2 (s, √ s, x)ds (1 + log 2 (T ))η 2 β (T ).
(2.3.113) By interpolating between (2.3.57) and the combination of (2.3.112) and (2.3.113) as well as using that r * (x) ∨ 1 ≤ 2r * (x) in the last line, we then obtain for all ε ∈ (0, 1) ) for the control of (x, ) → ´|∂ fct x, q r (T )| 2 dx (corresponding to the substeps 1 and 2 of the proof of Lemma 3).

ˆB (x) |∇u(T, y)| 2 dy = ˆB (x) |∇u(T, y)| 2 dy 1-ε ˆB (x) |∇u(T, y)| 2 dy ε (2.3.57),(2.3.113) d r * (x) ∨ 1 d 1 < √ T + 1 ≥ √ T T -2(1-ε) D 2ε (T, , x)(1 + log 2ε (T ))η 2ε β (T ) ≤ 2 d r d * (x)1 < √ T + 1 ≥ √ T D 2ε (T, , x)η 2 ε,β (
Step 2. We split this step into two parts, one for the improvement of (2.3.60) and an other for (2.3.75), treating separately the two regimes < √ T and ≥ √ T .

Substep 2.1. Improvement of (2.3.60). Proof that for all ε > 0 and where

< √ T ˆ|∂ fct x, q r (T )| 2 dx ≤ 2d r -d 1 + E ,ε (r, ) + +∞ j=0 2 j 2 η ε,β (2 j )F j, ,ε (r, )
J ,ε (x) := ˆ+∞ 1 16 D ε (t, , x)η ε,β (t)dt 2 .
Using the change of variables x → x r in the last r.h.s term of (2.3.117), it gives the term E ,ε (r, ) defined in (2.3.115). On the other hand, noticing that, by monotonicity of t ∈ R + * → η ε,β (t), for all j ∈ N, we have ˆ2j+1

2 j D 2ε (t, , x)η 2 ε,β (t)dt η 2 ε,β (2 j ) ˆ2j+1 2 j D 2ε (t, , x)dt, (2.3
1 B (x) (z)|∇v T (t, z)| 2 dz dy dt ≤ ˆ2j+1 2 j ˆB (x) B r * (y) (y) |∇v T (t, z)| 2 dz dy dt + ˆ2j+1 2 j ˆRd \B (x) B r * (y) (y) 1 B (x) (z)|∇v T (t, z)| 2 dz dy dt.
(2.3.121)

Next, we make use of (2.3.120) to bound the second r.h.s term of (2.3.121) with ˆ2j+1

2 j ˆRd \B (x) B r * (y) (y) 1 B (x) (z)|∇v T (t, z)| 2 dz dy dt (r * (x) ∨ ) d+ε ˆ2j+1 2 j ˆRd \B (x) |y -x| -d-ε B r * (y) (y) |∇v T (t, z)| 2 dz dy dt. (2.3.122)
The combination of (2.3.121) and (2.3.122) (where we bound (r * (x) ∨ ) d+ε d+ε r d+ε * (x)) as well as (2.3.73) (where we bound η 4r * (x) (x) r d * (x)(|x| + 1) -2d ) proves that the r.h.s integral of (2.3.119) is indeed bounded by 2 j r -d F j, ,ε (r, ) and concludes the argument for (2.3.114).

Substep 2.2. Improvement of (2.3.75). Proof that for all ε > 0 and ≥ √ T

ˆ|∂ fct x, q r (T )| 2 dx d D ,1 (r, ) + D ,2 (r) log 2 ( √ T r ) + +∞ j=0 2 j 2 η ε,β (2 j )H j, ,ε (r, ) 2 , (2.3.123)
with D ,1 (r, ) and D ,2 (r) defined respectively in (2.3.77) and (2.3.76),

H j, ,ε (r, ) = d H j, ,ε (r, , R d \B 4 ) + r d * (0)H j, ,ε (r, , R d ) + ˆB7 ˆ2j+1 2 j D 2ε (t, , x)dt 1 2 r d 2 * (0) log(1 + |x| r ) (|x| + r) d + r d 2 +1 * (x) (|x| + 1) d+1 + g 2r (x) dx, (2.3.124)
as well as for all open subsets U of R d

H j, ,ε (r, , U) = ˆU ˆ2j+1 2 j D 2ε (t, , x)dt B (x) r d * (0)r d log 2 (1 + |y| r ) (|y| + r) 2d dy + r d * (y) (|y| + 1) 2d + r d g 2 2r (y)dy + r d+ε * (x) ε ˆRd \B (x) |y -x| -d-ε r d * (0)r d log 2 (1 + |y| r ) (|y| + r) 2d dy + r d * (y) (|y| + 1) 2d + r d g 2 2r (y) dy dx 1 2
. for correlated coefficient fields

The estimates (2.3.78) and (2.3.79) are unchanged and give respectively the d D ,2 (r) log 2 ( √ T r ) and d D ,1 (r, ) contributions in the r.h.s of (2.3.123). We improve the estimate (2.3.93). We argue differently with the generic case ≥ r * (0) and the non-generic case < r * (0).

Regime ≥ r * (0). We have from ( Step 3. Proof of (2.1.22). We have from the logarithm Sobolev inequality in form of (2.4.1), for all p ∈ [1, +∞)

|q r (T ) -q r (T ) | p 1 p p 1 2 ˆ+∞ 1 -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p ≤ p 1 2 (I 1 √ T + I 2 √ T ), (2.3.128) 2.3. Proofs 115 with I 1 √ T := ˆ√T 1 -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p
, and

I 2 √ T := ˆ+∞ √ T -d π( ) ˆ|∂ fct x, q r (T )| 2 dx d p 2 1 p
.

We then treat separately the two terms above.

(i) In the regime < √ T we use (2.3.114) combined with the Minkowski inequality in L p • (Ω):

I 1 √ T (2.3.114) r -d 2 ˆ√T 1 d π( ) 1 + E ,ε (r, ) + +∞ j=0 2 j 2 η ε,β (2 j )F j, ,ε (r, ) 2 d p 2 1 p (2.1.6) ≤ r -d 2 ˆ√T 1 d-1-β 1 + E p ,ε (r, ) 1 p + +∞ j=0 2 j 2 η ε,β (2 j ) F 2p j, ,ε (r, ) 1 2p 2 d 1 2 r -d 2 µ β (T ) 1 + sup ≥1 E p ,ε (r, ) 1 2p + sup ≥1 +∞ j=0 2 j 2 η ε,β (2 j ) F 2p j, ,ε (r, ) 1 2p
,

I 2 √ T ˆ+∞ √ T π( ) D ,1 (r, ) + D ,2 (r) log 2 ( √ T r ) + +∞ j=0 2 j 2 η ε,β (2 j )H j, ,ε (r, ) 2 d p 2 1 p (2.1.6) ˆ+∞ √ T -1-β D p ,1 (r, ) 1 p + D p ,2 (r) 1 p log 2 ( √ T r ) + +∞ j=0 2 j 2 η ε,β (2 j ) H 2p j, ,ε (r, ) 1 2p 2 d 1 2 r -d 2 µ β (T ) ˆ+∞ 1 -1-β D p ,1 (r, √ T )) 1 p d 1 2
+ log(

√ T r ) sup r≥1 D p ,2 (r) 1 2p
+ sup

≥1 +∞ j=0 2 j 2 η ε,β (2 j ) H 2p j, ,ε (r, ) 1 2p
.

(2.3.130)

It remains to show that, for all ε ∈ (0, 1)

sup ≥1 E p ,ε (r, ) 1 2p + sup ≥1 +∞ j=0 2 j 2 η ε,β (2 j ) F 2p j, ,ε (r, ) 1 2p p 2 d+1 β∧d +ε (1 + log( √ T r )), (2.3.131)
and ˆ+∞

1 -1-β D p ,1 (r, √ T )) 1 p d 1 2
+ log(

√ T r ) sup r≥1 D p ,2 (r) 1 2p + sup ≥1 +∞ j=0 2 j 2 η ε,β (2 j ) H 2p j, ,ε (r, ) 1 2p p 2 d+1 β∧d +ε (1 + log 2 ( √ T r )). (2.3.132)
This is done the same way as for (2.3.100) and (2.3.101) using in addition (2.3.108) and +∞ j=0 2 j η ε,β (2 j ) ε 1. for correlated coefficient fields Proof of (2.1.20). We use the same notations as in the previous proof. We split the proof into two parts, treating separately the two regimes < r and ≥ r. We start with preliminary estimates.

Step 0. Preliminary. First, we will use several times the assumption (2.1.19)

r 2 log 2 (1 + |x|) (|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) +(1+log(r+1)1 d=2 )η 4r * (x) (x)+f 2 2r (x).
(2.3.137) Step 1. Regime < r. Proof that for all ε > 0 and < r

ˆ|∂ fct x, q(r 2 ) f r | 2 dx ≤ d r d * (0)(1 + log(r + 1)1 d=2 ) + 2 (1 + log( r + 1)1 d=2 ) + M ,1 (r, ) + M ,2 (r, ) + +∞ j=0 2 j 2 η ε,β (2 j )K j, ,ε (r, ) 2 , (2.3.138)
with

M ,1 (r, ) :=1 + N ,ε (r, , 1) + log 2 (3 ) n=0 2 n N 1 2 ,ε (r, , 2 n+1 ) 2 + d log 2 (r) n= log 2 ( ) 2 -n(d-2) N ,ε (r, , 2 n+1 ) + r 2 d +∞ n= log 2 (r) 2 -nd N ,ε (r, , 2 n+1 ) (2.3.139)
where for all ρ > 0, N ,ε (r, , ρ) = r d * (0)

´r2 1 D ε (t, ρ, 0)η ε,β (
(|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) dx + (1 + log(r)1 d=2 ) d ˆRd \B r d * (x) (|x| + 1) 2d dx + d ˆRd \B f 2 2r (x)dx + r d * (0) ˆB7 r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 dx 2 + (1 + log(r)1 d=2 ) ˆB7 r d 2 +1 * (x) (|x| + 1) d+1 dx 2 + ˆB7 f 2r (x)dx 2 .
(2.3.140)

as well as

K j, ,ε (r, ) := K j, ,ε (r, , R d \B 4 ) + r d 2 * (0)K j, ,ε (r, , R d ) + ˆB7 ˆ2j+1 2 j D 2ε (t, , x)dt 1 2 r d 2 (0) r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 + (1 + log 1 2 (r)1 d=2 ) r d 2 +1 * (x) (|x| + 1) d + f 2r (x) dx, (2.3.141)
with for all open set U of R d K j, ,ε (r, , U)

:= ˆU ˆ2j+1 2 j D 2ε (t, , x)dt 1 2 B (x) r d * (0) r 2 log 2 (1 + |y|) (|y| + 1) 2d ∧ 1 (|y| + 1) 2(d-1) + (1 + log(r)1 d=2 ) r d * (y) (|y| + 1) 2d + f 2 2r (y) + r d+ε * (x) ε ˆRd \B (x) |y -x| -d-ε r d * (0) r 2 log 2 (1 + |y|) (|y| + 1) 2d ∧ 1 (|y| + 1) 2(d-1) + (1 + log(r)1 d=2 ) r d * (y) (|y| + 1) 2d + f 2 2r (y) dy dx 1 2
.

We split this step into two parts. The first part is devoted to the control of the first fifth r.h.s terms of (2.2.50). x, (η 1 )(0)dx +

ˆ log 2 (3 ) n=0 2 n T x, (η 2 n+1 )(0) 2 dx + ˆ ˆRd \B 2 B (y) |f r (z)| 2 dz 1 2
T x, (η )(y)dy

2 dx d (1 + 2 (1 + log( r + 1)1 d=2 )), (2.3
1 + r d * (0) ˆr2 1 D ε (t, 1, 0)η ε,β (t)dt 2 ,
and for all n ∈ N Proof of (2.3.144). We start with the first l.h.s term. We distinguish between the generic case ≥ r * (0) and the non-generic case < r * (0).

B 2 n+1 ˆr2 0 ∇u(t, y)dt 2 dy ≤ 1 + r d (0) ˆr2 1 D ε (t, 2 n+1 , 0)η ε,β (t)dt 2 . ( 2
Regime ≥ r * (0). We split the integral into the far-field contribution |x| ≥ 4 and the nearfield contribution |x| < 4 . For the far-field contribution, we make use of the estimates (2.

d 2 r d 2 * (0) ˆB7 r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 dx + (1 + log 1 2 (r + 1)1 d=2 ) ˆB7 η 1 2 4r * (x) (x)dx + ˆB7 f 2r (x)dx ,
which gives the second term of (2.3.140) using that η

1 2 4r * (x) (x) r d 2 +1 * (x)(|x| + 1) -d-1 .
Regime < r * (0). We use the Cauchy-Schwarz inequality, the identity ´fflB (x) = ´and the localized energy estimate (2.

2 j 2 η ε,β (2 j )K j, ,ε (r, , R d \B 4 ) 2 .
For the near-field contribution, we make use of (

2 j D 2ε (t, , x)dt 1 2 r d 2 * (0) r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 + (1 + log 1 2 (r + 1)1 d=2 ) r d 2 +1 * (x) (|x| + 1) d + f 2r (x) dx 2 ,
where we used that +∞ j=0 2

j 2 η ε,β (2 j ) < +∞.
Regime < r * (0). We make use of (

+∞ j=0 2 j 2 η ε,β (2 j )K j, ,ε (r, , R d ) 2 .
Step 2. Regime ≥ r. Proof that for all ε > 0 and ≥ r

ˆ|∂ fct x, q(r 2 ) f r | 2 dx ≤ d r 2 + M ,3 (r) + M ,2 (r, ) + +∞ j=0 2 j 2 η ε,β (2 j )K j, ,ε (r, ) 2 , (2.3.149)
with

M ,3 (r) = ˆr2 1 s -1 2 (1 + log( r √ s )) B √ s r d * (y)dy + ˆRd \B 1 r d * (y)( √ sy)g √ 2 (y)dy 1 2 ds 2 .
The estimates (2.3.144) and (2.3.148) are unchanged. We provide the arguments for the two first and the fourth r.h.s terms of (2.2.50) and prove that

ˆ ˆr2 1 s -1 2 T x, (η √ s )(0)ds 2 dx d r 2 . ( 2
ˆr2 1 s -1 2 (1 + log( r √ s )) B √ s r d * (y)dy + ˆRd \B 1 r d * (y)( √ sy)g √ 2 (y)dy 1 2
ds.

(

.

Step 3. Proof of (2.1.20). We have from the logarithm Sobolev inequality in form of (2.4.1), for all p ∈ [1, +∞)

|q(r 2 ) f r -q(r 2 ) f r | p 1 p p 1 2 ˆ+∞ 1 -d π( ) ˆ|∂ fct x, q r (r 2 ) f r | 2 dx d p 2 1 p ≤ p 1 2 (I 1 r + I 2 r ), (2.3.153) with I 1 r := ˆr 1 -d π( ) ˆ|∂ fct x, q r (r 2 ) f r | 2 dx d p 2 1 p
, and

I 2 r := ˆ+∞ r -d π( ) ˆ|∂ fct x, q r (r 2 ) f r | 2 dx d p 2 1 p
.

We then treat separately the two terms above.

(i) In the regime < r we use (

I 1 r ≤ ˆr 1 -1-β r d * (0)(1 + log 2 (r + 1)1 d=2 ) + 2 (1 + log( r + 1)1 d=2 ) + M ,1 (r, ) + M ,2 (r, ) + +∞ j=0 2 j 2 η ε,β (2 j )K j, ,ε (r, ) 2 p 2 1 p p d β∧d 1 + χ d,β (r)) + ˆr 1 -1-β ( M p ,1 (r, ) 1 p + M p ,2 (r, ) 1 p + +∞ j=0 2 j 2 η ε,β (2 j ) K p j, ,ε (r, ) 1 p d 1 2
.

(ii) In the regime ≥ r we use (2.3.149) combined with the Minkowski inequality L p • (Ω):

I 2 r ≤ ˆ+∞ r -1-β r 2 + M ,3 (r) + M ,2 (r, ) + +∞ j=0 2 j 2 η ε,β (2 j )K j, ,ε (r, ) 2 d p 2 1 p r 1-β 2 + ˆ+∞ r -1-β M p ,3 (r, ) 1 p + M p ,2 (r, ) 1 p + +∞ j=0 2 j 2 η ε,β (2 j ) K p j, ,ε (r, ) 1 p d 1 2
.

It remains to prove that for all ε ∈ (0, 1)

ˆr 1 -1-β M p ,1 (r, ) 1 p + M p ,2 (r, ) 1 p + +∞ j=0 2 j 2 η ε,β (2 j ) K p j, ,ε (r, ) 1 p d p d+2 β∧d +ε χ 2 d,β (r), (2.3

.154) and ˆ+∞

r -1-β M p ,3 (r, ) 1 p + M p ,2 (r, ) 1 p + +∞ j=0 2 j 2 η ε,β (2 j ) K p j, ,ε (r, ) 1 p d p d+2 β∧d +ε χ 2 d,β (r).
( . Therefore, by making use of the triangle inequality, we get

ˆr 1 -1-β M p ,1 (r, ) 1 p d 1 + p d β∧d +ε(1+2 d+2 β∧d ) ˆr 1 -1-β (1 + 2 (1 + log( r + 1)1 d=2 ) + d r 2-d )d p d β∧d +ε(1+2 d+2 β∧d ) χ 2 d,β (r).
Secondly, from the triangle inequality, the moment bounds (2. we have for all < r M p ,2 (r, )

1 p p d β∧d d ˆBr\B 1 (|x| + 1) 2(d-1) dx + d ˆRd \Br r 2 log 2 (1 + |x|) (|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) dx + (1 + log 2 (r + 1)1 d=2 ) d ˆRd \B 1 (|x| + 1) 2d dx + d ˆRd \B |f 2r (y)| 2 dx + ˆB7 1 (|x| + 1) d-1 dx 2 + (1 + log 2 (r + 1)1 d=2 )p 2 β∧d ˆB7 1 (|x| + 1) d+1 dx 2 + ˆB7 f 2r (x)dx 2 (2.3.136),(2.3.147) p d β∧d 2 (1 + log( r + 1)1 d=2 ) + (1 + d r 2-d + p 2 β∧d )(1 + log 2 (r + 1)1 d=2 ) , which provides ˆr 1 M p ,2 (r, ) 1
p d p d+2 β∧d χ 2 d,β (r).
Finally, using the same decomposition as before and in addition the moment bound (2.3.108) of D we get

ˆr 1 +∞ j=0 2 j 2 η ε,β (2 j ) K p j, ,ε (r, ) 1 p d p d+2 β∧d +ε(1+2 d β∧d ) χ 2 d,β (r),
which concludes the proof of (2. Secondly, using the triangle inequality M p ,2 (r, )

1 p p d β∧d d ˆRd \B r 2 log 2 (1 + |x|) (|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) dx + (1 + log(r + 1)1 d=2 ) d ˆRd \B 1 (|x| + 1) 2d dx + d ˆRd \B |f 2r (x)| 2 dx + ˆB7 r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 dx 2 + (1 + log(r + 1)1 d=2 )p 2 β∧d ˆB7 1 (|x| + 1) d+1 dx 2 + ˆB7 |f 2r (x)|dx 2 p d+2 β∧d ˆRd \B r 2 log 2 (1 + |x|) (|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) dx + ˆB7 r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 dx 2 + 1 + log(r + 1)1 d=2 + r 2 .
(2.3.156)

We then argue differently, depending on the regime of β and d:

2.3. Proofs 125 (i) For β > 2, we use ˆRd \B r 2 log 2 (1 + |x|) (|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) dx + ˆB7 r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 dx 2 ≤ r 2 ˆRd \B log 2 (1 + |x|) (|x| + 1) 2d dx + r 2 ˆB7 log(1 + |x|) (|x| + 1) d dx 2 r 2 (1 + log 2 ( )), (2.3.157)
to deduce, combined with (2.3.156)

ˆ+∞ r -1-β M p ,2 (r, ) 1 p d ˆ+∞ r -1-β (r 2 log 2 ( ) + 1 + r 2 + log(r + 1)1 d=2 )d r -β (r 2 + 1 + log(r + 1)1 d=2 + log 2 (r + 1)) (2.3.158) 1,
where we used in the last line that β > 2.

(ii) For β ≤ 2 and d > 2, we use

ˆRd \B r 2 log 2 (1 + |x|) (|x| + 1) 2d ∧ 1 (|x| + 1) 2(d-1) dx+ ˆB7 r log(1 + |x|) (|x| + 1) d ∧ 1 (|x| + 1) d-1 dx 2 r 2 log 2 ( )∧ 2 ,
which yields combined with (2.3.156) and a dyadic decomposition ˆ+∞ r -1-β M p ,2 (r, ) We provide the argument for ∇u, the decay of u is proved the same way. We apply Lemma 3 and we make use of the Minkowski inequality in L p • (Ω) and the stationarity of q r (T, •) to the effect of:

1 p d log 2 (r) n=0 ˆ2n+1 r 2 n r -1-β ( 2 + 1 + r 2 + log(r + 1)1 d=2 )d + +∞ n= log 2 (r) ˆ2n+1 r 2 n r -1-β (r 2 log 2 ( ) + 1 + r 2 + log(r + 1)1 d=2 )d log 2 (r) n=0 (2 n r) 2-β 1 β<2 + 1 β=2 + r -β 2 -nβ (r 2 + log(r + 1)1 d=2 ) + +∞ n= log 2 (r) 2 -nβ (r 2-β log 2 (2 n+1 r) + 1 + r 2-β + log(r + 1)1 d=2 ) χ β,d (r).
for all p ∈ [1, +∞), T ≥ 4 and R ≥ √ T ˆη√ 2R ( y c )|∇u(T, y)| 2 dy p 2 1 p 1 T T 2 T 4 √ t 0 r √ t d 2 ˆη2R ( y c
)|q r (t, y) -q r (t, y) |dy dr dt

p 1 p 1 T T 2 T 4 √ t 0 r √ t d 2 |q r (t) -q r (t) | p 1 p dr dt.
Then, we split the integral over [0, √ t] into the contributions r ≤ 1 and 1 ≤ r ≤ √ t:

(i) For r ≤ 1 we use (2.2.43):

1 T T 2 T 4 1 √ t ˆ1 2 0 r √ t d 2 |q r (t) -q r (t) | p 1 p dr dt (2.2.43) p 1 2 d β∧d T -1-d 4 T 2 T 4 1 √ t ˆ1 0 r d 2 + log( √ t r ) dr dt p 1 2 d β∧d T -3 2 -d 4 log(T ).
(ii) For 1 ≤ r ≤ √ t we use Theorem 10 and the change of variable r → √ t r : for all α <

1 1 2 +2 d+1 β∨d 1 T T 2 T 4 1 √ t ˆ√t 1 r √ t d 2 |q r (t) -q r (t) | p 1 p dr dt (2.1.17) p 1 α T -1-d 4 µ β (T ) T 2 T 4 1 √ t ˆ√t 1 1 + log 2 ( √ t r
) dr dt

≤ p 1 α η β (T ) ˆ T 2 1 r -2 (1 + log 2 (r))dr p 1 α η β (T ),
where η β is defined in (2.1.24). It concludes the argument for (2.1.23).

The estimate (2.1.25) is a direct consequence of (2.1.23) and the stationarity of ∇u: for all x ∈ R d and T ≥ 1

|∇u(T, x)| 2 = ˆη√ 2R ( y c )|∇u(T, y)| 2 dy (2.1.23) T -1 η 2 β (T ) C 2 ,d,λ,β T -1 η 2 β (T ).
Proof of Corollary 2 : Bounds on the flux and gradient of correctors.

We split the proof into two steps. The first one gives a rigorous proof of the formula (2.1.31). The second step prove (2.1.32).

Step 1. We prove the two following integral formulas

∇φ(•) = ˆ+∞ 0 ∇u(t, •)dt, (2.3
B R |u(T, y)| 2 dy 1 2 ≤ B R u(T, y) - B R u(T, z)dz 2 dy 1 2 + B √ T u(T, y)dy 2 1 2 + B R u(T, y)dy - B √ T u(T, y)dy 2 1 2 R B R |∇u(T, y)| 2 dy 1 2 + B √ T |u(T, y)| 2 dy 1 2 + B R u(T, y)dy - B √ T u(T, y)dy 2 1 2 (2.1.23),(2.1.25) ( R √ T + 1)η β (T ) + B R u(T, y)dy - B √ T u(T, y)dy 2 1 2 . (2.3.162)
From the fundamental calculus theorem, the stationarity of ∇u and the application of (2.1.25), we also have

B R u(t, y)dy - B √ T u(t, y)dy 2 1 2 = ˆ√T R B 1 ∇u(t, τ z) • zdz dτ 2 1 2 (2.1.25) (1 - R √ T )η β (T ) ≤ η β (T ). Hence, since R < √ T , (2.3.162) turns into B R |u(T, y)| 2 dy 1 2 η β (T ), 128
Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields and yields ˆu(T, y)ψ(y)dy

2 1 2 ≤ R d 2 ψ L ∞ (R d ) B R |u(T, y)| 2 dy 1 2 (2.3.162) R d 2 ψ L ∞ (R d ) η β (T ).
Consequently, we have in particular, almost surely lim T →+∞ ˆu(T, y)ψ(y)dy = 0.

For the second l.h.s term of (2.3.161), we have directly using (2.1.25):

ˆ∇ψ(y) • a(y) ˆ+∞ T ∇u(s, y)ds dy 2 1 2 ≤ ∇ψ L ∞ (R d ) ˆBR ˆ+∞ T |∇u(s, y)| 2 1 2
ds dy Step 2. We prove (2.1.32) and we split the proof into three steps. For the rest of the proof, we fix α <

(2.1.25) R d ∇ψ L ∞ (R d ) T 1 2 η β (T ),
1 1 2 +2 d+1 β∨d .
Substep 2.1. We start with the control of the flux and we only treat the control of |q r -q r |, the control of |(q T ) r -(q T ) r | will be obtained the same way, using (2.3.160) instead of (2.3.159). We have directly by using the triangle inequality and Theorem 10 with T = r 2 , for all p ∈ [1, +∞)

|q r -q r | p 1 p ≤ |q r -(q(r 2 )) r | p 1 p + |(q(r 2 )) r -(q(r 2 )) r | p 1 p + | q r -(q(r 2 )) r | (2.1.17) |q r -(q(r 2 )) r | p 1 p + π -1 2 * (r)p 1 α .

Proofs 129

It remains to control the first r.h.s term of the above inequality. To this aim, we write by dominating the Gaussian kernel by the exponential kernel and using the Minkowski inequality in L p • (Ω) as well as the Jensen inequality

|q r -(q(r 2 )) r | p 1 p ≤ ˆgr (x) ˆ+∞ r 2 ∇u(t,
ˆRd \Br η r (x)|∇u(t, x)| 2 dx p 2 1 p ≤ +∞ n=1 ˆB(n+1)r \Bnr η r (x)|∇u(t, x)| 2 dx p 2 2 p 1 2 ≤ +∞ n=1 e -n n d B (n+1)r |∇u(t, x)| 2 dx p 2 2 p 1 2 (2.3.165) p 1 2 d β∧d + 1 α t -1 2 η β (t). ( 2

.3.167)

We conclude by plugging the two above inequalities into (2.3.164) and the fact ´+∞

r 2 t -1 2 η β (t)dt π - 1
2 (r). The bound on (q T ) r is obtained the same way since from (2.3.160) we have

(q T ) r = ˆ+∞ 0 e -t T q r (t, •)dt.
Substep 2.2. We prove the control of |∇φ r |. We first notice that by integrating the equation (2.1.10) in time and using that, by stationarity, ∇ • q = 0, we have for all t ≥ 0

u(t, •) = ∇ • (q(t, •) -q(t, •) ). for correlated coefficient fields
From the definition (2.1.14), we deduce that

(∇φ(r 2 )) r = ∇ ∇ • ˆr2 0 (q(s, •) -q(s, •) )ds r . ( 2

.3.168)

By noticing that, from the semigroup property g r = g 1

√ 2 r g 1 √ 2 r we have for all f ∈ H 2 loc (R d ) |(∇ 2 f ) r | = |(∇ 2 f 1 √ 2 r ) 1 √ 2 r | 1 r 2 |f 1 √ 2 r | 1 √ 2 r , (2.3.169)
we deduce, from (2.3.168), the stationarity of q r , (2.1.17) and (2.4.2) that, for all p ≥ 1

|(∇φ(r 2 )) r | p 1 p = ∇ ∇ • ˆr2 0 (q(s, •) -q(s, •) )ds r p 1 p (2.3.169) 1 r 2 ˆr2 0 |q 1 √ 2 r (s) -q 1 √ 2 r (s) | p 1 p ds (2.1.17),(2.4.2) p 1 α r -d 2 ˆr2 0 µ β (s)(1 + log 2 ( √ s r ))ds π -1 2 * (r)p 1 α . ( 2
T (σ T,i,j,k ) r -∆(σ T,i,j,k ) r = (e k • q e i ,T -e j • q e i ,T ) (∂ j g r -∂ k g r ).
Therefore, we may express ∇(σ T,i,j,k ) r with help of the Green function G T of the massive Laplace

operator 1 T -∆ in R d ∇(σ T,i,j,k ) r = ∇G T ((e k • q e i ,T -e j • q e i ,T ) (∂ j g r -∂ k g r )).
Then, using that there exists a constant C which depends on d such that

G T = C ˆ+∞ 0 e -s T g √ s ds,
and by noticing that from the stationarity of (e k • q e i ,T -e j • q e i ,T ) we have ˆ+∞

0 ˆˆe -s T |∇g √ s (x)||e k • q e i ,T (x -y) -e j • q e i ,T (x -y)||∂ j g r (y) -∂ k g r (y)|dy dx ds |e k • q e i ,T -e j • q e i ,T | ∂ j g r -∂ k g r L 1 (R d ) ˆ+∞ 0 e -s T ∇g √ s L 1 (R d ) ds |e k • q e i ,T -e j • q e i ,T | ∂ j g r -∂ k g r L 1 (R d ) ˆ+∞ 0 s -1 2 e -s T ds < +∞,

Proofs 131

we deduce from the Fubini theorem combined with the semigroup property g r g √ s = g √ s+r 2 that, almost surely

∇(σ T,i,j,k ) r = ˆ+∞ 0 e -s T (e k • q e i ,T -e j • q e i ,T ) (∇(∂ j g √ s+r 2 -∂ k g √ s+r 2 ))ds = ˆ+∞ 0 e -s T (∇(∂ j (e k • q e i ,T ) -∂ k (e j • q e i ,T ))) √ s+r 2 ds.
Consequently, by making once again use of the stationarity of q e i ,T as well as (2.3.169) and (2.1.32) proved for (q e i ,T ) r in substep 2.1, we obtain for all p ≥ 1

|∇(σ T,i,j,k ) r | p 1 p = C ˆ+∞ 0 e -s T (∇(∂ j (e k • q e i ,T ) -∂ k (e j • q e i ,T ))) √ s+r 2 ds p 1 p (2.3.169) ˆ+∞ 0 1 s + r 2 |(q e i ,T ) 1 √ 2 √ s+r 2 -(q e i ,T ) 1 √ 2 √ s+r 2 | p 1 p ds p 1 2 + 1 α ˆ+∞ 0 1 s + r 2 π -1 2 * ( s + r 2 )ds p 1 2 + 1 α π -1 2 * (r). ( 2

Proof of Corollary 3 : Growth of the extended corrector (φ, σ)

We only give the arguments for φ. For the bound on σ, we may rewrite averages ´∇σ i,j,k (x) • g(x)dx where g is assumed to be a gradient field; i.e., g = ∇θ for some potential θ, using the second line of (2.1.15) to obtain ˆ∇σ i,j,k (x) • g(x)dx = ˆq(x) • Sg(x)dx, with S =: e j ⊗ e k -e k ⊗ e j and q = ((q e i ) j ) i,j . Since the averaging field Sg inherits the decay properties of g, we then conclude using Theorem 10 for q and the arguments for φ.

Let α <

x ∈ R d (|φ -φ 1 (0)| 2 ) p 2 1 (x) 1 p |φ 1 (x) -φ 1 (0)| p 1 p + (|∇φ| 2 ) p 2 1 1 p . ( 2
1 + p 1 α .
Therefore, the estimate (2.3.172) turns into

(|φ -φ 1 (0)| 2 ) p 2 1 (x) 1 p |φ 1 (x) -φ 1 (0)| p 1 p + p 1 α + 1. (2.3.173)
On the other hand, setting R = |x| ≥ 1, we have by the triangle inequality

|φ 1 (x) -φ 1 (0)| p 1 p ≤ |φ R (x) -φ 1 (x)| p 1 p + |φ R (x) -φ R (0)| p 1 p + |φ R (0) -φ 1 (0)| p 1 p . (2.3.174)
The second r.h.s term of (2.3.174) is estimated via the fundamental calculus theorem combined with the Minkowski inequality in L p • (Ω), the stationary property of ∇φ and (2.1.32)

|φ R (x) -φ R (0)| p 1 p = |x| ˆ1 0 x |x| • ˆ∇φ(y + τ x)g R (y)dy dx p 1 p (2.1.32) p 1 α |x|π -1 2 (R).
(2.3.175) By stationarity, the first and the third r.h.s term of (2.3.174) are estimated the same way and we bound the third term in two different ways, depending on the regimes of β and d:

(i) We consider the regimes β < 2, β = d = 2 and β > 2, d > 2. Our main tool here are the moment bounds on the gradients of correctors (2.1.32). We write by the fundamental calculus theorem

φ R (0) -φ 1 (0) = ˆR 1 ∂ ∂τ φ τ (0)dτ = ˆR 1 ˆ∇φ(x) • g τ (x) x τ dx dτ. (2.3.176)
Then, by noticing that from the semigroup property of Gaussian field in form of

g τ = g τ √ 2 g τ √ 2
, writing x τ = y τ + x-y τ and applying the Fubini theorem, we have for all

τ ∈ [1, R] ˆ∇φ(x) • g τ (x) x τ dx = ˆˆ∇φ(x) • x τ g τ √ 2 (y)g τ √ 2 (x -y)dy dx = ˆˆ∇φ(x) • y τ g τ √ 2 (y)g τ √ 2
(x -y)dy dx

+ ˆˆ∇φ(x) • x -y τ g τ √ 2 (y)g τ √ 2 (x -y)dy dx =2 ˆy τ g τ √ 2 (y) • ˆ∇φ(x)g τ √ 2 (x -y)dx dy =2 ˆy τ g τ √ 2 (y) • ∇φ τ √ 2
(-y)dy, we deduce from the Minkowski inequality in L p • (Ω), the stationarity property of ∇φ, (2.3.176) and (2.1.32)

|φ R (0) -φ 1 (0)| p 1 p ˆR 1 |∇φ τ √ 2 | p 1 p dτ (2.1.32) p 1 α ˆR 1 π -1 2 ( τ √ 2 )dτ p 1 α ξ d,β (R
R (0) -φ 1 (0) = -ˆ∇H(x) • ∇φ(x)dx, (2.3.178)
with

H := ˆR2 1 g √ τ (•)dτ.
Indeed, using that for all τ > 0,

∂ τ g √ τ = ∆g √ τ , we have φ R (0)-φ 1 (0) = ˆ(g R (x)-g 1 (x))φ(x)dx = ˆˆR 2 1 ∂ τ g √ τ (x)dτ φ(x)dx = ˆˆR 2 1 ∆g √ τ (x)dτ φ(x)dx,
and thus (2.3.178) follows from an integration by parts (which is justified by the sub-linearity property of the corrector φ). Now, using the formula (2.3.159), we get

φ R (0) -φ 1 (0) = -ˆ∇H(x) • ˆ+∞ 0 ∇u(s, x)ds dx = -ˆ∇H(x) • ∇φ(R 2 , x)dx -ˆ∇H(x) • ˆ+∞ R 2 ∇u(s,
ˆ∇H(x) • ∇φ(R 2 , x)dx p 1 p p 1 α log 1 2 (R + 2).
For the second r.h.s term of (2.3.179), we make use of the combination of (2.3.165), (2.3.166) and (2.3.167) as well as the following bound on ∇H:

for all x ∈ R d |∇H(x)| ˆR2 1 |x|τ -d 2 -1 e -|x| τ dτ ≤ |x|e -|x| 2 2R 2 ˆR2 1 τ -d 2 -1 e -|x| 2 2τ dτ (|x| + 1) 1-d e -|x| 2 2R 2 Rg 2R (x),
to obtain for all p ∈ [1, +∞)

ˆ∇H(x) • ˆ+∞ R 2 ∇u(s, x)ds dx p 1 p ≤R ˆ+∞ R 2 ˆg2R (x)|∇u(s, y)| 2 p 2 1 p p 1 α R ˆ+∞ R 2
s -1 2 η β (s)ds 1. for correlated coefficient fields

Proof of Corollary 5: Sub-systematic error

We split the proof into two steps.

Step where (exp n (•, T )) n∈N is the Richardson extrapolation of exp 1 (•, T ) := e -• T . Note that the extrapolation has the effect that for all τ ≥ 0

|1 -exp n (τ, T )| n τ T n ∧ 1 and ∂ ∂τ exp n (τ, T ) n 1 T τ T n-1
.

(2.3.181)

We then split the integral (2.3.180) into three contributions. We start by the contribution on the interval (0, 1). We write by an integration by parts

ˆ1 0 (1 -exp n (τ, T ))∇u(τ )dτ = ˆ1 0 ∂ ∂τ exp n (τ, T ) ˆτ 0 ∇u(t)dt dτ + (1 -exp n (1, T )) ˆ1 0 ∇u(τ )dτ.
Thus, by the Minkowski inequality in L 2 • (Ω) and the stationarity of ∇u, we get ˆ1 0

(1 -exp n (τ, T ))∇u(τ )dτ

2 1 2 ˆ1 0 ∂ ∂τ exp n (τ, T ) 2 dτ ˆ1 0 ˆη√ τ (x) ˆτ 0 ∇u(t, x)dt 2 dx 1 2 dτ + |1 -exp n (1, T )| 2 ˆη1 (x) ˆ1 0 ∇u(t, x)dt 2 dx 1 2
.

Hence, using the localized energy estimate (2.2.32) combined with (2.3.181), we arrive at

ˆ1 0 (1 -exp n (τ, T ))∇u(τ )dτ 2 1 2 n T -n ,
which is of higher order than the r.h.s of (2.1.37). We turn now to the contributions on the intervals (1, T ) and (T, +∞), for which the estimate of the decay of the semigroup (2.1.25) and (2.3.181) yield

ˆ+∞ 1 (1 -exp n (τ, T ))∇u(τ )dτ 2 1 2 ˆT 1 τ T n τ -1-d 4 dτ + ˆ+∞ T τ -1-d 4 dτ T 1 2 η β (T ).
It concludes the proof of (2.1.37).

Step

: ζ ∈ (0, +∞) → 1 ζ , g 1 : ζ ∈ (0, +∞) → (ζ + µ) -1
, and (g n ) n∈N is the Richardson extrapolation of g 1 with respect to µ -1 . Then, by the spectral theorem, we have for all n ∈ N

∇(φ n e,µ -1 -φ e ) • a∇(φ n e,µ -1 -φ e ) = (φ n e,µ -1 -φ e )L(φ n e,µ -1 -φ e ) = ˆ+∞ 0 ζ(g n (ζ, µ -1 ) -g 0 (ζ)) 2 dν Θ (ζ) . ( 2
|g n (ζ, µ -1 ) -g 0 (ζ)| µ n ζ(ζ + µ) n , which we use in the form, for all ζ ≤ µ ζ(g n (ζ, µ -1 ) -g 0 (ζ)) 2 µ 2n ζ(ζ + µ) 2n . ( 2
F ∈ L p • (Ω) |F -F | p 1 p d √ p ˆ+∞ 1 -d π( ) ˆ|∂ fct x, F | 2 dx d p 2 1 p . (2.4.1)
The following standard lemma gives the link between algebraic moment and exponential moment for non-negative random variables. The short proof is included for completeness. Lemma 9. Let X : Ω → R + a non-negative random variable. We have the following equivalence:

∃C 1 > 0 such that exp 1 C 1 X ≤ 2 ⇔ ∃C 2 > 0 such that ∀q ≥ 1, X q 1 q ≤ qC 2 . (2.4.2)
Proof. Let us suppose that there exists C 2 > 0 such that for all q ≥ 1, X q 1 q ≤ qC 2 . We have, for all

C 1 > 0 exp 1 C 1 X = +∞ n=0 X n n!C n 1 ≤ +∞ n=0 ( C 2 C 1 n) n n! , we then choose C 1 such that +∞ n=0 ( C 2 C 1 n) n n! ≤ 2.
Let us now suppose that there exists C 1 > 0 such that exp 1 C 1 X ≤ 2. This implies that for all q ≥ 1, X q ≤ C q 1 q!. Since, from the Stirling formula, q! ≤ Cq q for some C > 0, we have for all q ∈ N, X q 1 q ≤ CC 1 q.

Large-scale regularity theory for parabolic system

In this section we recall the regularity theory for random parabolic operator of the form ∂ τ -∇ • a∇ developed in the papers [START_REF] Bella | A liouville theorem for stationary and ergodic ensembles of parabolic systems[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization and regularity theory of parabolic equations[END_REF] and draw some useful consequences. Here, we assume that a does not depend on time. However, the theory also holds with time dependent coefficients, using a time dependent minimal radius r * different from the one defined in Theorem 11 but this is not needed in this paper. We start by recalling the excess decay property, which can be found in [START_REF] Bella | A liouville theorem for stationary and ergodic ensembles of parabolic systems[END_REF]Prop.4] and the moment bound on r * which can be found in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]. We then prove large-scale C 0,1 estimates, following the arguments of [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF].

Theorem 11 (Excess decay).

There exists a 1 8 -Lipschitz stationary random field r * : Ω × R d → R + for which there exists a constant C < +∞ such that for all

x ∈ R d exp 1 C π * (r * (x)) ≤ 2, (2.4.3)
with for all r ≥ 1

π * (r) =    r β if β < d, r d log -1 (r) if β = d, r d if β > d.

Appendix 137

In addition, for all distributional solution of

∂ τ u -∇ • a∇u = 0 in C R for R ≥ r * ,
we have for all r ∈ [r * , R] and α ∈ (0, 1)

Exc(∇u, r) d,λ,α r R 2α Exc(∇u, R), (2.4.4)
with Exc(∇u, r) := inf ξ∈R d ffl Cr |∇u(t, y) -ξ -∇φ ξ (y)| 2 dy.

A direct consequence of the excess decay property of Theorem 11 is the following large-scale C 0,1 estimates, in the spirit of [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF], state in the parabolic setting.

Corollary 7 (Large-scale C 0,1 estimates). For the random field r * defined in Theorem 11, we have for all (s, x) ∈ R d+1 and weak solution u of

∂ τ u -∇ • (a(∇u + g)) = ∇ • h in C R (s, x) for R ≥ r * , with (g, h) ∈ L 2 loc (R d ), for all r ∈ [r * (x), R] and α > 0 Cr(s,x) |∇u(t, y)| 2 dt dy d,λ,α C R (s,x) |∇u(t, y)| 2 dt dy + sup ρ∈[r * ,R] R ρ 2α Cρ(s,x) h - Cρ(s,x) h 2 + g - Cρ(s,x) g 2 .
(2.4.5)

In particular, if g ≡ h ≡ 0, we have the following mean value property for a-caloric functions: for all r ∈ [r * (x), R]

Cr(s,x) |∇u(t, y)| 2 dt dy d,λ C R (s,x)
|∇u(t, y)| 2 dt dy.

(2.4.6)

Proof. Without loss of generality, we may assume that (s, x) = (0, 0). We split the proof into two steps.

Step 1. Proof of

sup r∈[r * ,R] 1 r 2α Exc(∇u + g, r) d,λ,α 1 R 2α Exc(∇u + g, R) + sup r∈[r * ,R]
∈ L 2 ((-ρ 2 , ρ 2 ), H 1 0 (B ρ )) ∩ H 1 ((-ρ 2 , ρ 2 ), H -1 (B ρ )) be the weak solution of ∂ t w + ∇ • a∇w = ∇ • (a(g -ξ) + h) in C ρ , w = 0 on ∂ p C ρ , (2.4.10)
where

∂ p C ρ = (∂B ρ × (-ρ 2 , 0)) ∪ B ρ × {0}. Then, because (t, x) ∈ R d+1 → u(t, x) -w(t, x) + ξ • x is a a-caloric
- ˆCρ w∂ τ w + ˆCρ ∇w • a∇w = ˆCρ h - Cρ h + ∇w • a(g -ξ) . Since - ˆCρ w∂ τ w = - ˆ0 -ρ 2 d dt w(t, •) 2 L 2 (Bρ) dt = w(-ρ 2 , •) 2 L 2 (Bρ) ≥ 0, this yields ˆCρ ∇w • a∇w ≤ ˆCρ ∇w • h - Cρ h + a(g -ξ) .
By uniform ellipticity assumption (2.1.2) on a, (2.4.12) follows. The combination of (2.4.11), (2.4.12) and the triangle inequality yields (2.4.9). Now, we conclude by a Campanato iteration. Setting 0 < θ = r ρ ≤ 1, we rewrite (2.4.9) as

Exc(∇u + g, θρ) ≤ C 1 θ 2α Exc(∇u + g, ρ) + θ -d-2 Cρ |h - Cρ h| 2 + |g - Cρ g| 2 .
We divide by (θρ) 2α and take the supremum over ρ ∈ [ r * θ , R] : (2.4.13)

sup r∈[r * ,θR] 1 r 2α Exc(∇u + g, r) ≤C 1 θ 2(α -α) sup r∈[r * ,R] 1 r 2α Exc(∇u + g, r) + θ -d-2-2α sup r∈[r * ,R]
We now choose θ = θ(d, λ, α) ≤ 1 so small that C 1 θ 2(α -α) ≤ 1 2 . By using

sup r∈[r * ,R] 1 r 2α Exc(∇u + g, r) ≤ sup r∈[θR,R]

Since

sup r∈[θR,R] 1 r 2α Exc(∇u + g, r) 1 R 2α sup r∈[θR,R] R d+2 r d+2 Exc(∇u + g, R) 1 R 2α Exc(∇u + g, R),
this yields (2.4.7) in the case R < +∞. In the case R = ∞ we obtain (2.4.8) in the limit R → +∞ by the square integrability of ∇u + g on R d+1 , in form of

lim sup R→+∞ Exc(∇u + g, R) ≤ lim sup R→+∞ C R |∇u + g| 2 = 0.
Step 2. Proof of (2.4.5). We split this step into two parts.

Substep 2.1. Proof that for all ρ > 0 there exists a unique

ξ ρ ∈ R d such that Exc(∇u + g, ρ) = Cρ |∇u + g -(ξ ρ + ∇φ ξρ )| 2 , (2.4.14)
and for all r

* ≤ r ≤ R |ξ r -ξ R | 2 sup ρ∈[r,R] R ρ 2α Exc(∇u+g, ρ)+ sup ρ∈[r * ,R] R ρ 2α Cρ h- Cρ h 2 + g- Cρ g 2 . (2.4.15)
We start by proving (2.4.14). Fix ρ > 0 and define

f : ξ ∈ R d -→ Cρ |∇u + g -(ξ + ∇φ ξ )| 2 .
f is a continuous function and the mean value property of φ, namely for all R ≥ r * :

B R |∇φ ξ + ξ| 2 ≥ 1 2 |ξ| 2 , (2.4.16)
shows that f is coercive. Consequently, ξ ρ in (2.4.14) exists. On the other hand, ξ ρ is unique. Indeed, suppose that (2.4.14) is verified for two vectors ξ 1 and ξ 2 . We have

Exc(∇u + g, ρ) = Cρ |∇u + g -(ξ 1 + ∇φ ξ 1 )| 2 = Cρ |∇u + g -(ξ 2 + ∇φ ξ 2 )| 2 ,
and in particular

2Exc(∇u + g, ρ) = Cρ |∇u + g -(ξ 1 + ∇φ ξ 1 )| 2 + |∇u + g -(ξ 2 + ∇φ ξ 2 )| 2 .
The parallelogram identity yields

2Exc(∇u + g, ρ) = Cρ 1 2 |ξ 1 -ξ 2 + ∇φ ξ 1 -ξ 2 | 2 + 2|∇u + g -( ξ 1 + ξ 2 2 + ∇φ ξ 1 +ξ 2 2 )| 2 .
140 Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

We infer that Exc(∇u + g, ρ)

≥ Cρ 1 4 |ξ 1 -ξ 2 + ∇φ ξ 1 -ξ 2 | 2 + Exc(∇u + g, ρ),

and so

Cρ

|ξ 1 -ξ 2 + ∇φ ξ 1 -ξ 2 | 2 = 0,
which gives ξ 1 = ξ 2 using the estimate (2.4.16).

We turn to the proof of (2.4.15). It is enough to prove that

∀r ≤ R ≤ 2r, |ξ r -ξ R | 2 Exc(∇u + g, R).
(2.4.17) Indeed, we argue by a dyadic argument. Let N ∈ N such that 2 -(N +1) R < r < 2 -N R. By (2.4.17), we have for all n ∈ {0..., N -1}

|ξ r -ξ 2 -N R | 2 Exc(∇u + g, 2 -N R) and |ξ 2 -(n+1) R -ξ 2 -n R | 2 Exc(∇u + g, 2 -n R).
Thus, by the triangle inequality followed by the excess decay (2.4.4) and the fact that +∞ n=0 2 -nα < +∞, we have

|ξ r -ξ R | 2 N n=0 Exc(∇u + g, 2 -n R) 2 (2.4.7) N n=0 2 -nα Exc(∇u + g, R) + sup ρ∈[r * ,R] R ρ 2α Cρ h - Cρ h 2 + g - Cρ g 2 1 2 2 Exc(∇u + g, R) + sup ρ∈[r * ,R] R ρ 2α Cρ h - Cρ h 2 + g - Cρ g 2 sup ρ∈[r,R] ( R ρ ) 2α Exc(∇u + g, ρ) + sup ρ∈[r * ,R] R ρ 2α Cρ h - Cρ h 2 + g - Cρ g 2 .
We now turn to the argument for (2.4.17). By (2.4.16) we have

|ξ r -ξ R | 2 Cr |(ξ ρ -ξ R ) + ∇φ ξr-ξ R | 2 ,
which, by linearity of ξ → φ ξ , we may rewrite as

|ξ r -ξ R | 2 Cr |(ξ ρ + ∇φ ξr ) -(ξ R + ∇φ ξ R )| 2 ,
so that, by the triangle inequality in L 2 (C ρ ) and using that r ≤ R ≤ 2r, we obtain

|ξ r -ξ R | 2 Cr |∇u -(ξ r + ∇φ ξr )| 2 + C R |∇u -(ξ R + ∇φ R )| 2 .
By definition of Exc and using once more that r ≤ R ≤ 2r, this turns as desired into |ξ r -ξ R | 2 Exc(∇u + g, r) + Exc(∇u + g, R) Exc(∇u + g, R).

f (y)dy dx ≤ ˆf (x)dx ≤ C ˆ B r * (x) (x)
f (y)dy dx.

(2.4.20)

Technical lemmas

We state here two technical lemmas. The first one is needed in the proof of Lemma 2. For a proof, see [8, Lem. C.6].

Lemma 11. Suppose that there exist A, ξ ≥ 0 and ρ :

[ 1 2 , 1) → R + which satisfies sup t∈[ 1 2 ,1) (1 -t) ξ ρ(t) < +∞,
and for all

1 2 ≤ s < t < 1, ρ(s) ≤ 1 2 ρ(t) + (t -s) -ξ A.
Then, there exists a constant C depending on ξ such that ρ( 12 ) ≤ CA. The second one is the Caccioppoli inequality for parabolic system. For a proof, see for instance [START_REF] Bella | A liouville theorem for stationary and ergodic ensembles of parabolic systems[END_REF]Lemma 5].

Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields Lemma 12 (Caccioppoli estimate). There exists a constant C depending on λ such that for every ρ ≤ R and weak solution u of

∂ τ u -∇ • a∇u = 0 in C R , we have ˆCρ |∇u(x)| 2 dx ≤ C (R -ρ) 2 ˆCR \Cρ u(x) - C R u(y)dy 2 dx.

Proof of Theorem 10 under a functional inequality with oscillation

We fix T ≥ 1, 1 ≤ r ≤ √ T and the unit vector e ∈ R d . We only give the argument for (2.1.17), (2.1.20) is obtained combining the ideas of this section and the proof of Section 2.3.4. We make for simplicity the two additional assumptions: (i) u e is real-valued and a is symmetric. We recall that it implies

∇u(t, •) L ∞ (R d ) t -1 , (2.4.21)
see Remark 4.

(ii) The coefficient field a takes the form, for some

χ ∈ C ∞ c (R d ) supported in B 1 , a := χ ã,
with a field ã : R d → R d×d which takes value into the set of uniformly elliptic and bounded matrices and with a probability law which satisfies the logarithm Sobolev inequality with oscillation (2.1.40). In this setting, ∇ • ae ∈ L ∞ (R d ) and

∇ • ae L ∞ (R d ) χ 1.
We recall that it implies the following energy estimate: for all R ≥ 1 and z ∈ R d

ˆ1 0 B R (z) |∇u(t, x)| 2 dx dt χ 1, (2.4.22)
where a proof can be found in [92, Lemma 3]. The first step is to estimate the derivative (x, ) ∈

R d × [1, +∞) → ∂ osc x, q r (T ). We claim that for all (x, ) ∈ R d × [1, +∞) |∂ osc x, q r (T )| 2 χ ( + 1) 2d (1 + log 2 (T )) B +1 (x) g 2 r (y)dy + B +1 (x) |∇v T (0, y)| 2 dy + log(T ) ˆT 1 t -1 B +1 (x) |∇v T (t, y)| 2 dy dt , (2.4.23)
with v T defined in (2.2.11).

We fix (x, ) ∈ R d × [1, +∞) and we consider ã and ã such that ã = ã = ã on R d \B (x).

We then set a := χ ã , a := χ ã and note that since χ is supported in B 1 a = a = a on R d \B +1 (x).

(2.4.24)

Appendix 143

Using the notation δu := u(a , •) -u(a , •), we have q r (a , T ) -q r (a , T ) = ˆgr (y)(a (y) -a (y))e dy + ˆgr (y)a (y) ˆT 0 ∇u(a , t, y)dt dy g 2 r (y)dy + log(T )

-
∂ τ δu -∇ • a∇δu = ∇ • (a -a )∇u(a , •) -∇ • (a -a )∇u(a , •) in (0, +∞) × R d , δu(0) = ∇ • (a -a )e. ( 2
ˆT 1 t -1 B +1 (x)
|∇v T (t, y)| 2 dy dt , which contributes to the first and third r.h.s term of (2.4.23) and concludes the proof.

We now control the entropy of q r (T ) by applying (2.1.40), using (2.4.23), the identity ´fflB +1 (x) dx = ´, ´g2 r (y)dy r -d and the plain energy estimate ´|∇v T (t, y)| 2 dy ´g2 r (y)dy,

Ent(q r (T )) χ ˆ+∞ 1 -d e -1 C β ( + 1) d (1 + log 2 (T )) ˆ B +1 (x) g 2 r (y)dy dx d + ˆ+∞ 1 -d e -1 C β ( + 1) d ˆ B +1 (x)
|∇v T (0, y)| 2 dy dx d

+ log(T ) ˆ+∞ 1 -d e -1 C β ( + 1) d ˆT 1 t -1 ˆ B +1 (x) |∇v T (t, y)| 2 dy dx dt d r -d (1 + log 2 (T )) ˆ+∞ 1 d e -1 C β d β,C r -d (1 + log 2 (T )).
To conclude, the log 2 (T ) correction may be removed following the argument of subsection 2.3.4, and the control of the entropy yields control of higher moments and provide stretched exponential moments.

Chapter 3

Quantitative homogenization theory for nonlinear elliptic equations and systems

Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this chapter we move forward to the nonlinear setting of monotone operators with p-growth. This first work is dedicated to a quantitative two-scale expansion result. Fluctuations will be addressed in companion articles. By treating the range of exponents 2 ≤ p < ∞ in dimensions d ≤ 3, we are able to consider genuinely nonlinear elliptic equations and systems such as -∇•A(x)(1+|∇u| p-2 )∇u = f (with A random, non-necessarily symmetric) for the first time. When going from p = 2 to p > 2, the main difficulty is to analyze the associated linearized operator, whose coefficients are degenerate, unbounded, and depend on the random input A via the solution of a nonlinear equation. One of our main achievements is the control of this intricate nonlinear dependence, leading to annealed Meyers' estimates for the linearized operator, which are key to the quantitative two-scale expansion result. This Chapter is based on the Preprint [START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF] jointly write with Antoine Gloria.

The chapter is organized as follows: In Section 3.1 we present the main results. In Section 3.3, we describe and prove the perturbative regularity theory for the linearized operator. Sections 3.4, 3.5 and 3.6 are devoted to the proof of the quantitative estimates on the correctors and the quantitative two-scale expansion.

Main results

In all the chapter, we decide to use scalar notations but the results also hold for sytem. For convenience, let us precisely recall the class of maps â : R d → R d that we consider in this chapter. We assume that â satisfies the following three properties: â(0) = 0 and there exist p ≥ 2 and C > 0 such that for all

ξ 1 , ξ 2 ∈ R d , |â(ξ 1 ) -â(ξ 2 )| ≤ C(1 + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 |, (3.1.1) (â(ξ 1 ) -â(ξ 2 )) • (ξ 1 -ξ 2 ) ≥ 1 C |ξ 1 -ξ 2 | p . ( 3
(R d ) → C 0 (R d ), v → ∇ • â(∇v)
. Such operators do not necessarily possess (as far as anyone can prove) the structural conditions sufficient for C 1,α -regularity, which is why we also consider two strengthenings of (3.1.2), which read for all

ξ 1 , ξ 2 ∈ R d (â(ξ 1 ) -â(ξ 2 )) • (ξ 1 -ξ 2 ) ≥ 1 C (|ξ 1 -ξ 2 | 2 + |ξ 1 -ξ 2 | p ), (3.1.3)
(â(ξ 1 ) -â(ξ 2 )) • (ξ 1 -ξ 2 ) ≥ 1 C (|ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 . (3.1.4)

Main results 147

We consider a map a : R d × R d → R d satisfying the strong monotonicity (3.1.1) and (3.1.4). For later references, we recall the equation for

1 u ε ∈ Ẇ 1,p (R d )/R -∇ • a( x ε , ∇u ε (x)) = ∇ • f (x). (3.1.5)
We also recall the homogenized equation with solution

ū ∈ Ẇ 1,p (R d )/R, -∇ • ā(∇ū(x)) = ∇ • f (x) (3.1.6)
as well as the formula for the homogenized operator

ā characterized in direction ξ ∈ R d by ā(ξ) = E[a(0, ∇φ ξ (0) + ξ)]
, where φ ξ is the corrector, defined as the unique almost sure distributional solution in

W 1,p loc (R d ) of -∇ • a(x, ∇φ ξ (x) + ξ) = 0, (3.1.7)
anchored at the origin via ´B φ ξ = 0, and the gradient ∇φ ξ is stationary, has vanishing expectation E[∇φ ξ ] = 0, and satisfies

E |∇φ ξ | 2 + |∇φ ξ | p |ξ| 2 + |ξ| p . (3.1.8)
For more precisions on the homogenization theory, we refer to Section 1.1.2 of the introduction of the thesis.

Growth conditions and classical regularity

In this section we recall what regularity theory one can expect for elliptic operators of the form ∇•â(∇) depending on properties of â, which we then apply both to the random and the homogenized operator. There are essentially two classes of results:

• Standard growth conditions: If ξ → â(ξ) is smooth, and â is strongly monotone (that is, it satisfies (3.1.1) and (3.1.4)), then ∇ • â(∇) possesses C k,α -regularity and nonlinear Calderón-Zygmund theory, cf. [START_REF] Kuusi | Guide to nonlinear potential estimates[END_REF];

• Non-standard growth conditions: If ξ → â(ξ) is smooth, and â is monotone and non-degenerate (that is, it satisfies (3.1.1) and (3.1.3)), then one expects that ∇•â(∇) possesses C k,α -regularity and nonlinear Calderón-Zygmund theory provided 2 ≤ p <2(d-1) d-3 (only active for d ≥ 4), cf. [START_REF] Bella | On the regularity of minimizers for scalar integral functionals with (p, q)-growth[END_REF] 2 .

On the one hand, strongly monotone operators are nicer since they possess regularity without restriction on the exponent p, but strong monotonicity is not stable by homogenization (in the sense that the homogenized operator might not be strongly monotone). On the other hand, although non-degenerate operators do only possess regularity if p is close enough to 2 in high dimensions (there is no restriction for d ≤ 3), this property is stable by homogenization.

As we shall see below, to establish quantitative homogenization results,

• The non-degeneracy condition (3.1.3) is needed for reasons that are independent of the regularity theory;

148 Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

• We need local regularity theory for the random operator, which (provided x → a(x, ξ) is smooth enough) is automatic for 2 ≤ p < 2(d-1) d-3 in view of the assumption (3.1.3), and follows from any p ≥ 2 if we further assume (3.1.4);

• In the random setting, it turns out that the condition 2 ≤ p < 2(d-1) d-3 is also needed in the proof of (large-scale) Meyers' estimates for the linearized operator.

For random operators, the assumptions we need in this paper to establish Meyers estimates for the linearized operator also imply regularity theory for the homogenized operator, and we refer the reader to Section 3.2.2 for a further discussion of this observation This is however not the case for periodic operators, and regularity theory for the homogenized operator might be an issue.

Quantitative assumptions and quantitative two-scale expansion

In view of the discussion above, to fix ideas and keep results and proofs readable, we consider the explicit class of p-Laplacians regularized at zero (see Section 3.2.3 for more general conditions). where A is a uniformly elliptic (non-necessarily symmetric) stationary ergodic matrix field. More precisely, we assume that A is smooth (uniformly wrt to the randomness) and satisfies the ellipticity conditions for some Let us now be more precise on the stochastic setting. It is convenient to define the probability space via Ω = {A : R d → M d (λ)}, endowed with some probability measure P. In this setting, a random variable Y can be seen as a (measurable) function of A → Y (A). We say that the measure P is ergodic if we have the implication:

0 < λ ≤ 1 ∀x, ξ ∈ R d : ξ • A(x)ξ ≥
Y (A(• + z)) = Y (A) for all z ∈ R d =⇒ Y = E[Y ] almost surely. We say that a random field X : R d × Ω → R k (for k ∈ N) is stationary if for all z ∈ R d and almost all x ∈ R d we have X(x + z, A) = X(x, A(• + z)), where A(• + z) : x → A(x + z).
(Note that the expectation E[X(x)] of a stationary random field X does not depend on x ∈ R d and we simply write E[X].) We use the notation L q (dP) for the space of q-integrable random variables.

In order to prove quantitative results, we need to quantify the ergodicity assumption, which we do by assuming Gaussianity of P and the integrability of the covariance function in the following sense. which we assume to be integrable on R d , and χ : R d → [0, 1] is a smooth compactly supported convolution kernel. In particular, A is smooth (uniformly wrt to the randomness). We further require 2 ≤ p < 2(d-1) d-3 in dimensions d ≥ 4.3 Our main achievement is an optimal quantitative corrector result, which extends the results of [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] to the genuinely nonlinear setting of p > 2. Following Dal Maso and Defranceschi [START_REF] Maso | Correctors for the homogenization of monotone operators[END_REF], we start with the suitable definition of the two-scale expansion. To this aim, we introduce a scale δ > 0 (which we should think of as being ε in the upcoming result), set K δ := δZ d and for all k ∈ K δ , we define the cube Q δ (k) = k + [-δ, δ) d centered at k and of sidelength 2δ. We also consider a partition (η k ) k∈K δ of unity on R d with the following properties:

0 ≤ η k ≤ 1, η k ≡ 1 on Q δ 2d (k), η k ≥ c on Q (1-1 3d )δ , supp η k ⊂ Q δ (k)
, and |∇η k | ≤ Cδ -1 (for some suitable c, C > 0 independent of δ). Given the solution ū of (3.1.6), we introduce local averages associated with the partition of unity in form for all k ∈ K δ of

(∇ū) k,δ := ´Rd η k ∇ū ´Rd η k ,
and define the two-scale expansion ū2s ε,δ associated with ū via

ū2s ε,δ := ū + ε k∈K δ η k φ (∇ū) k,δ ( • ε ), (3.1.11)
where φ ξ denotes the corrector in direction ξ ∈ R d (cf. (3.1.7)). This constitutes a convenient variant (introduced in [START_REF] Maso | Correctors for the homogenization of monotone operators[END_REF] to deal with monotone operators) of the classical two-scale expansion x → ū(x) + εφ ∇ū(x) ( x ε ), which may raise measurability issues. Based on this two-scale expansion, we have the following optimal convergence result.

∞ (R d ) and µ d ∇ 2 ū ∈ L 2 (R d ), then we have ∇u ε -∇ū 2s ε L 2 (R d ) ≤ C ε,ū εµ d ( 1 ε ), (3.1.13)
where C ε,ū denotes a random variable that satisfies

E exp(cC α ε,ū ) ≤ 2, (3.1.14)
for some exponent α > 0 depending on d, p, λ, and ∇ū L ∞ (R d ) , and some constant c further depending on µ d ∇ 2 ū L 2 (R d ) , but not on ε.

Some comments are in order:

• This result also holds for nonlinear systems with Uhlenbeck structure under the same assumptions on p.

Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

• In the periodic setting, Theorem 12 holds without restrictions on p ≥ 2 and with µ d ≡ 1 in any dimension. This result is sharper than [START_REF] Cherednichenko | On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems[END_REF], which contains the first quantitative two-scale expansion estimate for monotone periodic operators with p > 2 (there, one needs to know that ā satisfies (3.1.4) to construct a second-order two-scale expansion, which gives (3.1.13) after truncation and with a dependence of the constant on stronger norms of ū).

• The choice to work on the whole space with a right-hand side in divergence form allows one to avoid boundary layers (and therefore to truly focus on the homogenization error, in line with [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF]) and to treat all dimensions at once. In particular one could state and prove a similar result on a bounded domain with Dirichlet boundary conditions, in which case the bound would be of the order of the square root of that in (3.1.13).

• This result takes the same form (with the same optimal rates) as for the linear case [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF][START_REF] Armstrong | the additive structure of elliptic homogenization[END_REF] and for the nonlinear case [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] with p = 2. As opposed to the latter, the stretched exponential exponent α in Theorem 12 depends on ∇ū L ∞ (R d ) itself. This intricate dependence could be made explicit (provided we make the exponent and constants explicit in Gehring's lemma) and is reminiscent of the way we treat the non-degeneracy of the linearized equation (that is, perturbatively).

• This result makes the a priori assumption that ∇ū ∈ L ∞ (R d ) and

µ d ∇ 2 ū ∈ L 2 (R d ):
-In the scalar setting, under Hypothesis 3.1.2, since ā ∈ C 1,1 loc (cf. Corollary 16 below), the conditions ∇ū ∈ L ∞ (R d ) and µ d ∇ 2 ū ∈ L 2 (R d ) are not restrictive and hold under suitable assumptions on the right-hand side f , capitalizing on the results [START_REF] Bella | On the regularity of minimizers for scalar integral functionals with (p, q)-growth[END_REF] by Bella and Schäffner.

-Since the above result is local in nature, this estimate holds on domains of R d on which ū has the required regularity. In any case, if ∇ū develops some singularity somewhere, one does not expect the two-scale expansion to be accurate in that region. This applies in particular to systems and to the periodic setting if p > 2(d-1) d-3 .

• The restriction 2 ≤ p < 2(d-1) d-3 on the exponent p in Hypothesis 3.1.2 (which is only active in high dimensions d ≥ 4) is related to the perturbative regularity theory in the large that we develop for the linearized operator in Section 3.3. Indeed, the coefficient a ξ := Da(•, ξ + ∇φ ξ ) of the operator a linearized at ξ + ∇φ ξ scales like 1 + |ξ + ∇φ ξ | p-2 and therefore only satisfies E[|a ξ | p p-2 ] < ∞ a priori: as p increases, the stochastic integrability of the coefficients decreases. At some threshold (depending on dimension), this poor stochastic integrability cannot be compensated any longer by the Sobolev embedding -whence our restriction (even in dimension 3, the argument to get all the exponents 2 ≤ p < ∞ is not straightforward -see Sections 3.2 and 3.3).

Remarks on the strong monotonicity of ā

In this section we further investigate under which additional assumptions one could prove that ā satisfies (3.1.4) next to (3.1.2) and (3.1.3), which would ensure regularity theory for the homogenized operator (provided ā is smooth enough) independently of p and d. Let us first explain why this question is subtle, and assume for simplicity that A is symmetric. In what follows we denote by D i the derivative with respect to the i-th entry of the vector ξ ∈ R d (so that D i a(x, ξ) := ∇ ξ i a(x, ξ)). Informal computations (that are made rigorous in this paper) suggest that Dā(ξ) = āξ , where āξ is the homogenized matrix associated with the random coefficient field

a ξ := Da(•, ξ + ∇φ ξ ) = (1 + |ξ + ∇φ ξ | p-2 )A + (p -2)A (ξ + ∇φ ξ ) ⊗ (ξ + ∇φ ξ ) |ξ + ∇φ ξ | 2 |ξ + ∇φ ξ | p-2 .
As a first attempt to control āξ from below, we appeal to the Voigt-Reiss bounds (see [100, Section 1.6]), which yields after neglecting the second contribution to

a ξ āξ ≥ E a -1 ξ -1 ≥ AE (1 + |ξ + ∇φ ξ | p-2 ) -1 -1 ,
and amounts to controlling the harmonic average of 1 + |ξ + ∇φ ξ | p-2 from below (and therefore to have information on the critical set of the harmonic coordinate x → ξ • x + φ ξ (x)). A second attempt is to consider the specific direction ξ • āξ ξ. Starting point is the (informal) minimization problem

ξ • āξ ξ = inf suitable ∇ψ E (ξ + ∇ψ) • a ξ (ξ + ∇ψ) ≥ inf suitable ∇ψ E (ξ + ∇ψ) • A(1 + |ξ + ∇φ ξ | p-2 )(ξ + ∇ψ) .
Call ∇ψ * a minimizer. By minimality of ∇ψ * , the definition (3.1.9) of the monotone map a, and the corrector equation (3.1.7), we then have

ξ • āξ ξ ≥ E (ξ + ∇ψ * ) • A(1 + |ξ + ∇φ ξ | p-2 )(ξ + ∇ψ * ) = E (ξ + ∇φ ξ ) • A(1 + |ξ + ∇φ ξ | p-2 )(ξ + ∇ψ * ) = E (ξ + ∇ψ * ) • a(ξ + ∇φ ξ ) (3.1.7) = E (ξ + ∇φ ξ ) • A(1 + |ξ + ∇φ ξ | p-2 )(ξ + ∇φ ξ ) |ξ| 2 + |ξ| p .
In dimension d = 1, one directly has Dā(ξ) 1 + |ξ| p-2 , which yields (3.1.4). For d > 1 this is different since from the a priori estimate ξ • Dā(ξ)ξ |ξ| 2 (1 + |ξ| p-2 ) we cannot deduce e • Dā(ξ)e |e| 2 (1 + |ξ| p-2 ) for general e ∈ R d , unless combined with some isotropy arguments (which would ensure that controlling one direction is enough to control all).

The upcoming results follow both paths. First we show that ā satisfies (3.1.4) provided we have a quantitative version of unique continuation, at least for periodic coefficients (as essentially noticed by Cherednichenko and Smyshlyaev in [START_REF] Cherednichenko | On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems[END_REF]).

Theorem 13. Let A be a Q-periodic Lipschitz matrix field. For all ξ ∈ R d , denote by ψ ξ ∈ W 1,p per (Q) the unique weak solution of

-∇ • A(x)|∇ψ ξ + ξ| p-2 (∇ψ ξ + ξ) = 0.
Assume that for all ξ ∈ R d , there exists r > 0 such that the r-tubular neighborhood T r (ξ [2] (which shows that C(ξ) ∩ Q is indeed a finite union of points) -but, in this setting, regularity for the homogenized operator also holds because ā is non-degenerate, cf. [START_REF] Marcellini | Regularity and existence of solutions of elliptic equations with p, q-growth conditions[END_REF]. For d > 2 this is a widely open problem. For linear equations, this follows from [START_REF] Cheeger | Critical sets of elliptic equations[END_REF].

) = {x + B r | x ∈ C(ξ)} of the critical set C(ξ) = {x ∈ R d | ξ + ∇ψ ξ (x) = 0} is such that R d \ T r (ξ)
Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

In the random setting, we have a positive result assuming the statistical isotropy of A, which is new and holds in any dimension. These results are proved in Appendices 3.7.5 and 3.7.6. We suspect that Theorems 13 and 14 hold under weaker assumptions but we are currently unable to establish this (even using the quantitative estimates proved in this paper).

Strategy of the proof, extensions, and limitations

General strategy and auxiliary results

In this section, we motivate our general strategy by comparison to the linear setting. To be more precise, we also consider the linear homogenization problem on R d

-∇ • A( x ε )∇v ε (x) = ∇ • f (x), (3.2.1)
with the same assumptions on A as in Hypothesis 3.1.2.

We start by defining the notion of (nonlinear) flux corrector.

Definition 3.2.1. For all ξ ∈ R d , there exists a unique skew-symmetric random matrix field (σ ξ,ij ) 1≤i,j≤d , which solves almost surely in the distributional sense in R d the flux corrector equation

-σ ξ,ij = ∂ i (a(•, ξ + ∇φ ξ ) • e j ) -∂ j (a(•, ξ + ∇φ ξ ) • e i ), (3.

2.2)

which is anchored at the origin via ´B σ ξ = 0, and whose gradient ∇σ ξ is stationary, has vanishing expectation E[∇σ ξ ] = 0, and is bounded in the sense

E |∇σ ξ | p p-1 |ξ| 2 + |ξ| p .
In addition we have

∇ • σ ξ = a(•, ξ + ∇φ ξ ) -ā(ξ), (3.2.3)
where the divergence of a matrix field σ is understood as

(∇ • σ ξ ) i = d j=1 ∂ j σ ξ,ij .
The proof of existence and uniqueness of σ ξ is essentially the same as in the linear setting in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] when p = 2, provided the adaptations of [24, Lemma 1] for p > 2.

Note that in the linear case (3.2.1), we have the scaling relation (φ tξ , σ tξ ) = t(φ ξ , σ ξ ) (in the nonlinear setting, the homogeneities of φ ξ and of σ ξ with respect to ξ are different, and not explicit since a has no homogeneity due to the regularization at 0, cf. (3.1.9)).

The interest of the flux corrector is that it allows to put the remainder in the equation satisfied by the two-scale expansion error in conservative form (which is convenient to use energy estimates). More precisely, in the linear setting, the two-scale expansion of v ε takes the simpler form

v2s ε := v + εφ i ( • ε )∂ i v, (3.2.4)
where v solves the homogenized equation -∇• Ā∇v = ∇•f , φ i denotes the corrector in the canonical direction e i , and where we use implicit summation on the repeated index i.

Since φ i ( x ε )∂ i v(x) = φ ∇v(x) ( x ε ), (3.2.4
) is indeed an elementary version of (3.1.11). In this case, one can prove (making a crucial use of the skew-symmetry of σ -see the proof of Theorem 12 in our nonlinear setting) that v ε -v2s ε satisfies the equation

-∇ • A( • ε )∇(v ε -v2s ε ) = ε∇ • ((Aφ i -σ i )( • ε )∇∂ i v) (3.2.5)
(the factor ε comes by scaling since there are two gradients on v on the right-hand side). This yields the bound (3.1.13) by an energy estimate provided we control the growth of (φ, σ). In the nonlinear setting, we rather expect a term of the form a ξ φ ξ -σ ξ in the right-hand side of (3. (which has to be compared to the difference of scalings of φ ξ and σ ξ themselves, cf. Definition 3.2.1).

The main result on the extended nonlinear corrector (φ ξ , σ ξ ) is as follows. (Despite the above discussion, we do not make a difference in the scalings of φ ξ and σ ξ wrt ξ. This dependence is indeed not explicit as one could have expected, due to the nonlinear nature of the problem.) Theorem 15. Under Hypothesis 3.1.2, for all ξ ∈ R d , the stationary extended corrector gradient ∇(φ ξ , σ ξ ) satisfies for some exponent α > 0 depending on λ, p, and d, and some constant c ξ > 0 depending further on |ξ|,

E exp(c ξ |∇(φ ξ , σ ξ )| α ) ≤ 2. (3.2.6) For all g ∈ L 2 (R d ), averages of (∇φ ξ , ∇σ ξ,ij ) display the CLT scaling 4 in the form ˆg(∇φ ξ , ∇σ ξ ) ≤ C ξ,g ˆ|g| 2 1 2 , (3.2.7)
where C ξ,g is a random variable with finite stretched exponential moment

E exp(c ξ C α ξ,g ) ≤ 2,
for some exponent α > 0 depending on p, λ, and d, and some constant c ξ > 0 further depending on |ξ| (but all independent of g). This directly implies the following bounds on the growth of (φ ξ , σ ξ ):

For all x ∈ R d , |(φ ξ , σ ξ )(x)| ≤ C x,ξ µ d (x), (3.2.8)
where µ d is defined in (3.1.12) and C x,ξ is a random variable with the same stochastic integrability as C ξ,g .

Remark 3.2.1. Under Hypothesis 3.1.1, for Q-periodic matrix fields A, the nonlinear correctors are bounded in C 1,α (Q) (no restriction on p ≥ 2).

As pointed out in [5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF] and used in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] for p = 2, controlling the growth of correctors is not enough in the nonlinear setting. This should not come as a surprise when comparing (3.2.4) to (3.1.11). The additional gradient on ∇v in the right-hand side of (3.2.5) (at the origin of the factor ε) indeed comes for local differences

φ ∇v(x 1 ) -φ ∇v(x 2 ) = |x 1 -x 2 |φ ∇v(x 1 )-∇v(x 2 ) |x 1 -x 2 |
, when reformulated by taking advantage of the linearity of the corrector. In the nonlinear setting, this identity does not hold any longer for φ ξ (and even less for σ ξ by homogeneity). It is however replaced by the following Lipschitz-continuity results.

|∇(φ ξ 1 -φ ξ 2 , σ ξ 1 -σ ξ 2 )(x)| ≤ C x,K |ξ 1 -ξ 2 |, |(φ ξ 1 -φ ξ 2 , σ ξ 1 -σ ξ 2 )(x)| ≤ C x,K |ξ 1 -ξ 2 |µ d (x),
where C x,K is a random variable with finite stretched exponential moment depending only on d, p, λ, and K. In particular, ξ → ā(ξ) is locally C 1,1 .

Remark 3.2.2. Under Hypothesis 3.1.1, for Q-periodic matrix fields A, corrector differences are controlled by

|ξ 1 -ξ 2 | in C 1,α (Q) (no restriction on p ≥ 2)
, and ā is C 1,1 as well.

To prove such a result, we have to analyze the dependence of correctors with respect to the direction ξ, which leads us to the notion of linearized correctors. The control of (nonlinear) corrector differences is obtained as a corollary of bounds on these linearized correctors.

The following lemma (which is only used in an approximation argument) defines these linearized correctors. It is a consequence of [START_REF] Chiarini | Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium[END_REF]Section 4] and [24, Lemma 1] (which is devoted to the existence and uniqueness for linear corrector equations with prescribed in advance unbounded and degenerate coefficients). In the actual proofs, we shall only consider linearized correctors on bounded domains (see the discussion on periodization below), and the definition and control of the whole-space linearized corrector is given for completeness. anchored at the origin via ´B φξ,e = 0, and whose gradient ∇ φξ,e is stationary, has vanishing expectation E[∇ φξ,e ] = 0, and is bounded in the sense of

E |∇ φξ,e | 2 (1 + |ξ + ∇φ ξ |) p-2 (1 + |ξ| p-2 )|e| 2 .
In addition, there exists a skew-symmetric random matrix field (σ ξ,e,ij ) 1≤i,j≤d , which solves almost surely in the distributional sense in R d the linearized flux corrector equation where āξ e = E a ξ (e + ∇ φξ,e ) .

-σξ,e,ij = ∂ i (a ξ (e + ∇ φξ,e ) • e j ) -∂ j (a ξ (e + ∇ φξ,e ) • e i ), (3.2
The upcoming theorem gives further information on the linearized correctors, in line with Theorem 15 for the nonlinear correctors. where C ξ,g is a random variable with stretched exponential moments E exp(c ξ C α ξ ξ,g ) ≤ 2, for some exponent α ξ > 0 and some constant c ξ > 0 depending on p, λ, d, and |ξ|. This directly implies that for all x ∈ R d , we have |( φξ,e , σξ,e )(x)| ≤ C x,ξ µ d (x), where C x,ξ is a random variable with the same moment bounds as C ξ,g . Remark 3.2.3. Under Hypothesis 3.1.1, for Q-periodic matrix fields A, the linearized correctors exist and are bounded in C 1,α (Q) (no restriction on p ≥ 2).

The general strategy we described above is essentially the same as in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] for p = 2, itself very close to the strategy in the linear setting [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF]. In line with [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF][START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF][START_REF] Otto | Introduction to stochastic homogenization[END_REF][START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], we are after Meyers' type estimates for the linearized operator -∇ • a ξ ∇ (defined in Lemma 13) . The main difference between this work and [5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF][START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] is the way we obtain these estimates -that is, in the annealed version of Theorem 22. For p = 2, annealed Meyers estimates (even without loss of stochastic integrability) follow rather directly from the boundedness of a ξ from above and below. In our genuinely nonlinear setting, these estimates are difficult to establish since a ξ may be degenerate and unbounded (recall that the original proof of Meyers' estimates argues by perturbation and requires that inf a ξ sup a ξ > 0, whereas we have inf a ξ sup a ξ = 0 almost surely). There are two reasons why inf a ξ sup a ξ = 0: the degeneracy of a ξ and the unboundedness of a ξ . The upcoming technical discussion points out the difficulties in the analysis, give hints on how to treat them, and explains how it leads to the above results.

First, in view of the difficulty to control the critical set of harmonic coordinates, we have imposed a non-degeneracy condition from the very beginning and assumed that a(x, •) satisfies (3.1.4) for κ = 1 (which rules out the p-Laplacian, but not the p-Laplacian regularized at 0). Let us emphasize that this only yields the non-degeneracy of the linearized operator in a perturbative way (it disappears in the regime when the solution has a large gradient). Doing so, the main remaining (and most important) difficulty is the unboundedness of the coefficients of the linearized operator.

Meyers estimates are the object of Section 3.3. We start with Subsections 3. In dimension d = 3, the condition on p reads p < 6. To reach the larger range of exponents of Theorem 12, the main observation is that one can improve Caccioppoli's inequality by choosing wisely the cut-off function -following an idea by Bella and Schäffner [START_REF] Bella | Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations[END_REF] (see Lemma 17). Doing so, we are able to use the Sobolev embedding in dimension d -1 rather than d, and therefore treat all exponents p ≥ 2 in the physically-relevant dimension d = 3. This yields the large scale Meyers' estimates of Theorem 18 (with a condition on p in dimensions d > 3).

In the form of Theorem 18, the Meyers estimates are only useful if we have a good control of the Meyers minimal radius r (the larger r , the weaker the estimate), which we obtain in Subsection 3.3.3. Since r is a stationary random field, by control we mean moment bounds in probability. This is where our contribution further differs from other contributions of the literature on degenerate or unbounded coefficients: the statistics of a ξ (which drives the moments of r ) are not given a priori (as opposed to the percolation cluster in [START_REF] Armstrong | Elliptic regularity and quantitative homogenization on percolation clusters[END_REF], or to the moment bounds on a ξ in [START_REF] Bella | A Liouville theorem for elliptic systems with degenerate ergodic coefficients[END_REF]) but part of the problem -estimate (3.2.6), which is essentially equivalent to the control of r in Theorem 21, is the very output of the analysis. Indeed, the coefficients a ξ are a function of ∇φ ξ , which depends itself on the random input A as the solution of the nonlinear corrector equation (3.1.7) (and therefore far from explicit).

Here comes the second ingredient to our approach: sensitivity calculus and concentration of measures (see Appendix 3.7.7). On the one hand, using the Meyers estimate in the large (in its improved weighted form of Theorem 19 based on the hole-filling estimate) and sensitivity calculus, we control the stochastic moments of averages of ∇φ ξ by the CLT scaling and moments of r -see Proposition 5. On the other hand, by Caccioppoli's inequality for the nonlinear corrector equation, we control super level sets of r by moments of averages of ∇φ ξ . The desired control of the moments of r of Theorem 21 follows by combining these two nonlinear estimates and taking advantage of the CLT scaling and the small room given by the hole-filling to buckle and single out r .

Once we have good control of r , the quenched large scale Meyers estimates of Theorem 18 can finally be upgraded to the annealed Meyers estimates of Theorem 22 (as introduced by Duerinckx and Otto in [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF], using Shen's lemma [START_REF] Shen | The l p boundary value problems on lipschitz domains[END_REF]), cf. Subsection 3.3.4.

Having these estimates at hand, Theorem 15 follows from another application of sensitivity calculus, cf. Section 3.4. The proof of Theorem 17 is similar, cf. Section 3.5. Although we have Theorem 22, we cannot use the elegant and efficient buckling argument of [START_REF] Otto | Introduction to stochastic homogenization[END_REF] for the linearized corrector either (due to unboundedness), and we have to pass again via the super level sets of another minimal radius. We then conclude with the routine proof of Theorem 12 in Section 3.6.

In order to establish these estimates on nonlinear and linearized correctors, we first use an approximation argument which allows us to discard the long-range correlations induced by the elliptic character of the equation, and actually define the associated approximations of the nonlinear flux corrector and of the linearized correctors by elementary deterministic arguments. In this contribution, we proceed by periodization in law, which has the advantage to keep differential relations neat in the approximation (in particular the identity (3.2.3)). For all L > 0, we introduce in Definition 3.7.2 (see Appendix 3.7.7) a probability measure

P L taking values in Q L = [-L 2 , L
2 ) d -periodic functions. The associated maps x → a(x, ξ) are therefore Q L -periodic P L -almost surely, and the corrector equations are posed on the bounded domain Q L . The coupling between P and P L given in Lemma 29 then allows us to infer results on P from corresponding results on P L , see in particular Proposition 10. The choice of periodization in law is convenient but not essential. In the linear setting one often adds a massive term to the equation (which yields an exponential cut-off for long-range interactions) [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization[END_REF][START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF] or disintegrates scales via a semi-group approach [START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF][START_REF] Clozeau | Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields[END_REF]. All our estimates are proved for fixed periodization and the above results follow by letting the periodization parameter go to infinity.

Towards large-scale regularity and a nonlinear theory of fluctuations

As discussed in Chapter 1 and in Subsection 6.2.1 in Chapter 6 (see the paragraph on fluctuations), there are three main types of results in (quantitative) stochastic homogenization of the linear elliptic equation (3.2.1):

• Control of oscillations of the solution via a quantitative two-scale expansion (here in form of Theorem 12 in the nonlinear setting);

• Large-scale regularity for the operator -∇ • A( x ε )∇; • Control of the fluctuations of observables of the form ´g • ∇u ε and ´g • A( • ε )∇u ε (using the so-called homogenization commutator).

Let us start with large-scale regularity. The general principle [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF][START_REF] Avellaneda | L p bounds on singular integrals in homogenization[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF][START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] is that the heterogeneous equation should possess the same regularity properties as the homogenized operator at the scale at which homogenization kicks in (characterized by the size of the corrector). In our range 2 ≤ p <

2(d-1)

d-3 of exponents (for which we already control the growth of correctors) and if we restrict ourselves systems to scalar equations, the results [START_REF] Bella | On the regularity of minimizers for scalar integral functionals with (p, q)-growth[END_REF] by Bella and Schäffner (combined with the C 1,1 loc regularity of ξ → ā(ξ)) should ensure that -∇ • ā(∇) does possess nice regularity theory (both in terms C 1,α regularity and nonlinear Calderón-Zygmund theory), so that there is no obstruction to largescale regularity in this setting. Such large-scale regularity would typically allow to upgrade the quantitative two-scale expansion of (3.1.13) stated in L 2 (R d ) to any L q (R d ) with 1 < q < ∞ (with the same convergence rate in ε), and in particular cover the natural L p (R d )-norm.

Next to the large-scale regularity for the nonlinear operator -∇•a(•, ∇), one may wish to establish large-scale regularity for the linearized operator -∇ • a ξ ∇. In Section 3.3 below, large-scale Meyers estimates are proved (in their convenient annealed form of Theorem 22). Although this perturbative result is enough to prove the quantitative two-scale expansion of Theorem 12, Theorem 22 should hold for all exponents 1 < q < ∞ by adapting the arguments of [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] to mildly unbounded coefficients (a ξ indeed has finite stretched exponential moments by Lemma 21 and Theorem 21) and using the bounds of Theorem 17 on the linearized correctors (to control the scale at which homogenization kicks in).

We now turn fluctuations, the second main topic of stochastic homogenization. In the linear setting, the theory of fluctuations relies on the quantity Ξ ε := (A( • ε ) -Ā)∇u ε , called the homogenization commutator in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] (this object first appeared in a slightly different form in [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF] in the context of large-scale regularity). The homogenization commutator is a natural object to consider since observables of Ξ ε can be post-processed into observables of the field and flux (the two cornerstones of homogenization), that is, ´g • ∇u ε and ´g • A( • ε )∇u ε -see below for the argument in the nonlinear setting. The theory splits into two parts. On the one hand, fluctuations of the homogenization commutator Ξ ε can be accurately described by the fluctuations of its two-scale expansion based on the standard commutator Ξ := (A -Ā)(∇φ + Id), cf. [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Robustness of the pathwise structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]. On the other hand, the standard commutator Ξ behaves (in law) on large scales as a Gaussian random field, cf. [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Scaling limit of fluctuations in stochastic homogenization[END_REF]. The proofs of these results in the linear setting make heavy use of large-scale regularity. In order to extend this theory to the nonlinear setting, one mainly needs large-scale regularity for the linearized operator and a suitable notion of nonlinear homogenization commutator. A naïve guess would be to define the nonlinear commutator as a ε (∇u ε ) -ā(∇u ε ). This quantity however does not weakly converge to zero since ā is nonlinear. We have to devise a quantity that is compatible with weak convergence and still encapsulates the diagram of Figure 1.1.2. To this aim we reformulate the constitutive law by linearizing (assuming that ā is differentiable): Since a ε (0) = ā(0) = 0, we then have

a( • ε , ξ) = a( • ε , ξ) -a( • ε , 0) = ˆ1 0 Da( • ε , tξ)dt ξ, ā(ξ) = ā(ξ) -ā(0) = ˆ1 0 Dā(tξ)dt ξ,

Gradient field

Constitutive law Flux Conservation law In this way, homogenization can be concisely reduced to the single condition ( ´1 0 Da( • ε , t∇u ε )dt)∇u ε -( ´1 0 Dā(t∇ū)dt)∇u ε 0, and we define the nonlinear homogenization commutator of u ε as

∇u ε multiply by ´1 0 Da( • ε ,t∇uε)dt -→ q ε = ( ´1 0 Da( • ε , t∇u ε )dt)∇u ε -∇ • q ε = ∇ • f ↓ ↓ ↓ ∇ū multiply by ´1 0 Dā(t∇ū)dt -→ q = ( ´1 0 Dā(t∇ū)dt)∇ū -∇ • q = ∇ • f
Ξ ε (f ) := ˆ1 0 Da( • ε , t∇u ε )dt ∇u ε - ˆ1 0 Dā(t∇ū)dt ∇u ε . (3.2.14)
Let us argue that, as in the linear setting, the homogenization commutator contains both the fluctuations of the field ∇u ε and of the flux a( • ε , ∇u ε ). We start with ∇u ε and consider fluctuations of the observable ´g • ∇u ε . We introduce the auxiliary map v solution of the linear equation

∇ • ˆ1 0 Dā(t∇ū)dt * ∇v = ∇ • g, (3.2.15)
(which we assume to be well-posed in this discussion -the notation * is used for the transposition).

Then we have ˆg

• ∇u ε (3.2.15) = - ˆ ˆ1 0 Dā(t∇ū)dt * ∇v • ∇u ε (3.2.14) = -ˆ∇v • (Ξ ε (f ) -a( • ε , ∇u ε )) (3.1.5) = -ˆ∇v • Ξ ε (f ) -ˆ∇v • f,
so that the fluctuations of ´g • ∇u ε are given by those of -´∇v • Ξ ε (f ) (since the additional term -´∇v • f is deterministic). Likewise, for the flux we introduce the solution w of

∇ • ˆ1 0 Dā(t∇ū)dt * ∇ w = ∇ • ˆ1 0 Dā(t∇ū)dt * g, (3.2.16) and obtain ˆg • a( • ε , ∇u ε ) (3.1.5) = ˆ(g -∇ w) • a( • ε , ∇u ε ) -ˆ∇ w • f (3.2.14) = ˆ(g -∇ w) • Ξ ε (f ) + ˆ(g -∇ w) • ˆ1 0 Dā(t∇ū)dt ∇u ε -ˆ∇ w • f (3.2.16) = ˆ(g -∇ w) • Ξ ε (f ) -ˆ∇ w • f, so that the fluctuations of ´g • a( • ε , ∇u ε ) are given by those of -´(g -∇ w) • Ξ ε (f ) (since the additional term -´∇ w • f is deterministic).
Next to the homogenization commutator of the solution, we introduce the standard homogenization commutator, associated with the corrector. For all ξ ∈ R d , we define

Ξ ξ := ˆ1 0 Da(•, t(ξ + ∇φ ξ ))dt (ξ + ∇φ ξ ) - ˆ1 0 Dā(tξ)dt (ξ + ∇φ ξ ). (3.2.17)
The pathwise structure of fluctuations (in the form of a quantitative two-scale expansion of commutators in the spirit of (3.1.11) and accurate in the fluctuation scaling -see [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Robustness of the pathwise structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF] in the linear setting) and the scaling limit of the standard commutator (see [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Scaling limit of fluctuations in stochastic homogenization[END_REF] in the linear setting) will be the object of companion articles.

Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

Extensions and limitations

Hypothesis 3.1.2 makes several assumptions on the monotone operator and the randomness:

• The underlying probability law is Gaussian with integrable correlations;

• The monotone map a(x, ξ) is a multiple of (1 + |ξ| p-2 )ξ, the randomness is multiplicative (in form a random matrix field), and the admissible range of p depends on d;

• The spatial dependence x → a(x, ξ) is smooth on a deterministic level;

• If it admits a variational form, the operator is associated with a convex energy functional.

Several of these assumptions can be slightly relaxed, while others are crucial. They are discussed in the following paragraphs.

Probability laws

Consider our multiplicative model. Our approach is based on a sensitivity calculus which allows us to linearize quantities with respect to the randomness (say, wrt A) and on functional inequalities which allow us to control variances using this sensitivity calculus. In Hypothesis 3.1.2 we consider a Gaussian random field with integrable covariance function, and one might wonder to what extent Gaussianity and the integrability of the covariance functions are necessary. Our argument strongly relies on the CLT scaling r -d 2 of spatial averages ffl Br ∇φ ξ of the corrector gradient, which essentially follow from the same property for a(x, ξ) -E a(x, ξ) . On the one hand, sensitivity calculus, functional inequalities, and CLT scaling are not limited to Gaussian fields: they can be developed as soon as the stationary field A is constructed via a "hidden" product structure. In particular, the random checkerboard and various Poisson-based processes also enjoy such tools, and we refer the reader to [START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF] for a systematic study of sensitivity calculus and (multiscale) functional inequalities for random fields commonly used in the mechanics of composite materials [START_REF] Torquato | Random heterogeneous materials[END_REF]. Such models could be considered here as well. On the other hand, the CLT scaling indeed requires the integrability of the covariance function. (Since there is some little room in the argument, one could consider a covariance function such that ´Rd |c(x)|(1 + |x|) -β dx < ∞ provided 0 < β 1, but this is detail.) In order to address Gaussian coefficients with heavier tail, one would first need to establish (nonlinear and linear) large-scale regularity for the random operator (and its linearized version), as in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative estimates in stochastic homogenization for correlated coefficient fields[END_REF][START_REF] Clozeau | Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields[END_REF]. By [START_REF] Bella | On the regularity of minimizers for scalar integral functionals with (p, q)-growth[END_REF] and (3.1.3), there is no obstruction to this approach in the range 2 ≤ p < 2(d-1)

d-3
for scalar equations.

Another setting would ensure that spatial averages of a(x, ξ) -E a(x, ξ) decay at the CLT scaling: if A has finite range of dependence -as addressed in [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF]5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF] for p = 2. The quenched Meyers estimates of Theorem 18 (and its weighted version of Theorem 19) proved below do hold for general stationary ergodic coefficients -and therefore in the setting of finite range of dependence. They are however of little use without a good control of the Meyers minimal radius (provided by Theorem 21 for Gaussian coefficients with integrable covariance). In [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF]5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF] for p = 2, estimates of moments of the corrector gradient (which would control the Meyers minimal radius) are obtained by combining a rate of convergence (any would do) for the Dirichlet problem with a Campanato argument based on C 1,α -regularity for the homogenized operator. In particular, one would have to adapt the duality arguments of [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF]5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF] to p > 2 to prove convergence rates. Since the natural object considered in [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF]5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF] is close to the homogenization commutator for p = 2, the nonlinear commutators we introduced in (3.2.14) and (3.2.17) might be good objects to start with for p > 2.

Form of the monotone map

There are three different assumptions when considering a monotone map of the form (x, ξ) → a(x, ξ) = A(x)(1 + |ξ| p-2 )ξ: coercivity conditions, regularity with respect to ξ, and multiplicative character of the randomness. To start with, we must assume that ξ → a(x, ξ) is twice-differentiable (for all x) in order to apply sensitivity calculus to the linearized corrector.

Multiplicative models. The form of a is such that one can easily differentiate a with respect to the randomness. This is not strictly necessary but quite convenient. Any model having such a property would do, and we can consider coefficients of the form a(x, ξ) = ρ(A(x), ξ)ξ provided

M → ρ(M, •) satisfies |D M ρ(M, ξ)| 1 + |ξ| p-1 and |D M ∂ ξ ρ(M, ξ)| 1 + |ξ| p-2 . This holds for instance for a(x, ξ) = χ(x)a 1 (ξ) + (1 -χ(x))a 2 (ξ), (3.2.18)
where χ : R d → [0, 1] is a smooth random field (with a sensitivity calculus and a suitable functional inequality) and a 1 and a 2 are two given (suitable) monotone maps. This model is more in line with composite materials.

Coercivity conditions. What is crucial is the strong monotonicity and non-degeneracy (3.1.4) for some κ > 0 (which we take to be 1 without loss of generality). Whereas local regularity would hold with κ = 0, the choice κ > 0 is forced upon us to rule out the degeneracy of the linearized operator (cf. the critical set of ∇φ ξ ). In particular, this condition does not hold for the p-Laplacian, to which our results do not apply. To our opinion, relaxing this condition constitutes a very challenging problem.

Restriction on p. Our assumptions (3.1.1) and (3.1.4) make the the operator have p-growth from above and below. In dimensions d > 3, we further impose the condition 2 ≤ p < 2(d-1) d-3 . Doing so allows us to prove a quenched Meyers estimate in the large with an almost surely finite radius r . In order to relax this condition, a possible starting point could be to introduce (yet another) minimal radius, say ρ , which would be defined such that ffl Br r (x) q dx 1 for some suitable q depending on d, p and all r ≥ ρ . The additional difficulty is that one would have to deal with the three quantities r , ρ , ∇φ ξ together rather than the two quantities r , ∇φ ξ . Since there is only little room in the exponents for the buckling argument (see proof of Theorem 21), it is not clear to us whether this might indeed work.

Local regularity

It is quite tempting to assert that quantitative homogenization is a matter of large scales (or say, low frequencies), and that local regularity assumptions might be convenient but are not necessary. This is indeed quite relevant provided small scales do not interact with large scales. A convincing counterexample of that is the quasiperiodic (and almost periodic) setting, where small and large scales indeed interact via a weak Poincaré inequality in a high-dimensional torus, cf. [START_REF] Armstrong | Bounded correctors in almost periodic homogenization[END_REF]. In our nonlinear setting, local regularity is not so much needed for the nonlinear correctors, but it seems unavoidable for the linearization part. This regularity requirement could be weakened in several directions:

• Only a local C α -control of the spatial dependence is needed for some α > 0, and the control of this local norm can be random itself provided the latter has good moment bounds. In particular, with the same notation as in Hypothesis 3.1.2, this is the case for coefficients of the form A(y) = B(G(y)) provided the (non-negative) Fourier transform ĉ of the covariance function satisfies ĉ(k) ≤ (1+|k|) -d-2α (for some α > α). Then x → A C α (B(x)) is stationary systems and has finite Gaussian moments (as a slight quantification of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Appendix A.3] shows). All our arguments can be adapted to this setting.

• In the proofs we use local regularity to control pointwise values of the (nonlinear and linear) corrector gradient by its local averages, and therefore control a local supremum by a local C α -norm. Such a control would also follow from a local broken C α -norm, so that one could in principle be able to deal with some A (or χ in (3.2.18)) that would be piecewise smooth (and a fortiori piecewise constant with smooth boundaries, covering the case of smooth inclusions in a background material). This constitutes a question of classical regularity theory. For linear equations and systems, this is proved in [110,[START_REF] Li | Estimates for elliptic systems from composite material[END_REF] and for monotone operators and p = 2 in [START_REF] Neukamm | Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains[END_REF]. The case p > 2 constitutes an interesting independent problem.

• The state of the art of local regularity is as follows. For scalar equations, the structure can be quite general, and only requires the Hölder continuity of the map x → a(x, •) in the sense (see [START_REF] Kuusi | Guide to nonlinear potential estimates[END_REF]Theorem 13])

sup r>0 ˆr 0 (ω(ρ)) 2 p ρ α dρ ρ < +∞, (3.2.19)
for

ω : r ∈ (0, +∞) → sup z∈R d ,Br(x)⊂R d Br(x) a(y, z) -(a) x,r (z) (|z| + 1) p-1 2 dy 1 2 ,
for some α > 0 and (a) x,r (z) := ffl Br(x) a(y, z)dy. For systems however, we are restricted to quasi-diagonal structures of the form a(x, ξ) = ρ(x, |ξ|)ξ, for some ρ : R d × R d → R (the so-called Uhlenbeck structure, see [START_REF] Uhlenbeck | Regularity for a class of non-linear elliptic systems[END_REF]).

Non-convex energy functionals?

It would be natural to try to extend these results to the setting of nonlinear elasticity, for which a large part of the qualitative theory has been established (cf. [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF][START_REF] Braides | Homogenization of some almost periodic coercive functional[END_REF][START_REF] Messaoudi | Stochastic homogenization of nonconvex integral functionals[END_REF], and [START_REF] Duerinckx | Stochastic homogenization of nonconvex unbounded integral functionals with convex growth[END_REF] for the most general results in this context). Besides the much more delicate regularity theory (cf. [START_REF] Mingione | Regularity of minima: an invitation to the dark side of the calculus of variations[END_REF]), non-convexity essentially prevents us from using the corrector equation efficiently (cf. the counterexamples to the cell formula in the periodic setting by Müller [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF], see also [START_REF] Barchiesi | New counterexamples to the cell formula in nonconvex homogenization[END_REF]), and may cause loss of ellipticity upon linearization (see [START_REF] Geymonat | Homogenization of non-linearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity[END_REF] at the nonlinear level, and [START_REF] Gutiérrez | Laminations in linearized elasticity: the isotropic non-very strongly elliptic case[END_REF][START_REF] Briane | Loss of ellipticity through homogenization in linear elasticity[END_REF][START_REF] Francfort | Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity[END_REF][START_REF] Gloria | Loss of strong ellipticity through homogenization in 2D linear elasticity: a phase diagram[END_REF] at the linear level) -except in the vicinity of the identity (cf. [START_REF] Müller | On the commutability of homogenization and linearization in finite elasticity[END_REF][START_REF] Gloria | Commutability of homogenization and linearization at identity in finite elasticity and applications[END_REF], and the further use of rigidity [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity[END_REF] to establish quantitative results in this regime [START_REF] Neukamm | Quantitative homogenization in nonlinear elasticity for small loads[END_REF]). Hence, quantitative results in homogenization of nonlinear nonconvex models of elasticity remain widely out of reach today.

Perspectives

We refer to Section 6 for some possible other extension and open questions about this work.

Perturbative regularity theory for the linearized operator

In this section we consider periodized random operators a L distributed according to the law P L given in Definition 3.7.2. In particular, for all L ≥ 1, a L is almost surely Q L -periodic in its space variable, and remains random and stationary (this owes to the fact that we use periodization in law rather than naive periodization, cf. Appendix 3.7.7). This implies that φ ξ and σ ξ are necessarily Q L -periodic fields almost surely, so that the equations (3.1.7) and (3.2.2) can be posed on Q L rather 3.3. Perturbative regularity theory for the linearized operator 163 than R d -and likewise for the linearized correctors. For all L ≥ 1 we use the notation

H 1 per (Q L ) (resp. W 1,p per (Q L )) for Q L -periodic fields of H 1 loc (R d ) (resp. W 1,p loc (R d ))
with vanishing average. Our aim is to prove regularity statements and bounds that are uniform in the periodization parameter L ≥ 1.

The Meyers minimal radius

In this paragraph we introduce the notion of Meyers minimal radius, a stationary random field which quantifies the scale at which Meyers' estimates hold for the linearized operator. We start with a definition.

Definition 3.3.1 (Meyers minimal radius). Let ξ ∈ R d , L ≥ 1 and c > 0. If it exists, the (Q L - periodic) minimal radius r ,ξ,L (•, c) is defined for all x ∈ R d via r ,ξ,L (•, c) : x ∈ R d → inf y∈R d r ,ξ,L (y, c) + |x -y| , ( 3
∈ R d r ,ξ,L (y, c) := inf r=2 N ,N ∈N ∀R ≥ r, B R (y) |∇φ ξ | p ≤ c(1 + |ξ| p ) . (3.3.2)
We now argue that r ,ξ,L (•, c) is a well-defined bounded random field if c is chosen large enough. Proof. Without loss of generality, we may assume that x = 0. We start with the proof of (3.3.4), and then turn to the proof of (3.3.3). We let c denote a constant depending only on d, λ, and p, that may change from line to line.

Step 1. Proof of (3.3.4).

From the defining equation (3.1.7) for φ ξ , we have

-∇ • (a(•, ξ + ∇φ ξ ) -a(•, ξ)) = ∇ • a(•, ξ) in Q L ,
so that by testing the equation with φ ξ and using the monotonicity (3.1.4) and boundedness (3.1.1), we obtain for some constant c depending on λ and d

Q L |∇φ ξ (x)| 2 (1 + |ξ| p-2 + |ξ + ∇φ ξ (x)| p-2 )dx ≤ c Q L |ξ|(1 + |ξ| p-2 )|∇φ ξ |.
By absorbing part of the right-hand side into the left-hand side, this yields

Q L |∇φ ξ (x)| 2 (1 + |ξ| p-2 + |ξ + ∇φ ξ (x)| p-2 )dx ≤ c Q L |ξ| 2 (1 + |ξ| p-2 ). systems
By the triangle inequality in form of |ξ + ∇φ ξ (x)| p-2 |∇φ ξ (x)| p-2 -|ξ| p-2 , and using the above twice, we obtain

Q L |∇φ ξ (x)| 2 (1 + |∇φ ξ (x)| p-2 )dx ≤ c Q L |ξ| 2 (1 + |ξ| p-2 ) ≤ c(1 + |ξ| p ).
Assume that L is dyadic. Given now R ≥ L, we cover B R by N L,R ≤ c d ( R L ) d translations of Q L (where c d only depends on dimension), which we denote by

Q j L for 1 ≤ j ≤ N R,L . This yields B R (y) |∇φ ξ | 2 + |∇φ ξ | p ≤ L d |B R | N R,L j=1 Q j L |∇φ ξ | 2 + |∇φ ξ | p ≤ c d R d L d L d |B R | c(1 + |ξ| p ) = c 1 (1 + |ξ| p )
for the choice c 1 := c d |B| -1 , which only depends on d and λ. This yields (3.3.4). If L is not dyadic, we cover B R by cubes of sidelength 2 l with l such that 2 l ≤ L < 2 l+1 , and obtain the result at the price of increasing c 1 .

Step 2. Proof of (3.3.3). By definition (3.3.1) of r ,ξ,L , we have r ,ξ,L (0) ≤ r ,ξ,L (0) by testing the infimum problem with y = 0. Let us now prove that there exists c 2 such that for all R ≥ 1 we have the implication

r ,ξ,L (0, c 1 ) ≤ R =⇒ r ,ξ,L (0, c 2 ) ≤ R, from which we deduce (3.3.3). By definition (3.3.1) of r ,ξ,L , if r ,ξ,L (0, c 1 ) ≤ R, there exists y ∈ R d such that |y| ≤ R and r ,ξ,L (y, c 1 ) ≤ R. This implies that B R ⊂ B R(y) with R := ( 1 + 1)R so that B R |∇φ ξ | p ≤ ( R R ) d B R |∇φ ξ | p ≤ ( 1 + 1) d c 1 .
Hence, with c 2 := ( 1 + 1) d c 1 , this yields r ,ξ,L (y, c 2 ) ≤ R, and therefore (3.3.3).

In the rest of the paper, the notation r ,ξ,L refers to the minimal scales r ,ξ,L (•, c 1 ) for which Lemma 14 holds. When no confusion occurs, we simply write r for r ,ξ,L , and use the short-hand notation B (x) for B r ,ξ,L (x) (x).

We conclude this paragraph by showing that the Meyers minimal radius controls local averages of the nonlinear corrector.

Lemma 15 (Control of averages of the nonlinear correctors).

There exists a nonlinear hole-filling exponent 0 < δ ≤ d depending on d, p, and λ such that for all (x, ξ) ∈ R d × R d , we have for all r > 0

Br(x) |ξ + ∇φ ξ | 2 + |ξ + ∇φ ξ | p d,λ,p (1 + |ξ| p ) r (x) ∨ r r d-δ . (3.3.5)
Proof. Without loss of generality, we may assume that x = 0. We use the short-hand notation ρ := r ∨ r ≥ r . By the hole-filling estimate (3.7.3) applied to the defining equation (3.1.7) for φ ξ , there exists δ > 0 depending on d and λ such that

Br |ξ + ∇φ ξ | 2 + |ξ + ∇φ ξ | p ρ r d-δ Bρ |ξ+∇φ ξ | 2 +|ξ+∇φ ξ | p ρ r d-δ Bρ |ξ| 2 +|ξ| p +|∇φ ξ | 2 +|∇φ ξ | p .
Theorem 18 relies on the combination of Lemma 16 with Gehring's inequality in form of Lemma 18 (Gehring's lemma). Let s > 1, and let f and h be two non-negative measurable functions in L q loc (R d ) such that there exists C > 0 for which for all r > 0 and x ∈ R d Br(x)

f s 1 s ≤ C B 2r (x) f + B 2r (x) h s 1 s .
Then, there exists s > s depending on q and C such that for all r > 0 and x ∈ R d , we have

Br(x) f s 1 s B 2r (x) f + B 2r (x) h s 1 s .
We are now in the position to prove Theorem 18.

Proof of Theorem 18. Let 1 ≤ q * < 2 be given by (3.3.14). We first prove that for all r > 0

Br B (x) |∇u| 2 µ ξ dx B 2r B (x) |∇u| 2 µ ξ q * 2 dx 2 q * + B 2r B (x) |g| 2 dx. (3.3.19)
If r ≤ 3r (0) this estimate follows from Lemma 33, and it remains to treat the case r ≥ 3r (0). We first use (3.7.62) with f = |∇u| 2 µ ξ to the effect of ˆBr

B (x) |∇u| 2 µ ξ dx B 67 48 r |∇u| 2 µ ξ . (3.3.20)
Then, by the reverse Hölder inequality (3.3.13) followed by (3.7.63), we obtain

(1 + |ξ| p ) p p-2 B 2r B (x) |∇u| q * dx 2 q * + B 2r B (x) |g| 2 dx. (3.3.21)
We then slightly reformulate the first right-hand side term using Jensen's inequality in the inner integral (since q * < 2) and the lower bound µ ξ ≥ 1, so that

B 2r B (x) |∇u| q * dx 2 q * ≤ B 2r B (x) |∇u| 2 q * 2 dx 2 q * ≤ B 2r B (x) |∇u| 2 µ ξ q * 2 dx 2 q * . (3.3.22)
The combination of (3.3.20)

f : x → B (x) |∇u| 2 µ ξ q * 2 , h : x → B (x) |g| 2 q * 2 , s = 2 q * > 1.
This yields (3.3.10), whereas (3.3.11) follows by applying (3.3.10) for B r with r = √ d 2 L and using the periodicity of the quantities involved together with the plain energy estimate ´QL |∇u| 2 µ ξ ´QL |g| 2 . systems

Quenched weighted Meyers' estimate in the large

The main result of this paragraph is the following upgrade of Theorem 18.

Theorem 19 (Quenched weighted Meyers estimates in the large). Under Hypothesis 3.1.2, for all ξ ∈ R d , there exists β > 0 depending only on |ξ| and d such that for all 2 ≤ m ≤ m (cf. Theorem 18), 0 ≤ 2ε ≤ β and all Q L -periodic fields g and u related via (3.3.9), we have ˆQL ω ε,r (x)

B (x) |∇u| 2 µ ξ m 2 dx |ξ| ˆQL ω 2ε,r (x) B (x) |g| 2 m 2 dx, (3.3.23)
where for all

x ∈ Q L ω ε,r (x) := 1 + |x| + r (0) r ε . (3.3.24)
The same result holds with a ξ replaced by a * ξ (the pointwise transpose field).

We proceed in two steps: From Theorem 18 we first prove a suitable hole-filling estimate which we use in turn to upgrade Theorem 18.

Corollary 20 (Linear hole-filling estimate in the large). Under Hypothesis 3.1.2, for all ξ ∈ R d there exist an exponent β > 0, depending only on d, p and |ξ|, and a constant c d ≥ 1 with the following properties. Let u be a

Q L -periodic function which is a ξ -harmonic in Q R (x) for some x ∈ R d and L ≥ R ≥ c d r (x), that is -∇ • a ξ ∇u = 0 in Q R (x).
Then for all r (x) ≤ r ≤ R, ˆQr(x)

|∇u| 2 µ ξ |ξ| ( r R ) β ˆQR (x) |∇u| 2 µ ξ . (3.3.25)
The same result holds with a ξ replaced by a * ξ (the pointwise transpose field).

Proof of Corollary 20 . Without loss of generality, we may assume that x = 0, r ≥ r (0), and that 2cr ≤ R 4 with c = 3∨

:= d(1 -1 m ) ˆQr |∇u| 2 µ ξ ≤ ˆBcr |∇u| 2 µ ξ (3.7.63) r d B 2cr B (x) |∇u| 2 µ ξ dx ≤ r d B 2cr B (x) |∇u| 2 µ ξ mdx 1 m ≤ r d ( R r ) d m B R 4 B (x) |∇u| 2 µ ξ mdx 1 m (3.3.10) r d ( R r ) d m B R 4 B (x) |∇u| 2 µ ξ dx (3.7.62) ( r R ) β ˆB R 2 |∇u| 2 µ ξ ≤ ( r R ) β ˆQR |∇u| 2 µ ξ .
We now prove Theorem 19.

Proof of Theorem 19. We split the proof into four steps. In the first step, we show that for compactly supported right-hand sides g the solution gradient decays algebraically away from the source in L 2 , based on hole-filling. We then upgrade this L 2 estimate into an L m estimate for some m > 2 using Meyers' estimate (3.3.10). In the third step, we remove the assumption that g be compactly supported by using a dyadic decomposition of scales. In the last step we exploit the algebraic decay to add the desired weight. Since the proof relies on a dyadic decomposition of the torus, it is convenient to work with cubes rather than balls when taking averages (which makes constants slightly cumbersome).

Step 1. L 2 algebraic decay rate. We prove that there exists δ > 0, depending on d, p and |ξ|, such that for all L ≥ R ≥ r ≥ c d r (0) and all g compactly supported in Q r we have

ˆQL \Q R |∇u| 2 µ ξ ( r R ) β ˆQr |g| 2 . (3.3.26)
We proceed by duality in form of

ˆQL \Q R |∇u| 2 µ ξ = sup h ˆQL \Q R h • ∇u √ µ ξ , (3.3.27)
where the supremum runs over functions

h ∈ L 2 (Q L \Q R ) d with h L 2 (Q L \Q R ) d = 1.
Consider such a test function h (implicitly extended by zero on Q R ) and denote by v the unique weak solution in

H 1 per (Q L ) of -∇ • a * ξ ∇v = ∇ • (h √ µ ξ ), (3.3.28)
which is well-posed since µ ξ is bounded on Q L almost surely by Lemma 25. By testing (3.3.28) with u and (3.3.9) with v, we obtain by Cauchy-Schwarz' inequality and the support condition on g Step 2. L m algebraic decay rate for 2 ≤ m ≤ m.

ˆQL h • ∇u √ µ ξ = ˆQL g • ∇v √ µ ξ ≤ ˆQr |g| 2 1 2 ˆQr |∇v| 2 µ ξ 1 2 . (3.3.29) Since h vanishes on Q R , v is a * ξ -harmonic in Q R ,
ˆQr |∇v| 2 µ ξ ( r R ) β ˆQR |∇v| 2 µ ξ ( r R ) β ˆQL |h| 2 = ( r R ) β . ( 3
In this step, we prove that, with C d = 4C ∨ c d ∨ 16 (with C ≥ 1 as in (3.7.64)), for all L > R ≥ 2r with r ≥ C d r (0), and all g compactly supported in Q r , we may upgrade (3.3.26) to

ˆQL \Q R B (x) |∇u| 2 µ ξ m 2 dx R d(1-m 2 ) ( r R ) δ m 2 ˆQr |g| 2 m 2 . (3.3.31)
for all 2 ≤ m ≤ m, where m is the Meyers exponent of Theorem 18. systems

Let J ∈ N be such that 2 J R < L ≤ 2 J+1 R. By writing Q L \ Q R = (Q L \ Q 2 J R ) ∪ ∪ J j=1 (Q 2 j R \ Q 2 j-1 R
) (with the convention that the second union is empty if J = 0), it is enough to prove that for all 1 ≤ j ≤ J + 1, we have

ˆQ(2 j R)∧L \Q 2 j-1 R B (x) |∇u| 2 µ ξ m 2 dx (2 j R) d(1-m 2 ) ( r 2 j R ) β m 2 ˆQr |g| 2 m 2 . (3.3.32)
Indeed, for all m ≥ 2, d(1 -m 2 ) -δ m 2 ≤ -δ, so that the dyadic terms sum to (3.3.31). To start with, reverting from balls to cubes, one may reformulate Theorem 18 with cubes instead of balls, and replace B r and B 2r by Q r and Q C 1 r , respectively (for some C 1 depending only dimension). Let 1 ≤ j ≤ J be fixed (the case j = J + 1 can be treated similarly). We partition

Q 2 j R \ Q 2 j-1 R into the union of cubes {Q k } k=1,...,N of side-length 1
C 2 2 j R for some C 2 to be fixed later (the number N of such cubes then depends on d and C 2 , but not on j or R), to the effect that for all m > 2 we have

ˆQ2 j R \Q 2 j-1 R B (x) |∇u| 2 µ ξ m 2 dx = N k=1 ˆQk B (x) |∇u| 2 µ ξ m 2 dx. (3.3.33) By Theorem 18, for all 2 ≤ m ≤ m and 1 ≤ k ≤ N , Q k B (x) |∇u| 2 µ ξ m 2 dx Qk B (x) |∇u| 2 µ ξ dx m 2 + Qk B (x) |g| 2 m 2 dx, (3.3.34)
where

Qk ⊃ Q k denotes the cube of side-length C 1 C 2 2 j R centered at the center x k ∈ Q 2 j R \ Q 2 j-1 R of Q k .
We now control the two right-hand side terms of (3.3.34). On the one hand, by the -Lipschitz property of r and the assumption R ≥ 2C d r (0), for all x ∈ Qk , we have

|x| ≤ |x k | + √ d 2 C 1 C 2 2 j R ≤ √ d 2 (1 + C 1 C 2 )2 j R, and therefore r (x) ≤ r (0) + |x| ≤ R( 1 2C d + √ d 2 2 j (1 + C 1 C 2 )). (3.3.35)
Recall the constant C in (3.7.64) and that C 1 only depends on dimension. We now choose C 2 := 8C 1 .

For our choice

C d = 4C ∨ c d ∨ 16 and R ≥ 2C d r (0), and since 0 < = 1 9C √ d ∧ 1 16 , we have C 1 C 2 2 j R = 2 j-3 R and C( 1 2C d + √ d 2 2 j (1 + C 1 C 2 ) ≤ C( 1 8C + 1 18C 2 j (1 + 1 8 )) ≤ 2 j-3 , which, by (3.3.35), entails C 1 C 2 2 j R ≥ Cr (x), condition under which (3.7.64) yields ˆQ k B (x) |∇u| 2 µ ξ dx ˆQ k |∇u| 2 µ ξ ,
where Qk denotes the cube of side-length

C 1 C 2 2 j+1 R = 2 j-2 R centered at the x k , so that Qk mod LZ d ⊂ Q 2 j+1 R∧L \Q 2 j-2 R . Hence, by (3.3.26), ˆQ k B (x) |∇u| 2 µ ξ dx ˆQ2 j+1 R∧L \Q 2 j-2 R |∇u| 2 µ ξ ( r 2 j R ) β ˆQr |g| 2 . (3.3.36)
On the other hand, the same argument implies

ˆQ k B (x) |g| 2 dx ˆQ2 j+1 R∧L \Q 2 j-2 R |g| 2 = 0, (3.3.37)
where we used that g is supported in Q r and r ≤ R Step 3. Extension to general g. In this step, we relax the support assumption on g in (3.3.31), and claim that for all L ≥ R ≥ 2C d r (0) and all 2 ≤ m ≤ m,

ˆQL \Q R B (x) |∇u| 2 µ ξ m 2 dx 1 m ˆQL \Q R 4 B (x) |g| 2 m 2 dx 1 m + ˆQR ( |x|+r (0) R ) βm 4 B (x) |g| 2 m 2 dx 1 m . (3.3.38) Let N ∈ N be such that 2 N C d r (0) ≤ R < 2 N +1 C d r (0) (note that N ≥ 0 since R ≥ 2C d r (0)). We decompose g as g = N i=0 g i with g 0 := g1 Q C d r (0) , g i := g1 Q 2 i C d r (0) \Q 2 i-1 C d r (0) for all 1 ≤ i ≤ N -1, and g N := g1 Q L \Q 2 N -1 C d r (0)
. By linearity (and uniqueness of the solution) of the equation, we have u = N i=0 u i where u i denotes the (unique) weak solution in

H 1 per (Q L ) of -∇ • a ξ ∇u i = ∇ • (g i √ µ ξ ).
By the triangle inequality, we then have for 2 ≤ m ≤ m,

ˆQL \Q R B (x) |∇u| 2 µ ξ m 2 dx 1 m ≤ N i=0 ˆQL \Q R B (x) |∇u i (y)| 2 µ ξ m 2 dx 1 m . (3.3.39)
We start by estimating the term for i = N , for which we use the Meyers estimate (3.3.11) to the effect that ˆQL

B (x) |∇u N | 2 µ ξ m 2 dx 1 m (3.3.11) ˆQL B (x) |g N | 2 m 2 dx 1 m .
We then reformulate the right-hand side using the support condition on g N . For

x ∈ Q 2 N -2 C d r (0) , since = 1 9C √ d ∧ 1 16 , C ≥ 1, N ≥ 0, and C d ≥ 16, we have r (x) ≤ r (0) + |x| ≤ r (0)(1 + √ d 2 2 N -2 C d ) ≤ 2 N -2 C d r (0)( 1 4 + 1 9 ) ≤ 2 N -3 C d r (0),
so that we have the implication

y ∈ B (x) =⇒ y ∈ Q 2 N -2 C d r (0) (x) =⇒ y ∈ Q 2 N -1 C d r (0) (0) =⇒ g N (y) = 0. Since R < 2 N +1 C d r (0) = 4 2 N -1 C d r (0), Q R 4 ⊂ Q 2 N -1 C d r (0)
, and the above implies

ˆQL B (x) |∇u N | 2 µ ξ m 2 dx 1 m ˆQL \Q R 4 B (x) |g| 2 m 2 dx 1 m . ( 3

.3.40) systems

We then turn to the contributions for 0 ≤ i ≤ N -1, for which we appeal to (3.3.31) with r =

2 i C d r (0) ≥ C d r (0) and R ≥ 2 N C d r (0) ≥ 2r, and obtain ˆQL \Q R B (x) |∇u i | 2 µ ξ m 2 dx (3.3.31) R d(1-m 2 ) ( r R ) β m 2 ˆQr\Qr/2 |g| 2 m 2 R d(1-m 2 ) ( r R ) β m 4 ˆQr\Qr/2 ( |y|+r (0) R ) β 2 |g(y)| 2 dy m 2 .
We then appeal to (3.7.65) (which holds for r since r = 2 i C d r (0) ≥ 2 i Cr (0) by definition of C d ), to Jensen's inequality, and to the Lipschitz regularity of r in form of r (x) ≤ r (0) + |x|, and get

ˆQL \Q R B (x) |∇u i | 2 µ ξ m 2 dx 1 m (3.7.65) R d 2-m 2m ( r R ) β 4 ˆQ2r B (x) ( |y|+r (0) R ) β 2 |g(y)| 2 dydx 1 2 ≤ R d 2-m 2m ( r R ) β 4 (2r) d( 1 2 -1 m ) ˆQ2r B (x) ( |y|+r (0) R ) β 2 |g(y)| 2 dy m 2 dx 1 m ( r R ) β 4 +d m-2 2m ˆQR ( |x|+r (0) R ) βm 4 B (x) |g| 2 m 2 dx 1 m ≤ (2 β 4 +d m-2 2m ) i-N ˆQR ( |x|+r (0) R ) βm 4 B (x) |g| 2 m 2 dx 1 m . (3.3.41)
The claimed estimate (3.3.38) then follows from (

Step 4. Proof of (3.3.23).

If L ≤ 2C d r (0) ≤ 2C d L, then the weight is essentially constant, for all x ∈ Q L , ω r,ε (x) (1 + L r ) ε , and the conclusion (3.3.23) is obviously satisfied. In the rest of this step we thus assume that L > 2C d r (0). Let 2C d r (0) < r ≤ L (the case 0 < r ≤ 2C d r (0) reduces to the case r = 2C d r (0) by homogeneity). Let N ∈ N be such that

2 N C d r (0) ≤ L < 2 N +1 C d r (0) and let N 0 ≤ N be such that 2 N 0 C d r (0) ≤ r < 2 N 0 +1 C d r (0). We then have ˆQL ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx = ˆQ2 N 0 C d r (0) ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx + N -1 i=N 0 ˆQ2 i+1 C d r (0) \Q 2 i C d r (0) ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx + ˆQL \Q 2 N C d r (0) ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx. (3.3.42)
We then control each right-hand side term separately. For the first term, we have sup

Q 2N 0 C d r (0) ω ε 2 ,r ω ε 2 ,r (0) ω ε 2 ,r (x) ∀ x ∈ Q L , so that by Theorem 18 ˆQ2N 0 C d r (0) ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx ω ε 2 ,r (0) ˆQL B (x) |g| 2 m 2 dx ˆQL ω ε 2 ,r (x) B (x) |g| 2 m 2 dx. (3.3.43) For all N 0 ≤ i ≤ N -1, we combine the bound ω ε 2 ,r | Q 2 i+1 C d r (0) \Q 2 i C d r (0) 2 ε 2 (i-N 0 ) with (3.3.38) to the effect that (using that 2ε ≤ β) ˆQ2 i+1 C d r (0) \Q 2 i C d r (0) ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx 1 m 2 ε 2m (i-N 0 ) ˆQL \Q 2 i C d r (0) B (x) |∇u| 2 µ ξ m 2 dx 1 m (3.3.38),2ε≤δ 2 ε 2m (i-N 0 ) ˆQL \Q 2 i-2 C d r (0) B (x) |g| 2 m 2 dx 1 m +2 ε 2m (i-N 0 ) ˆQ2 i C d r (0) ( |x|+r (0) 2 i C d r (0) ) εm 2 B (x) |g| 2 m 2 dx 1 m . (3.3.44)
For the first right-hand side term, we use that for all

x ∈ Q L \ Q 2 i-2 C d r (0) we have 2 ε 2 (i-N 0 ) 2 -ε 2 (i-N 0 ) ω ε,r (x), so that 2 ε 2m (i-N 0 ) ˆQL \Q 2 i-2 C d r (0) B (x) |g| 2 m 2 dx 1 m 2 -ε 2m (i-N 0 ) ˆQL ω ε,r B (x) |g| 2 m 2 dx 1 m .
(3.3.45) For the second term, we rather use that for all x ∈ Q 2 i C d r (0) we have by definition of N 0 and since m ≥ 2

2 ε 2 (i-N 0 ) ( |x|+r (0) 2 i C d r (0) ) εm 2 2 ε 2 (i-N 0 ) ( |x|+r (0) 2 i C d r (0) ) ε 2 ε 2 (i-N 0 ) 2 -ε(i-N 0 ) ( |x|+r (0) r ) ε 2 -ε 2 (i-N 0 ) ω ε,r (x), so that 2 ε 2m (i-N 0 ) ˆQ2 i C d r (0) ( |x|+r (0) 2 i C d r (0) ) εm 2 B (x) |g| 2 m 2 dx 1 m 2 -ε 2m (i-N 0 ) ˆQL ω ε,r B (x) |g| 2 m 2 dx 1 m . (3.3.46) Summing (3.3.44)-(3.3.46) over i form N 0 to N -1 we then obtain N -1 i=N 0 ˆQ2 i+1 C d r (0) \Q 2 i C d r (0) ω ε 2 ,r (x) B (x) |∇u| 2 µ ξ m 2 dx ˆQL ω ε,r B (x)
|g| 2

Control of the Meyers minimal radius: sensitivity estimate and buckling

The main result of this section is the following control of the Meyers minimal radius.

Theorem 21. Under Hypothesis 3.1.2, for all ξ ∈ R d , there exist an exponent γ > 0 depending on d, λ, and p, and a constant c ξ depending additionally on |ξ| (and all independent of L ≥ 1) such that

E L exp(c ξ r γ ,ξ,L ) ≤ 2. (3.3.48)
Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

The proof of Theorem 21 relies on the combination of the following sensitivity estimate (based on the quenched weighted Meyers estimate of Theorem 19) with the Caccioppoli inequality via a buckling argument.

Proposition 5. Under Hypothesis 3.1.2, for all ξ ∈ R d , denote by m > 2, δ > 0, and β > 0 the Meyers and nonlinear and linear hole-filling exponents, respectively (cf. Lemma 15 and Theorem 19). Then, for all r ≥ 1 and 0 < τ < 1, the random variable F := ffl Br ∇φ ξ satisfies

E L |F| 2q 1 q |ξ| qr -d E L r d-δ 1-τ q ,ξ,L 1-τ 2 q (3.3.49)
for all q ≥ 1 + d+1 ε , where

ε := ( β 2 ) ∧ ( (d+1)( m-2) 2 ) ∧ ( τ (d-δ) 4(1-τ ) ). (3.3.50)
We start with the proof of Theorem 21, and then turn to the proof of Proposition 5.

Proof of Theorem 21. We use the short-hand notation r := r ,ξ,L (0, c 1 ) (cf. (3.3.2) and Lemma 14). We split the proof into two steps. In the first step, we control the probability of the level set {r = R} for all dyadic R ∈ [1, L] using averages of ∇φ ξ , which we combine with Proposition 5 and the bound r ≤ r to buckle on moments of r , and therefore on r in the second step.

Step 1. We claim that there exist θ ∈ (0, 1) and c > 0, depending on p, d and λ such that for all dyadic R ∈ [1, L] and all exponents q ≥ 1,

P L [r = R] ≤ c q (1 + |ξ| p ) -q E L B θR ∇φ ξ pq . (3.3.51)
Assume that r = R. By the definition (3.3.2) of r , we then have

c 2 (1 + |ξ| p ) ≥ B 2R |∇φ ξ | p , (3.3.52)
B R 2 |∇φ ξ | p > c 2 (1 + |ξ| p ). (3.3.53)
By the Caccioppoli inequality (3.7.2), (3.3.53) turns into

inf η∈R B R 1 R 2 |φ ξ (x) -η| 2 + 1 R p |φ ξ (x) -η| p 1 + |ξ| p , (3.3.54)
which we shall use in the stronger form Let θ ∈ (0, 1) (the value of which we shall choose below), and set c R := ffl B R ffl B θR (x) φ ξ (y)dy dx. By the triangle inequality and Poincaré's inequality in L p (B R ), we obtain Using now (3.3.52), we may absorb the first right-hand side term into the left-hand side for θ small enough (independent of R), and therefore conclude that for some c > 0 (depending only on d, p, λ)

inf η∈R B R 1 R p |φ ξ (x) -η| p 1 + |ξ| p . ( 3
inf η∈R B R 1 R p |φ ξ -η| p B R 1 R p φ ξ (x) - B θR (x) φ ξ p dx + B R 1 R p B θR (x) φ ξ -c R p dx θ p B 2R |∇φ ξ (x)| p dx + B R B θR (x) ∇φ ξ p dx. ( 3
{r = R} ⊂ B R B θR (x) ∇φ ξ p dx ≥ 1 c (1 + |ξ| p ) ,
which yields (3.3.51) by Markov's inequality and the stationarity of ∇φ ξ .

Step 2. Buckling argument.

Fix

τ := 1 -d-δ d-δ 2 = δ 2d-δ > 0, to the effect that d -δ 1 -τ = d - δ 2 , 1 - τ 2 = 1 - δ 2(2d -δ) , ε := ( (d+1)( m-2) 2 ) ∧ ( δ 8 ).
For all dyadic 1 ≤ R ≤ L, by (3.3.51) and by Proposition 5 with this choice of τ and r = θR, we obtain for all q with q p 2 ≥ 1 + d+1 ε ,

P L [r = R] ≤ c q ξ q p 2 q R -d p 2 q E L r (d-δ 2 ) p 2 q 1- δ 2(2d-δ) (3.3.3) ≤ c q ξ q p 2 q R -d p 2 q E L r (d-δ 2 ) p 2 q 1- δ 2(2d-δ) .
(3.3.58) Therefore, using a dyadic decomposition (the sum is actually finite since r ≤ L), we deduce that (up to changing the value of c ξ )

E L r (d-δ 2 ) p 2 q ≤ 1 + +∞ n=1 (2 n ) (d-δ 2 ) p 2 q P[r = 2 n ] (3.3.58) ≤ 1 + c q ξ q q p 2 E L r (d-δ 2 ) p 2 q 1- δ 2(2d-δ) +∞ n=1 2 (d-δ 2 )q p 2 n 2 -dq p 2 n ≤ 1 + c q ξ q q p 2 E L r (d-δ 2 ) p 2 q 1- δ 2(2d-δ) .
Since both terms of this inequality are finite, this gives by Young's inequality provided

q p 2 ≥ 1 + d+1 ε E L r (d-δ 2 ) p 2 q 1 q c ξ q p 2d-δ δ ,
from which the stretched exponential moment bound (3.3.48) follows with γ := δ 8 (cf. Lemma 31), which is not expected to be sharp.

We conclude this section with the proof of Proposition 5.

Proof of Proposition 5. We split the proof into three steps. In the first step, we compute the functional derivative of F, in the sense of (3.7.43), and apply the logarithmic-Sobolev inequality in the second step to control moments of F. In the third step, we then control these moments by suitable moments of r using the quenched weighted Meyers estimate in the large of Theorem 19.

Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

Step 1. Sensitivity calculus. In this step, we take a slightly more general version of F (this will be further used in the proof of Theorem 15), which we define, for some given g ∈ L 2 (Q L ) d (extended by periodicity on R d ), by

F := ˆQL ∇φ ξ • g.
We then argue that for all x ∈ Q L ,

∂ x F = ˆB(x) |a • (ξ + ∇φ ξ ) ⊗ ∇u|, (3.3.59)
with the short-hand notation a • (ζ) := (1 + |ζ| p-2 )ζ and where u is the unique weak Q L -periodic solution (with zero average) of

-∇ • a * ξ ∇u = ∇ • g, (3.3.60)
we recall that a * ξ is bounded from above and below, since a is assumed to be smooth, and satisfies (3.3.6), cf. Lemma 25. Let denote by h a sequence that goes to zero and by δA a coefficient field supported in B(x) (and extended by Q L -periodicity) such that δA L ∞ (R d ) ≤ 1. We let h be small enough so that A + hδA is uniformly elliptic, and define

δ h F := F(A + hδA) -F(A) h , δ h φ ξ := φ ξ (A + hδA) -φ ξ (A) h , (3.3.61)
a h ξ := ˆ1 0 Da(•, ξ + t∇φ ξ (A + hδA) + (1 -t)∇φ ξ (A))dt. (3.3.62)
By the definition of F, we have δ h F = ´QL ∇δ h φ ξ • g, and we need to characterize δ h φ ξ . By the defining equation (3.1.7), we obtain

-∇ • a(•, ξ + ∇φ ξ + h∇δ h φ) -a(•, ξ + ∇φ ξ ) = h∇ • δAa • (ξ + ∇φ ξ (A + hδA)), (3.3.63)
which we rewrite, by the fundamental theorem of calculus and the definition of a h ξ , as

-∇ • a h ξ ∇δ h φ ξ = ∇ • δAa • (ξ + ∇φ ξ (A + hδA)). (3.3.64)
Assume that δ h φ ξ converges weakly in

H 1 per (Q L ) to the solution δφ ξ ∈ H 1 per (Q L ) of -∇ • a ξ ∇δφ ξ = ∇ • δAa • (ξ + ∇φ ξ ). ( 3
ˆQL |∇φ ξ (A + hδA) -∇φ ξ (A)| 2 + |∇φ ξ (A + hδA) -∇φ ξ (A)| p h 2 , so that ∇φ ξ (A + hδA) → ∇φ ξ (A) in L p (Q L ), and therefore φ ξ (A + hδA) → φ ξ (A) in C 1,α (Q L ) by
Arzela-Ascoli's theorem as claimed.

Step 2. Application of the logarithmic-Sobolev inequality: For all q ≥ 1,

E L F 2q 1 q q(1 + |ξ| p )E L ˆQL r (x) d-δ ˆB(x) |∇u| 2 µ ξ dx q 1 q . ( 3
E L F 2q 1 q qE L ˆQL ˆB(x) |a • (ξ + ∇φ ξ )||∇u| 2 dx q 1 q . (3.3.67)
By Cauchy-Schwarz' inequality, the definition (3.3.7) of µ ξ , and (3.3.5), we have for all

x ∈ Q L ˆB(x) |a • (ξ + ∇φ ξ )||∇u| 2 ˆB(x) |ξ + ∇φ ξ | 2 (1 + |ξ + ∇φ ξ | p-2 ) ˆB(x) |∇u| 2 µ ξ (3.3.5) (1 + |ξ| p )r (x) d-δ ˆB(x) |∇u| 2 µ ξ .
The claim (3.3.66) then follows in combination with (3.3.67).

Step 3. Proof of (3.3.49). For 0 < τ < 1 given, we define ε as in (3.

|∇u| 2 µ ξ dx ˆQL r (x) d-δ B (x)
|∇u| 2 µ ξ dx.

Inserting the weight (1

+ |x| r ) 2ε m (1 + |x| r ) -2ε
m , and using Hölder's inequality in space with exponents ( m 2 , m m-2 ) followed by Hölder's inequality in probability with exponents

( 1 1-τ , 1 τ ), (3.3.66) turns into 1 q(1 + |ξ| p ) E L F 2q 1 q (3.3.68) E L ˆQL (1 + |x| r ) -d-1 r (x) m m-2 (d-δ) dx q m-2 m ˆQL (1 + |x| r ) ε B (x) |∇u| 2 µ ξ q m 2 dx 2 m 1 q ≤ E L ˆQL (1 + |x| r ) -d-1 r (x) m m-2 (d-δ) dx q m-2 m(1-τ ) 1-τ q E L ˆQL (1 + |x| r ) ε B (x) |∇u| 2 µ ξ m 2 dx q 2 mτ τ q .
By the change of variables x r

x, Jensen's inequality in space provided q ≥ m m-2 = 1 + d+1 ε , and the stationarity of r , we control the first right-hand side term of (3.3.68) by

E L ˆQL (1 + |x| r ) -d-1 r (x) m m-2 (d-δ) dx q m-2 m(1-τ ) 1-τ q r d m-2 m E L r q d-δ 1-τ 1-τ q . (3.3.69)
For the second right-hand side term of (3.3.68), we appeal to the quenched weighted Meyers estimate (3.3.23), which we may apply to equation (3.3.60) (rewriting the right-hand side as

1 √ µ ξ g √ µ ξ )
with weight ω ε,r since ε ≤ β 2 . By stationarity of r , this yields

E L ˆQL (1 + |x| r ) ε B (x) |∇u| 2 µ ξ m 2 dx 2 mτ q τ q |ξ| E L ˆQL ω 2ε,r (x) B (x) |g| 2 1 µ ξ m 2 dx 2 mτ q τ q r -2d+ 2 m d E L r 4ε mτ q τ q , (3.3.70)
where we used that g = |B r | -1 1 Br , that µ ξ ≥ 1, and (3.7.65). By (3.3.50) and our choice m = 2+ 2ε d+1 ,

4ε mτ ≤ d-δ 1-τ , to the effect that E L r 4ε mτ q τ q ≤ E L r q d-δ 1-τ 4ε(1-τ )
qm(d-δ) by Hölder's inequality. Using (3.3.50) again, this time in form of 4ε(1-τ ) m(d-δ) ≤ τ 2 , and the lower bound r ≥ 1, (3.3.70) turns into

E L ˆQL (1 + |x| r ) ε B (x) |∇u| 2 µ ξ m 2 dx 2 mτ q τ q |ξ| r -2d+ 2 m d E L r q d-δ 1-τ τ 2q . ( 3

Annealed Meyers' estimate

The annealed Meyers (or perturbative Calderón-Zygmund) estimates recently introduced by Duerinckx and Otto in [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF] (see also [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]) constitute a very versatile upgrade of their quenched counterpart in stochastic homogenization. In the present setting the annealed Meyers estimates take the following form. > 0 (where m is the Meyers exponent of Theorem 18), for all Q L -periodic random fields g and u related via (3.3.9), we have for all exponents 2 -κ ≤ q, m ≤ 2 + κ and 0 < δ ≤ 1 2 , ˆQL

E L B(x) |∇u| 2 µ ξ q 2 m q dx |ξ| δ -1 4 | log δ| 1 2 ˆQL E L B(x) |g| 2 q(1+δ) 2 m q(1+δ) dx. (3.3.72)
The same result holds with a ξ replaced by a * ξ (the pointwise transpose field). The proof is based on the quenched Meyers estimate in the large of Theorem 18, on the moment bounds of Theorem 21 on the Meyers minimal radius (which allows us to use duality at the price of a loss of stochastic integrability), real interpolation, and the following refined dual version of the Calderón-Zygmund lemma due to Shen [147, Theorem 3.2], based on ideas by Caffarelli and Peral [START_REF] Caffarelli | On W 1,p estimates for elliptic equations in divergence form[END_REF]. Lemma 19 ([40,[START_REF] Shen | The l p boundary value problems on lipschitz domains[END_REF]). Given 1 ≤ q < m ≤ ∞, let F, G ∈ L q ∩ L m (Q L ) be nonnegative Q L -periodic functions and let C 0 > 0. Assume that for all balls D (of radius L) there exist measurable functions F D,1 and F D,2 such that F ≤ F D,1 + F D,2 and F D,2 ≤ F + F D,1 on D, and such that

D F q D,1 1 q ≤ C 0 C 0 D G q 1 q , 1 C 0 D F m D,2 1 m ≤ C 0 D F q D,2 1 q .
Then, for all q < s < m, ˆQL F s

1 s C 0 ,q,s,m ˆQL G s 1 s .
Before we prove Theorem 22, let us note that Theorem 21 allows one to pass from averages on B (x) to averages on B(x) using [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]Lemma 6.7] in the (slightly more general) form of Lemma 20. Let r be a stationary random field satisfying E L exp(cr α ) ≤ 2 for some α > 0 and c 1.

Set B (x) := B r (x) (x) for all x ∈ Q L . For all f ∈ C ∞ per (Q L ; L ∞ (Ω)) and 1 ≤ q 1 ≤ q 2 < ∞, we have (i) for all r > q 1 , ˆQL E L B (x) |f | 2 q 1 2 q 2 q 1 dx 1 q 2 ( 1 q 1 -1 r ) -( 1 q 1 -1 2 ) + ζ( 1 q 1 -1 r ) 1 q 1 -1 q 2 ˆQL E L B(x) |f | 2 r 2 q 2 r 1 q 2 ;
(ii) for all r < q 1 , ˆQL

E L B (x) |f | 2 q 1 2 q 2 q 1 dx 1 q 2 ( 1 r -1 q 1 ) ( 1 2 -1 q 2 ) + ζ( 1 r -1 q 1 ) -( 1 q 1 -1 q 2 ) ˆQL E L B(x) |f | 2 r 2 q 2 r 1 q 2 ;
where we have set ζ(t) := log(2 + 1 t ), and the multiplicative constants depend on q 1 , q 2 , α. The proof of this result is identical to that of [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]Lemma 6.7], noting that the assumption

E L exp( 1 C r d ) ≤ 2 can be weakened to E L exp( 1 C r α ) ≤ 2
for any α > 0 at the price of adding a dependence on α in the multiplicative factors in the estimates, and R d can be replaced by Q L .

Proof of Theorem 22. We split the proof into three steps. In the first step, we upgrade Theorem 18 by adding expectations using Lemma 19 in a suitable way. At the price of a loss of stochastic integrability we then remove the local averages at scale r in Step 2 by using Lemma 20. The formulation with local averages at unit scale allows us to conclude using a standard duality argument, and real interpolation.

Step 1. Proof that for all 2 ≤ q < m < m, we have ˆQL

E L B (x) |∇u| 2 µ ξ q 2 m q dx |ξ| ˆQL E L B 5 (x) |g| 2 q 2 m q dx (3.3.73)
with the short hand notation

B 5 (x) := B 5r (x) (x). Let 2 ≤ q 1 ≤ m 1 ≤ m. Let D be a ball centered at x ∈ Q L and of radius 0 < r D L, we define D := B r D ∨(2r (x)) (x)
, and let N be the smallest integer so that D ⊂ Q N L . We then decompose u

as u = u D,1 + u D,2 , where u D,1 is the Q N L -periodic solution of -∇ • a ξ ∇u D,1 = ∇ • g √ µ ξ 1 D . Note that u D,2 is a ξ -harmonic on D . We start with the control of u D,1 and claim that ˆD E L B (y) |∇u D,1 | 2 µ ξ q 1 2 dy E L ˆ8D B 5 (y) |g| 2 q 1 2 dy . (3.3.74)
Assume first that r D ≥ 2r (x), so that D = D. By taking the expectation in Theorem 18, we have

ˆD E L B (y) |∇u D,1 | 2 µ ξ q 1 2 dy ≤ E L ˆQNL B (y) |∇u D,1 | 2 µ ξ q 1 2 dy (3.3.11) E L ˆQNL B (y) |g| 2 1 D q 1 2 dy .
|∇u D,1 | 2 µ ξ q 1 2 dy E L ˆ2D B (y) |g| 2 q 1 2 dy . If r D ≤ 2r (x), then sup D r inf D r , D = B 2r (x) (x) =: B 2 (x)
, and a plain energy estimate yields

ˆD E L B (y) |∇u D,1 | 2 µ ξ q 1 2 dy E L |D|r (x) -d q 1 2 ˆQNL |∇u D,1 | 2 µ ξ q 1 2 E L |D|r (x) -d q 1 2 ˆD |g| 2 q 1 2 E L |D| B 2 (x) |g| 2 q 1 2 ,
and it remains to turn the right-hand side into an integral over D. For all y ∈ D, we have r (y) ≥ r (x) -1 16 r D ≥ 7 8 r (x), and therefore for all z ∈ B 2 (x), |z -y| ≤ |z -x| + |x -y| ≤ 4r (x) ≤ 5r (y), to the effect that B 2 (x) ⊂ B 5 (y). Recalling that sup D r inf D r , this implies the following stronger form of (3.3.74)

ˆD E L B (y) |∇u D,1 | 2 µ ξ q 1 2 dy E L ˆD B 5 (y) |g| 2 q 1
2 dy .

We now turn to the control of u D,2 , and claim that

1 8 D E L B (y) |∇u D,2 | 2 µ ξ q 1 2 m 1 q 1 dy 1 m 1 D E L B (y) |∇u D,2 | 2 µ ξ q 1 2 dy 1 q 1 . (3.3.75)
The starting point is the Minkowski inequality: Since m 1 q 1 ≥ 1,

1 8 D E L B (y) |∇u D,2 | 2 µ ξ q 1 2 m 1 q 1 dy 1 m 1 ≤ E L 1 8 D B (y) |∇u D,2 | 2 µ ξ m 1 2 dy q 1 m 1 1 q 1 .
(3.3.76) We then appeal to the local Meyers estimate (3.3.10) to bound the right-hand side

1 8 D B (y) |∇u D,2 | 2 µ ξ m 1 2 dy |ξ| 1 4 D B (y) |∇u D,2 | 2 µ ξ dy m 1 2 + 1 4 D B (y) |g| 2 (1-1 D ) m 1 2 dy.
Since for all y ∈ 1 4 D, one has r (y) ≤ r (x) + 1 16 1 4 r D ≤ 3 4 (r D ∨ (2r (x))), B (y) ⊂ D and the second right hand side term vanishes identically. Combined with (3.3.76) and Jensen's inequality in space (using that q 1 2 ≥ 1), this entails We are in the position to conclude. Setting F :

1 8 D E L B (y) |∇u D,2 | 2 µ ξ q 1 2 m 1 q 1 dy 1 m 1 ≤ E L 1 4 D B (y) |∇u D,2 | 2 µ ξ dy q 1 2 1 q 1 ≤ E L 1 4 D B (y) |∇u D,2 | 2 µ ξ q 1 2 dy 1 q 1 , 3.3.
x → E L ffl B (x) |∇u| 2 µ ξ q 1 2 1 q 1 , G : x → E L ffl B 5 (x) |g| 2 q 1 2 1 q 1 , F D,1 : x → E L ffl B (x) |∇u D,1 | 2 µ ξ q 1 2 1 q 1 , and F D,2 : x → E L ffl B (x) |∇u D,2 | 2 µ ξ q 1 2 1 q 1 ,
the assumptions of Lemma 19, and the claimed estimate (3.3.73) follows.

Step 2. Reformulation of (3.3.73). Since both r and 5r satisfy stretched exponential moment bounds, Lemma 20 allows us to reformulate (3.3.73) as: For all 2 ≤ q < m < m and 0 < r

≤ 1 2 , ˆQL E L B(x) |∇u| 2 µ ξ q 2 m q dx |ξ|,r r -m-2 2m | log r| 2(m-q) qm ˆQL E L B(x) |g| 2 q+r 2 m q+r dx. (3.3.77)
Step 3. Proof of (3.3.72). First, we show that for all m < m < q ≤ 2 and 0 < r 1, ˆQL

E L B(x) |∇u| 2 µ ξ q 2 m q dx |ξ| r -2-m 2m | log r| 2(q-m) qm ˆQL E L B(x) |g| 2 q-r 2 m q-r dx. (3.3.78)
Indeed, by duality we have ˆQL

E L B(x) |∇u| 2 µ ξ q 2 m q dx 1 m = sup h E L ˆQL ∇u • h √ µ ξ ,
where the supremum runs over maps

h ∈ C ∞ per (Q L , L ∞ (dP L )) d such that ´QL E L ffl B(x) |h| 2 q 2 m q dx = 1. For such h, denote by v h the unique Q L -periodic solution of -∇ • a * ξ ∇v h = ∇ • (h √ µ ξ ).
Testing this equation with u and the defining equation (3.3.9) for u by v h , we obtain (using periodicity in the last equality)

ˆQL ∇u • h √ µ ξ = ˆQL ∇v h • g √ µ ξ = ˆQL B(x) ∇v h • g √ µ ξ dx.
By Cauchy-Schwarz' inequality on B(x), followed by Hölder's inequality with exponents (q-r, q-r q-r-1 ) on Q L and with exponent (m, m ) in probability, this yields

E L ˆQL ∇v h • g √ µ ξ ≤ ˆQL E L B(x) |g| 2 q-r 2 m q-r dx 1 m ˆQL E L B(x) |∇v h | 2 µ ξ (q-r) 2 m (q-r) dx 1 m .
Since (q -r) -q = r (q-1)(q-1-r) , we may apply (3.3.77) to ∇v h to the effect that

E L ˆQL ∇v h • g √ µ ξ r -m -2 2m | log r| 2(m -q ) q m ˆQL E L B(x) |g| 2 q-r 2 m q-r dx 1 m ˆQL E L B(x) |h| 2 q 2 m q dx 1 m ,
from which (3.3.78) follows by the arbitrariness of h and the identities m -2 2m = 2-m 2m and 2(m -q ) q m = 2(q-m) qm . systems

Replacing r by qr in (3.3.77) and (3.3.78), and using the bounds m-2 2m ≤ 1 4 and 2(m-q) qm ≤ 1 2 for 2 ≤ q ≤ m ≤ 3 and 2-m 2m ≤ 1 4 and 2(q-m) qm ≤ 1 2 for 3 2 ≤ m ≤ q ≤ 2, we have thus proved that (3.3.72) holds for all 2 ≤ q < m < m ∧ 3 and for all m ∨ 3 2 < m < q ≤ 2. By choosing κ = ( m-2)∧1 8 , the validity of (3.3.72) in the full range of exponents 2 -κ ≤ m, q ≤ 2 + κ then follows by real interpolation.

We conclude this subsection by the annealed version of the maximal regularity for the Laplacian.

Theorem 23. Let L ≥ 1. For all Q L -periodic random fields g and u related via

-u = ∇ • g,
we have for all exponents 1 < m, q < ∞, ˆQL

E L B(x) |∇u| 2 q 2 m q dx 1 m m,q ˆQL E L B(x) |g| 2 q 2 m q dx 1 m . (3.3.79)
A proof of this result can be found in [101, Section 7.1]. A simpler argument (based on CZ estimates for Hilbert-valued operators and interpolation) would show that the multiplicative constant in (3.3.79) is of the order m + m + q + q (this finer result will not be used here).

Control of correctors: Proof of Theorem 15

The proof relies on the following upgrade of Proposition 5 based on Theorem 21 and on Theorem 22.

Corollary 24. Under Hypothesis 3.1.2, there exists γ > 0 such that for all ξ ∈ R d , L ≥ 1, and all g ∈ L 2 (R d ) compactly supported in Q L , the random field F := ´QL g(∇φ ξ , ∇σ ξ ) satisfies for all q ≥ 1

E L |F| 2q 1 q |ξ| q γ ˆQL |g| 2 . (3.4.1)
For future reference, we state the following consequence of local regularity and of the hole-filling estimate.

Lemma 21. Under Hypothesis 3.1.2, with 0 < δ ≤ d the nonlinear hole-filling exponent of Lemma 15, we have for all ξ ∈ R d and

x ∈ R d ξ + ∇φ ξ C α (B(x)) (1 + |ξ|)(r (x)) d-δ p . (3.4.2)
Proof. By the deterministic regularity theory of Lemma 25 applied to the equation (3.1.7) combined with the estimate (3.3.5), we indeed have

ξ + ∇φ ξ C α (B(x)) A C 0,α (R d ) B 2 (x) |ξ + ∇φ ξ | p 1 p (3.3.5) ≤ A C 0,α (R d ) (1 + |ξ|)(r (x)) d-δ p .
Before we turn to the proof of Corollary 24, let us quickly argue that it yields Theorem 15. q ´QL |ξ + ∇φ ξ | q 1 q , so that assumption (3.7.49) for ∇σ ξ follows from taking the expectation of the q-th power of this estimate and using the stationarity of the extended corrector gradient together with the moment bound on ∇φ ξ . By (3.7.50), we can then pass to the limit L ↑ +∞ in the moment bounds on the extended corrector gradient for the periodized ensemble, and obtain (3.2.6). Likewise, the claimed estimate (3.2.7) follows from Corollary 24 for g compactly supported by passing to the limit L ↑ ∞ using (3.7.50). The result for general g ∈ L 2 (R d ) is then obtained by approximation.

The control (3.2.8) of the growth of the extended corrector is a direct consequence of (3.2.7) by "integration" (see for instance [54, Proof of Theorem 4.2, Step 3] -the argument is also displayed in the proof of Corollary 16).

It remains to prove Corollary 24.

Proof of Corollary 24. We split the proof into two steps, first treat averages of ∇φ ξ and then turn to averages of ∇σ ξ .

Step 1. Averages of ∇φ ξ . In this step we set

F := ´QL g • ∇φ ξ for some g ∈ L 2 (R d ) d compactly supported in Q L .
The starting point is the estimate (3.3.66) in the proof of Proposition 5, which takes the form for all q ≥ 1 of

E L F 2q 1 q q(1 + |ξ| p )E L ˆQL r (x) d-δ ˆB(x) |∇u| 2 µ ξ dx q 1 q ,
where u is the unique weak Q L -periodic solution (with zero average) of (3.3.60), that is, -∇•a * ξ ∇u = ∇ • g. By duality, we may reformulate the right-hand side as

E L ˆQL r (x) d-δ ˆB(x) |∇u| 2 µ ξ dx q 1 q = sup E L [|X| 2q ]=1 E L ˆQL r (x) d-δ ˆB(x) |∇Xu| 2 µ ξ dx ,
where the supremum runs over random variables X ∈ L 2q (dP L ) which are independent of the space variable. Let 0 < η < 1 be some exponent (to be fixed later) small enough so that q 1+η > 1. We then appeal to Hölder's inequality with exponents ( q q -1-η , q 1+η ) and to the stationarity of r to the effect that

E L ˆQL r (x) d-δ ˆB(x) |∇Xu| 2 µ ξ dx ≤ E r q q -1-η (d-δ) q -1-η q ˆQL E L B(x) |∇Xu| 2 µ ξ q 1+η 1+η q dx.
Provided 2q ≤ 2 + κ, we may appeal to Theorem 22 on the second right-hand side factor, which yields (recall that X does not depend on the space variable, that E L [|X| 2q ] = 1 and that µ ξ ≥ 1)

ˆQL E L B(x) |∇Xu| 2 µ ξ q 1+η 1+η q dx η -1 4 | log(η)| 1 2 ˆQL E L |X| 2q B(x) |g| 2 1 µ ξ q 1 q dx η -1 4 | log(η)| 1 2 ˆ|g| 2 .
184 Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

The choice η = 1 2 (q -1) = 1 2(q-1) is legitimate provided q 1, in which case the above combined with the moment bound on r of Theorem 21 yields

E L F 2q 1 q q ν (1 + |ξ| p ) ˆ|g| 2 .
for some exponent ν > 0 independent of q. This entails (3.4.1) for ∇φ ξ for a suitable exponent γ > 0 (depending only on ν).

Step 2. Averages of ∇σ ξ . Fix 1 ≤ i, j ≤ d. We proceed as for ∇φ ξ : We first derive a representation formula for the sensitivity of F := ´QL g•∇σ ξ,ij with respect to changes of the coefficient A, and then use the annealed estimates of Theorems 22 and 23, and the moment bounds on r to conclude.

-σ ξ,ij = ∂ i (a(•, ξ + ∇φ ξ ) • e j ) -∂ j (a(•, ξ + ∇φ ξ ) • e i ).
As in Step 1 of the proof of Proposition 5, we proceed by duality. This time we introduce two auxiliary functions u 1 and u 2 as Q L -periodic solutions of

-u 1 = ∇ • g, -∇ • a * ξ ∇u 2 = ∇ • a * ξ (∂ i u 1 e j -∂ j u 1 e i ),
and claim that

δ x F = ˆB(x) |a • (ξ + ∇φ ξ ) ⊗ (∇u 2 + ∂ i u 1 e j -∂ j u 1 e i )|. (3.4.3)
Let us quickly argue in favor of (3.4.3). With the notation of Step 1 of the proof of Proposition 5, and δA an increment of A localized in B(x), we have by the defining equations for σ ξ,ij and u 1

δ h F := F(A + hδA) -F(A) h = ˆ(∂ i u 1 e j -∂ j u 1 e i ) • δ h a(ξ + ∇φ ξ ) ,
where

δ h a(ξ + ∇φ ξ ) = (A + hδA)a • (ξ + ∇φ ξ (A + hδA)) -Aa • (ξ + ∇φ ξ ) h = δAa • (ξ + ∇φ ξ (A + hδA)) + a h ξ ∇δ h φ ξ .
Passing to the limit h ↓ 0, and testing the equation for u 2 with δφ ξ and equation (3.3.65) with u 2 , we obtain

δF = lim h↓0 δ h F = ˆ(∂ i u 1 e j -∂ j u 1 e i ) • δAa • (ξ + ∇φ ξ ) + a ξ ∇δφ ξ = ˆ(∇u 2 + ∂ i u 1 e j -∂ j u 1 e i ) • δAa • (ξ + ∇φ ξ ),
and the claim follows by taking the supremum over δA.

Substep 2.2. Proof of (3.4.1). Combining (3.4.3) with the logarithmic-Sobolev inequality, we obtain for all q ≥ 1

E L |F| 2q 1 q qE L ˆQL B(x) |a • (ξ + ∇φ ξ )||∇u 2 + ∂ i u 1 e j -∂ j u 1 e i | 2 dx q 1 q .
3.4. Control of correctors: Proof of Theorem 15 185

We treat differently the terms involving u 1 and u 2 . For u 2 we proceed as in Step 3 of the proof of Proposition 5 (using the definition (3.3.7) of µ ξ and (3.3.5)), whereas for u 1 we directly use (3.4.2). This yields

E L |F| 2q 1 q |ξ| qE L ˆQL r (x) d-δ ˆB(x) |∇u 2 | 2 µ ξ dx q 1 q + qE L ˆQL r (x) 2(p-1) p (d-δ) ˆB(x) |∇u 1 | 2 dx q 1 q .
As in Step 1, this entails

E L |F| 2q 1 q |ξ| q sup E L [|X| 2q ]=1 E L ˆQL r (x) d-δ ˆB(x) |∇Xu 2 | 2 µ ξ dx + q sup E L [|X| 2q ]=1 E L ˆQL r (x) 2(p-1) p (d-δ) ˆB(x) |∇Xu 1 | 2 dx .
For the second right-hand side term, we proceed as in Step 1 (using Theorem 23 in place of Theorem 22), and it remains to treat the first right-hand side term. We then use Hölder's inequality with exponents

( q q -(1+η) 2 , q (1+η
) 2 ) for some 0 < η < 1 (so that q > (1 + η) 2 ) to be chosen below to the effect that

E L ˆQL r (x) d-δ ˆB(x) |∇Xu 2 | 2 µ ξ dx ≤ E r q q -(1+η) 2 (d-δ) q -(1+η) 2 q ˆQL E L B(x) |∇Xu 2 | 2 µ ξ q (1+η) 2 (1+η) 2 q dx.
We then appeal to the annealed Meyers estimate of Theorem 22 under the condition that 2 ≤ 2q (1+η) 2 ≤ 2 + κ, and obtain ˆQL

E L B(x) |∇Xu 2 | 2 µ ξ q (1+η) 2 (1+η) 2 q dx η 1 4 | log η| 1 2 ˆQL E L B(x) |µ ξ ∇Xu 1 | 2 1 µ ξ q 1+η 1+η q dx
since under the assumption 0 < η < 1 2 , we have

(1 + η) 2 -1 η. Bounding µ ξ by r p-2 p (d-δ)
(cf. Lemma 15) and using Hölder's inequality with exponents ( 1+η η , 1 + η), the integral in the righthand side is controlled by

ˆQL E L B(x) |µ ξ ∇Xu 1 | 2 1 µ ξ q 1+η 1+η q E L r q η p-2 p (d-δ) η q ˆQL E L B(x) |∇Xu 1 | 2 q 1 q .
We finally estimate the integral term by Theorem 23, which yields (since there is no loss in the stochastic exponent, g is deterministic, and

1 ≤ q ≤ 2) ˆQL E L B(x) |∇Xu 1 | 2 q 1 q E L |X| 2q ˆQL |g| 2 = ˆQL |g| 2 .
The conclusion follows by choosing η = 1 4 (q -1) and q 1, and using the moment bound on r of Theorem 21.

Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems 3.5 Control of corrector differences: Proof of Theorem 17

Reduction argument

As for nonlinear correctors, by Proposition 10 it is enough to prove estimates for L-periodic ensembles that are uniform with respect to L. We split the version of Corollary 24 for the linearized corrector into two statements: Proposition 6 below shows that averages of the gradient of the extended linearized corrector decay at the CLT scaling provided we have good control of moments of ∇ φξ,e , whereas Proposition 7 provides the latter.

Proposition 6. Under Hypothesis 3.1.2, for all ξ ∈ R d and all 0 < θ < 1 there exists γ > 0 (depending on |ξ| and θ) such that for all L ≥ 1, all g ∈ L 2 (R d ) compactly supported in Q L , and all unit vectors e ∈ R d , the random field F := ´QL g(∇ φξ,e , ∇σ ξ,e ) satisfies for all q ≥ 1 such that 2q ≤ 2 + κ (where κ > 0 is as in Theorem 22)

E L |F| 2q 1 q |ξ|,θ q γ E L sup B |∇ φξ,e + e| 2 µ ξ q(1+θ) 1 q(1+θ) ˆQL |g| 2 (3.5.1)
The proof of Proposition 6 relies on a sensitivity estimate by duality combined with the annealed Meyers estimate of Theorem 22.

Proposition 7 (Control of moments). Under Hypothesis 3.1.2, for all ξ ∈ R d , there exists γ > 0 (depending on |ξ|) such that for all L ≥ 1 and all unit vectors e ∈ R d we have

E L sup B |∇ φξ,e + e| 2 µ ξ q 1 q q γ . (3.5.2)
The proof of Proposition 7 is based on Proposition 6 and a buckling argument. Because the linearized corrector equation has unbounded coefficients, we cannot use the elegant approach of [START_REF] Otto | Introduction to stochastic homogenization[END_REF] (see also [54,Proposition 4.5]) to buckle on moments of ∇ φξ,e themselves. Instead, as we did for r ,ξ,L , we have to go through the super levelsets of some minimal radius controlling the growth of averages of |∇ φξ,e | 2 µ ξ .

Before we turn to the proofs, let us show how bounds on linearized correctors allow us to derive bounds on nonlinear corrector differences in form of Corollary 16.

Proof of Corollary 16. For simplicity, we only treat φ ξ .

Step 1. Statement for differences of corrector gradients. By (3.7.50) in Proposition 10 in form of (note the difference of expectations)

E |∇(φ ξ -φ ξ )| q 1 q = lim L↑+∞ E L |∇(φ ξ -φ ξ )| q 1 q ,
it suffices to prove the statement for the periodized ensemble. By Lemma 27, P L -almost surely, ξ → ∇φ ξ is differentiable and we have by the fundamental theorem of calculus for all e ∈ R d e • (∇φ ξ -∇φ ξ ) = ˆ1 0 ∇ φξ+t(ξ -ξ),e • (ξ -ξ)dt, (3.5.3) so that by taking the q-th moment and using Proposition 7, one obtains

E L |∇φ ξ -∇φ ξ | q 1 q ≤ |ξ -ξ | i ˆ1 0 E L |∇φ ξ+t(ξ -ξ),e i | q 1 q q γ |ξ -ξ |, (3.5.4)
3.5. Control of corrector differences: Proof of Theorem 17 187 which yields the claim by taking the limit L ↑ ∞.

Step 2. Statement for corrector differences. By (3.7.51), since ´B φ ξ = 0, for all x ∈ R d we have for all q ≥ 1

E ˆB(x) |φ ξ -φ ξ | 2 q 2 1 q = lim L↑∞ E L ˆB(x) φ ξ -φ ξ - B φ ξ -φ ξ 2 q 2 1
q .

(3.5.5)

To control the right-hand side, we shall bound moments of periodic random fields ζ by moments of averages of their gradients ∇ζ. Indeed, by Poincaré's inequality on B(x) for x ∈ Q L , we have for

c = ffl B ζ E L ˆB(x) (ζ -c) 2 q 2 1 q E L |∇ζ| q 1 q + E L B(x) ζ -c q 1
q , (3.5.6)

and it remains to estimate the second right-hand side term. For that purpose, we write

B(x) ζ - B ζ = ˆQL ∇ζ • ∇h x ,
where h x denotes the unique weak periodic solution in

Q L of -h x = 1 |B| (1 B(x) -1 B ).
We apply this to ζ = φ ξ -φ ξ and rewrite the gradient as e • ∇ζ = ´1 0 ∇φ ξ+t(ξ -ξ),e • (ξ -ξ)dt, to the effect that (with implicit sum on the repeated index i)

B(x) (φ ξ -φ ξ ) - B (φ ξ -φ ξ ) = (ξ -ξ) • ˆ1 0 ˆQL ∇φ ξ+t(ξ -ξ),e i ∇ i h x .
Using Propositions 6 and 7, this yields

E L B(x) (φ ξ -φ ξ ) - B (φ ξ -φ ξ ) q 1 q ≤ |ξ|,|ξ | |ξ -ξ |q γ ˆQL |∇h x | 2 1 2 .
A direct computation with Green's kernel gives ∇h x L 2 (Q L ) µ d (x), and thus

E L B(x) (φ ξ -φ ξ ) - B (φ ξ -φ ξ ) q 1 q ≤ |ξ|,|ξ | |ξ -ξ |q γ µ d (x).
Combined with (3.5.6), (3.5.5), and (3.5.4), this entails

E ´B(x) |φ ξ -φ ξ | 2 q 2 1 q
|ξ -ξ |q γ µ d (x), from which the claim follows using local regularity in form of Lemma 25 and (3.5.4) in the limit L ↑ +∞.

Step 3. Regularity of ξ → ā(ξ). The starting point is the definition ā(ξ) := E a(ξ + ∇φ ξ ) = E A(0)a • (ξ + ∇φ ξ (0)) and of its approximation by periodization āL (ξ) := E L A(0)a • (ξ + ∇φ ξ (0)) for all L ≥ 1. Since āL (ξ) → ā(ξ) as L ↑ +∞, it is enough to prove that Dā L is Lipschitz-continuous uniformly with respect to L and given for all ξ, e ∈ R d by Dā L (ξ)e := āL,ξ e = E L A(0)Da • (ξ + ∇φ ξ (0))(e + ∇ φξ,e (0)) .

The differentiability of ξ → āL (ξ) and the formula for Dā L (ξ) follow from (3.5.3), the continuity of ξ → ∇ φξ,e , and the moment bounds on ∇ φξ,e . It remains to argue that ξ → Dā L (ξ) is Lipschitzcontinuous. Since ξ → ∇φ ξ is continuously differentiable with stretched exponential moment bounds, systems it is enough to prove that ξ → ∇ φξ,e is itself Lipschitz-continuous in L 2 (dP). This is a direct consequence of the defining equation (3.2.9) in the form for all ξ, ξ ∈ R d of -∇ • Da(•, ξ + ∇φ ξ )∇( φξ,eφξ ,e ) = ∇ • (Da(•, ξ + ∇φ ξ ) -Da(•, ξ + ∇φ ξ ))(e + ∇ φξ ,e ) combined with the differentiability of ξ → ∇φ ξ , uniform moment bounds on ∇ φξ ,e and ∇φ ξ , and an energy estimate.

CLT-scaling: Proof of Proposition 6

In this paragraph, we fix e and ξ, and use the short-hand notation r for r ,ξ,L , φ for φ ξ , µ for µ ξ , φ for φξ,e , σ for σξ,e . We split the proof into three steps. In the first two steps, we derive sensitivity estimates for averages of ∇ φ and of ∇σ, respectively, and then conclude in the third step using Theorems 22 and 23.

Step 1. Sensivity formula for ∇ φ: The random variable F 1 := ´QL g • ∇ φ (where g abusively denotes ge for some unit vector e ∈ R d ) satisfies for all

x ∈ Q L δ x F 1 = ˆB(x) |Da • (ξ + ∇φ)(e + ∇ φ) ⊗ ∇u 1 + a • (ξ + ∇φ) ⊗ ∇u 2 |, (3.5.7)
where we recall that a

• : ξ ∈ R d → (1 + |ξ| p-2 )ξ,
δ h F 1 := F 1 (A + hδA) -F 1 (A) h = ˆQL g • ∇δ h φ, δ h φ := φ(A + hδA) -φ(A) h , b h ξ := A ˆ1 0 D 2 a • (ξ + t∇φ(A + hδA) + (1 -t)∇φ(A))dt,
and recall the notation (3.3.61) and (3.3.62). By the defining equation (3.2.9) for the linearized corrector, we obtain

-∇ • a ξ ∇δ h φ =∇ • δADa • (ξ + ∇φ(A + hδA))(e + ∇ φ(A + hδA)) + 1 h ∇ • A(Da • (ξ + ∇φ(A + hδA)) -Da • (ξ + ∇φ(A))(e + ∇ φ(A + hδA)),
which we rewrite, by the fundamental theorem of calculus and the definition of b h ξ , as

-∇ • a ξ ∇δ h φ = ∇ • δADa • (ξ + ∇φ(A + hδA))(e + ∇ φ(A + hδA)) + ∇ • b h ξ ∇δ h φ(e + ∇ φ(A + hδA)).
As in Step 1 of the proof of Proposition 5, we can pass to the limit as h ↓ 0 and obtain that The combination of (3.5.11) and (3.5.12) then entails the claim (3.5.7) by taking the supremum over δA.

δ h φ converges in C 1,α (Q L ) to the solution δ φ ∈ H 1 per (Q L ) of -∇ • a ξ ∇δ φ = ∇ • δADa • (ξ + ∇φ)(e + ∇ φ) + ∇ • b ξ ∇δφ(e + ∇ φ), (3.5
Step 2. Sensitivity formula for ∇σ ij (for i, j fixed): The random variable

F 2 := ´QL g • ∇σ ij satisfies for all x ∈ Q L δ x F 2 = ˆB(x) |Da • (ξ + ∇φ)(e + ∇ φ) ⊗ (∇w 1 + ∂ i ve j -∂ j ve i ) + a • (ξ + ∇φ) ⊗ ∇w 2 |, (3.5.13)
where the functions v, w 1 , w 2 ∈ H 1 per (Q L ) solve (with an implicit sum over the repeated index k)

-v = ∇ • g, (3.5.14)
-∇ • a * ξ ∇w 1 = ∇ • a * ξ (∂ i ve j -∂ j ve i ), (3.5.15)
-∇ • a * ξ ∇w 2 = ∂ k D 2 a(ξ + ∇φ)(e + ∇ φ)e k • (∇w 1 + ∂ i ve j -∂ j ve j ) . (3.5.16)
We only display the algebra of the argument (passing already to the limit h ↓ 0, which entails that δ = lim h↓0 δ h satisfies the Leibniz rule). Recall the defining equation for σij with the notation

a ξ = Da(ξ + ∇φ) -σij = ∂ i (a ξ (e + ∇ φ) • e j ) -∂ j (a ξ (e + ∇ φ) • e i ).
First, by (3.5.14),

δF 2 = ˆ(∂ i ve j -∂ j ve i ) • δ Da(ξ + ∇φ)(e + ∇ φ) .
Since δ satisfies the Leibniz rule, we have

δ Da(ξ + ∇φ)(e + ∇ φ) = δADa • (ξ + ∇φ)(e + ∇ φ) + D 2 a(ξ + ∇φ)∇δφ(e + ∇ φ) + a ξ ∇δ φ.
The first right-hand term directly gives the right-hand side contribution of (3.5.13) involving ∇v.

For the second term, we introduce the solutions w 2,1 and

w 2,2 of -∇•a * ξ ∇w 2,1 = ∂ k D 2 a(ξ +∇φ)(e+ ∇ φ)e k • (∂ i ve j -∂ j ve j ) and -∇ • a * ξ ∇w 2,2 = ∂ k D 2 a(ξ + ∇φ)(e + ∇ φ)e k
• ∇w 1 to the effect that w 2 = w 2,1 + w 2,2 . By using (3.3.65), we obatin

ˆ(∂ i ue j -∂ j ue i ) • D 2 a(ξ + ∇φ)∇δφ(e + ∇ φ) = ˆ∇w 2,1 • δAa • (ξ + ∇φ).
This yields part of the right-hand side contribution of (3.5.13) involving ∇w 2 . We conclude with the third term. Using first (3.5.15) we obtain ˆ(∂ i ue j -∂ j ue i ) • a ξ ∇δ φ = -ˆ∇w 1 • a ξ ∇δ φ, systems and therefore using (3.5.10)

ˆ(∂ i ue j -∂ j ue i ) • a ξ ∇δ φ = ˆ∇w 1 • δADa • (ξ + ∇φ)(e + ∇ φ) + D 2 a(ξ + ∇φ)∇δφ(e + ∇ φ) .
The first right-hand side term yields the right-hand side contribution of (3.5.13) involving ∇w 1 . For the second term, we use w 2,2 , and conclude using (3.3.65) that

ˆ∇w 1 • D 2 a(ξ + ∇φ)∇δφ(e + ∇ φ) = ˆ∇w 2,2 • δAa • (ξ + ∇φ).
This gives the second part of the right-hand side contribution of (3.5.13) involving ∇w 2 , recalling that ∇w 2 = ∇w 2,1 + ∇w 2,2 .

Step 3. Proof of (3.5.1).

From the logarithmic-Sobolev inequality, and Steps 1 and 2, we deduce by the triangle inequality that for all q ≥ 1,

E L |F| 2q 1 q qE L ˆQL ˆB(x) |Da • (ξ + ∇φ)||e + ∇ φ|(|∇u 1 | + |∇v| + |∇w 1 |) 2 dx q 1 q =: I 1 + qE L ˆQL ˆB(x) |a • (ξ + ∇φ)|(|∇u 2 | + |∇w 2 |) 2 dx q 1 q =: I 2 .
To control these terms we proceed as in the proof of Corollary 24: using duality in probability and Theorems 22 and 23. We treat the two right-hand sides separately. (In what follows, γ denotes finite positive exponents independent of q, the precise value of which we are not interested in.)

Substep 3.1. Proof of

I 1 q γ E L ˆB(x) |e + ∇ φ| 2 µ ξ q(1+θ) 1 q(1+θ) ˆQL |g| 2 .
(3.5.17)

The most technical term to treat is the one involving w 1 (which is defined by solving two equations successively, whereas u 1 and v are defined by solving one equation only). By Cauchy-Schwarz' inequality, and the definitions of a • and µ ξ ,

E L ˆQL ˆB(x) |Da • (ξ + ∇φ)||e + ∇ φ||∇w 1 | 2 dx q 1 q E L ˆQL ˆB(x) |e + ∇ φ| 2 µ ξ ˆB(x) |∇w 1 | 2 µ ξ dx q 1 q .
By duality (in probability), this entails where the supremum runs over random variables X (independent of the space variable) such that

E L ˆQL ˆB(x) |Da • (ξ + ∇φ)||e + ∇ φ||∇w 1 | 2 dx q 1 q sup X E L ˆQL ˆB(x) |e + ∇ φ| 2 µ ξ ˆB(x) |∇Xw 1 | 2 µ ξ dx ,
E |X| 2q = 1.
To obtain the claimed dependence on the moments of ´B(x) |e + ∇ φ| 2 µ ξ , we set

η • := θ (1+θ)(q-1)
, to the effect that q > 1 + η • and q q -(1+η•) = q(1 + θ), and use Hölder's inequality with exponents ( q q -(1+η•) , q 1+η• ), so that the above turns into

E L ˆQL ˆB(x) |Da • (ξ + ∇φ)||e + ∇ φ||∇w 1 | 2 dx q 1 q E L ˆB(x) |e + ∇ φ| 2 µ ξ q(1+θ) 1 q(1+θ) sup X ˆQL E L ˆB(x) |∇Xw 1 | 2 µ ξ q 1+η• 1+η• q dx.
For convenience, we rewrite 1 + η • as (1 + η) 2 , and apply Theorem 22 to (3.5.15), which yields provided 2q ≤ 2 + κ,

ˆQL E L ˆB(x) |∇Xw 1 | 2 µ ξ q (1+η) 2 (1+η) 2 q dx ζ(η • ) ˆQL E L ˆB(x) |∇Xv| 2 µ ξ q 1+η 1+η q dx, where ζ : t → t -1 4 | log t| 1 2 (since for 0 < η • < 1 2 , ζ(η) = ζ( √ 1 + η • -1) ζ(η • ))
. By the bound

µ ξ r (d-δ) p-2 p
and Hölder's inequality with exponents ( 1+η η , 1 + η), followed by Theorem 23 applied to (3.5.14) (with exponent q 1) we further have

ˆQL E L ˆB(x) |∇Xv| 2 µ ξ q 1+η 1+η q dx E L r q η (d-δ) p-2 p η q ˆQL E L ˆB(x) |∇Xv| 2 q 1 q dx E L r q η (d-δ) p-2 p η q E L |X| 2q 1 q ˆQL |g| 2 = E L r q η (d-δ) p-2 p η q ˆQL |g| 2 ,
where we used that g is deterministic and E |X| 2q = 1. We have thus proved that

E L ˆQL ˆB(x) |Da • (ξ + ∇φ)||e + ∇ φ||∇w 1 | 2 dx q 1 q E L ˆB(x) |e + ∇ φ| 2 µ ξ q(1+θ) 1 q(1+θ) ζ(η • )E L r q √ 1+η•-1 p-2 p (d-δ) √ 1+η•-1 q ˆQL |g| 2 . Since η • = θ (1+θ)(q-1)
, by definition of ζ and by the moment bound on r of Theorem 21,

qζ(η • )E L r q √ 1+η•-1 p-2 p (d-δ) √ 1+η•-1 q q γ
for some exponent γ > 0 independent of q. This entails the claimed estimate (3.5.17).

Substep 3.2. Proof of

I 2 q γ E L sup B {|e + ∇ φ| 2 µ ξ } q(1+θ) 1 q(1+θ) ˆQL |g| 2 .
(3.5.18)

Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and systems

We only display the argument for the term involving ∇w 2 , which is defined by solving three equations successively (which will compel us to appeal to Theorem 22 twice in a row, and then to Theorem 23). By Cauchy-Schwarz' inequality, and the definition of a • and µ ξ ,

E L ˆQL ˆB(x) |a • (ξ + ∇φ)||∇w 2 | 2 dx q 1 q E L ˆQL ˆB(x) µ ξ ˆB(x) µ ξ |∇w 2 | 2 dx q 1 q .
By duality and the bound µ ξ r (d-δ) p-2 p , we have

E L ˆQL ˆB(x) |a • (ξ + ∇φ)||∇w 2 | 2 dx q 1 q sup X E L ˆQL r (d-δ) p-2 p ˆB(x) µ ξ |∇Xw 2 | 2 dx ,
where the supremum runs over random variables X (thus independent of the space variable) such that E |X| 2q = 1. We now introduce exponents: η 2 := 1 q-1 θ 8(1+θ) and η 1 := 1 (q-1)(1+η 2 ) 2 (1+θ) which are chosen so that q (1+η 2 ) 2 η 1 = q(1 + θ) and q (1+η 2 ) 3 (1+η 1 ) > 1. Let us quickly check the second property:

(1+η 2 ) 3 (1+η 1 ) = (1+η 2 ) 3 + 1 + η 2 (q -1)(1 + θ) ≤ 1+(7+ 1 (q -1)(1 + θ) )η 2 + 1 (q -1)(1 + θ) < 1+ 1 q -1 = q .
With these exponents at hands, we first use Hölder's inequality with exponents ( q q -(1+η 2 ) 3 (1+η 1 ) , q (1+η 2 ) 3 (1+η 1 ) ) together with the stationarity of r , and obtain

E L ˆQL r (d-δ) p-2) p ˆB(x) µ ξ |∇Xw 2 | 2 dx E L r q q -(1+η 2 ) 3 (1+η 1 ) (d-δ) p-2 p q -(1+η 2 ) 3 (1+η 1 ) q ˆQL E B(x) µ ξ |∇Xw 2 | 2 q (1+η 2 ) 3 (1+η 1 ) (1+η 2 ) 3 (1+η 1 ) q dx.
Provided 2q ≤ 2 + κ, Theorem 22 applied to (3.5.16) yields

ˆQL E B(x) µ ξ |∇Xw 2 | 2 q (1+η 2 ) 3 (1+η 1 ) (1+η 2 ) 3 (1+η 1 ) q dx ζ(η 2 ) ˆQL E L B(x) µ -1 ξ |D 2 a(ξ+∇φ)| 2 |e+∇ φ| 2 (|∇Xw 1 | 2 +|∇Xv| 2 ) q (1+η 2 ) 2 (1+η 1 ) (1+η 2 ) 2 (1+η 1 ) q dx. Since µ ξ ≥ 1 and |D 2 a(ξ + ∇φ)| ≤ µ ξ , this yields ˆQL E B(x) µ ξ |∇Xw 2 | 2 q (1+η 2 ) 3 (1+η 1 ) (1+η 2 ) 3 (1+η 1 ) q dx ζ(η 2 ) ˆQL E L sup B(x) {|e+∇ φ| 2 µ ξ } q (1+η 2 ) 2 (1+η 1 ) B(x) (|∇Xw 1 | 2 +|∇Xv| 2 ) q (1+η 2 ) 2 (1+η 1 ) (1+η 2 ) 2 (1+η 1 ) q dx.
We only treat the term involving w 1 , which is the most subtle of the two. We then apply Hölder's inequality with exponents ( 1+η 1 η 1 , 1 + η 1 ), and use the stationarity of x → sup B(x) {|e + ∇ φ| 2 µ ξ } and {|e + ∇ φ| 2 µ ξ } q (1+η 2 ) 2 (1+η 1 )

B(x) (|∇Xw 1 | 2 + |∇Xv| 2 ) q (1+η 2 ) 2 (1+η 1 ) (1+η 2 ) 2 (1+η 1 ) q dx ≤ E L sup B {|e + ∇ φ| 2 µ ξ } q(1+θ) 1 q(1+θ) ˆQL E L B(x) |∇Xw 1 | 2 q (1+η 2 ) 2 (1+η 2 ) 2 q dx.
In view of equation (3.5.15), one may appeal to Theorem 22, and obtain

ˆQL E L B(x) |∇Xw 1 | 2 q (1+η 2 ) 2 (1+η 2 ) 2 q dx ζ(η 2 ) ˆQL E L B(x) µ ξ |∇Xv| 2 q 1+η 2 1+η 2 q dx.
We finally bound µ ξ using r , use Hölder's inequality with exponents ( 1+η 2 η 2 , 1 + η 2 ) and we apply Theorem 23 to equation (3.5.14)

ˆQL E L B(x) µ ξ |∇Xv| 2 q 1+η 2 1+η 2 q dx ≤ E L r q η 2 (d-δ) p-2 p η q ˆQL E L B(x) |∇Xv| 2 q 1 q dx E L r q η (d-δ) p-2 p η 2 q E L |X | 2q ˆQL |g| 2 = E L r q η 2 (d-δ) p-2 p η 2 q ˆQL |g| 2 .
As in Substep 3.1, the above estimates combine to (3.5.18) using Theorem 21 and our choice of η 2 .

Control of level sets: Proof of Proposition 7

As mentioned above, we do not buckle on moments of ∇ φξ,e but rather on a minimal scale that controls the growth of R → ffl Proof. We split the proof into two steps. In the first step, we control the C α -norm of a ξ that we use in the second step to control the linearized corrector via classical Schauder theory for elliptic systems. W.l.o.g we may assume that x = 0.

and all q ≥ 1 E L [χ q ] 1 q |ξ| q γ . ( 3
Step 1. Proof that

a ξ C α (B) ≤ Cr (d-δ) p-2 p , ( 3
Da • (ξ + ∇φ) C α (B) ≤ Da • (ξ + ∇φ) L ∞ (B) + D 2 a • (ξ + ∇φ) L ∞ (B) ξ + ∇φ C 0,α (B) (3.5.24
)

1 + ξ + ∇φ p-2 C α (B) (3.4.2)
A C 0,α (R d ) (1 + |ξ|) p-2 r (d-δ) p-2 p , (3.5.25)
from which the claim (3.5.22) follows since

a ξ C α (B) ≤ A C α (B) Da • (ξ + ∇φ) C α (B) .
Step 2. Proof of (3.5.20)

rd-β r β B r |e + ∇ φ| 2 µ rd-β r β B r µ ξ rd-β r β .
We now argue that (3.5.26) entails (3.5.20). By the Schauder estimate [START_REF] Giaquinta | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs[END_REF]Theorem 5.19] applied to (3.2.9) (for which the constant depends algebraically on the ellipticity ratio and the C 0,α -seminorm of the coefficients, which we may encapsulate in the C α -norm since µ ξ ≥ 1), and the bound (3.5.22) on the coefficient and (3.5.26), there is some γ > 0 (depending on α and d) such that The claim now follows from Markov' inequality.

e + ∇ φ L ∞ (B) a ξ γ C α (B) B 2 |e + ∇ φ| 2
Step 2. Control of the right-hand side of (3.5.27): For all 0 < ε, κ, θ < 1, and all dyadic R and exponents q ≥ 1

P L [r = R] ≤ c q q γ (R -(d-β+2(1-κ)-ε)q + R -(dκ-ε)q )E L r(d-β)(1+θ) 3 q 1 (1+θ) 3 , (3.5.30)
for some constant c > 0 depending on |ξ|, p, d, ε, κ, θ, but not on R and q. Since r ≤ L, it suffices to establish the statement for dyadic R ≤ L. By Lemma 32 and Theorem 21, there exists γ > 0 such that for all q ≥ d ε and R ≥ 1, we have

E L [C q ,R ] 1 q q γ , (3.5.31)
where the multiplicative constant does not depend on R and ε. By Hölder's inequality with exponents ( 1+θ θ , 1 + θ), we then get for the first right hand side term of (3.5.27)

E L C q ,R r(d-β)q 1 q ≤ E L C q 1+θ θ ,R θ 1+θ E L r(d-β)(1+θ)q 1 q(1+θ) θ q γ E L r(d-β)(1+θ)q 1 q(1+θ) . (3.5.32)
We turn to the second right hand side term of (3.5.27). By Hölder's inequality with exponents ( 1+θ θ , 1 + θ), stationarity of ∇ φ, and (3.5.31), we first have

E L C q ,R B R B R κ (x) ∇ φ 2 dx q 1 q ≤ E L C q 1+θ θ ,R θ q(1+θ) E L B R B R κ (x) ∇ φ 2 dx q(1+θ) 1 q(1+θ) θ q γ E L B R κ ∇ φ 2q(1+θ) 1 q(1+θ) .
Then, by Proposition 6 applied to g = |B R κ | -1 1 B R κ , followed by Lemma 22, by Hölder's inequality with exponent ( 1+θ θ , 1 + θ), and (3.5.21), we have

E L B R κ ∇ φ 2q(1+θ) 1 q(1+θ) |ξ|,θ q γ E L sup B |∇ φξ,e + e| 2 µ ξ q(1+θ) 2 1 q(1+θ) 2 ˆQL |g| 2 q γ E L χ q(1+θ) 2 r(d-β)q(1+θ) 2 1 q(1+θ) 2 R -dκ q γ R -dκ E L r(d-β)q(1+θ) 3 1 q(1+θ) 3
(where we changed the value of γ from one line to the other). Combined with (3.5.32), this entails (3.5.30) by redefining γ once more.

Step 3. Buckling argument. Recall that all the quantities we consider are finite since r ≤ L. We now express moments of r using its level sets and obtain by (3.5.30) for some K > 1 to be fixed below and all q ≥ 1 We now choose the exponents. We first fix 0 ≤ κ < 1 so that d(1 -κ) = β 2 , and then set ε := β 5d and 1 K := 1 -1 5d , to the effect that

E L rq(d-β K ) ≤ 1 + ∞ n=1 2 nq(d-β K ) P L [r = 2 n ] (3.5.30) ≤ 1 + ∞ n=1 2 nq(d-β K ) c q q γ (2 -nq(d-β+2(1-κ)-ε) + 2 -nq(dκ-ε) )E L rq(d-β)(1+θ) 3 1 (1+θ) 3 ≤ 1 + E L rq(d-β)(1+θ) 3 1 (1+θ) 3 c q q γ ∞ n=1 (2 nq(-β K +β-2(1-κ)+ε) + 2 nq(d(1-κ)+ε-β K ) ).
1 2 (2 nq(-β K +β-2(1-κ)+ε) + 2 nq(d(1-κ)+ε-β K ) ) ≤ 2 -nq β 5d .
With this choice, the series is summable and the above turns into

E rq(d-β K ) ≤ 1 + c q q γ E L rq(d-β)(1+θ) 3 1 (1+θ) 3
for some redefined constant c. We may then absorb part of the right-hand side into the left-hand side by Young's inequality upon choosing 0 < θ < 1 so small that (d -β)(1 + θ) 3 < d -β K (which is possible since K > 1), and the claimed moment bound follows for some suitable choice of γ > 0.

Quantitative two-scale expansion: Proof of Theorem 12

We assume δ ≤ 1, and split the proof into four steps. In the first step, we show that the two-scale expansion error satisfies a nonlinear PDE in conservative form (crucially using the flux corrector). In the second step we give a bound for the H -1 (R d )-norm of the right-hand side, the moments of which we control in the third step. We then conclude in the fourth step by using the monotonicity of the heterogeneous operator a ε . In the following, we use the short-hand notation ξ k := (∇ū) k,δ .

Step 1. Equation for the two-scale expansion error:

-∇ • (a( x ε , ∇ū 2s ε,δ (x)) -a( x ε , ∇u ε (x))) = ∇ • R ε,δ (x), (3.6.1)
where

R ε,δ (x) = k∈δZ d η k (x)(ā(ξ k ) -ā(∇ū(x)) - k∈δZ d εσ ξ k ( x ε )∇η k (x) + k∈δZ d η k (x)(a( x ε , ∇ū(x) + ∇φ ξ k ( x ε )) -a( x ε , ξ k + ∇φ ξ k ( x ε ))) + a x ε , ∇ū(x) + k∈δZ d ∇φ ξ k ( x ε )η k (x) - k∈δZ d η k (x)a( x ε , ∇ū(x) + ∇φ ξ k ( x ε )) + a x ε , ∇ū(x) + k∈δZ d ∇φ ξ k ( x ε )η k (x) + εφ ξ k ( x ε )∇η k (x) -a x ε , ∇ū(x) + k∈δZ d ∇φ ξ k ( x ε )η k (x) .
To start with, we expand ∇ū 2s ε,δ as

∇ • a( x ε , ∇ū 2s ε,δ ) = ∇ • a x ε , ∇ū(x) + k∈δZ d εφ ξ k ( x ε )∇η k (x) + ∇φ ξ k ( x ε )η k (x)
, systems which we rewrite in the form of the telescopic sum (using that k∈δZ d η k ≡ 1)

∇ • a( x ε , ∇ū 2s ε,δ (x)) -∇ • ā(∇ū(x)) = ∇ • k∈δZ d η k (x)(ā(ξ k ) -ā(∇ū(x)) + ∇ • k∈δZ d η k (x)(a( x ε , ξ k + ∇φ ξ k ( x ε )) -ā(ξ k )) +∇ • k∈δZ d η k (x)(a( x ε , ∇ū(x) + ∇φ ξ k ( x ε )) -a( x ε , ξ k + ∇φ ξ k ( x ε ))) +∇ • a x ε , ∇ū(x) + k∈δZ d ∇φ ξ k ( x ε )η k (x) - k∈δZ d η k (x)a( x ε , ∇ū(x) + ∇φ ξ k ( x ε )) +∇ • a x ε , ∇ū(x) + k∈δZ d ∇φ ξ k ( x ε )η k (x) + εφ ξ k ( x ε )∇η k (x) -a x ε , ∇ū(x) + k∈δZ d ∇φ ξ k ( x ε )η k (x) .
First, using (3.1.5) and (3.1.6) we may replace -∇ • ā(∇ū(x)) by -∇ • a( x ε , ∇u ε (x)) in the left-hand side. In the right-hand side, all the terms obviously converge strongly to zero in H -1 (R d ) (and are present in the definition of R ε,δ ) except the second term, which we need to reformulate. More precisely, using the flux corrector σ (see Definition 3.2.1) in form of the property (3.2.3), we have for all k ∈ δZ d (implicitly summing on the repeated indices ij)

∇ • η k (x)(a( x ε , ξ k + ∇φ ξ k ( x ε )) -ā(ξ k )) = ∇ • (η k (x)∇ • σ ξ k ( x ε )) = ∂ j (η k (x)∂ i σ ξ k ,ji ) = ε∂ i (∂ j η k (x)σ ξ k ,ji ( x ε )) -∂ i ∂ j (η k (x))σ ξ k ,ji ( x ε ), (3.6.2)
where the last term vanishes thanks to the skew-symmetry of (σ ξ k ,ji ) j,i and the symmetry of

(∂ i ∂ j η k ) j,i . By the skew-symmetry of σ ξ , one has ε∂ i (∂ j η k (x)σ ξ k ,ji ( x ε )) = -ε∇ • (σ ξ k ( x ε )∇η k (x)
), and we thus deduce

∇ • k∈δZ d η k (x)(a( x ε , ξ k + ∇φ ξ k ( x ε )) -ā(ξ k )) = -ε∇ • k∈δZ d σ ξ k ( x ε )∇η k (x) .
This yields (3.6.1).

Step 2. Control by continuity of the operators: The remainder R ε,δ satisfies ˆRd

|R ε,δ | 2 k ˆRd η k |ξ k -∇ū| 2 (1 + |ξ k | + |∇ū| + |∇φ ξ k ( • ε )|) 2(p-2) + ˆRd k η k k ε(φ ξ k -φ ξ k , σ ξ k -σ ξ k )( • ε )∇η k 2 1 + |∇ū| + k ∇φ ξ k ( • ε )η k 2(p-2) + ˆRd k η k k ε(φ ξ k -φ ξ k )( • ε )∇η k 2(p-1) + k ˆRd η k k ∇(φ ξ k -φ ξ k )( • ε )η k 2 (1 + |∇ū|) 2(p-2) + k∈δZ d ˆRd η k k ∈δZ d ∇(φ ξ k -φ ξ k )( • ε )η k 2(p-1)
.

(3.6.3)

This estimate directly follows from the definition of R ε,δ together with the continuity of the operator in form of

|ã(ξ 1 ) -ã(ξ 2 )| C|ξ 1 -ξ 2 |(1 + |ξ 1 | + |ξ 1 -ξ 2 |) p-2
for ã = a ε and ã = ā, and with the observation that k ∇η k = 0 so that for all maps (ζ k ) k one has

k ζ k ∇η k = k (ζ k -ζ k )∇η k , 3
= ε(φ ξ k , σ ξ k )( • ε ). Step 3. Control of moments of ´Rd |R ε,δ | 2 : For all q ≥ 1, E ˆRd |R ε,δ | 2 q 2 1 q ≤ C q γ (ε + δ)µ d ( 1 ε ) µ d ∇ 2 ū L 2 (R d ) , (3.6.4)
for some constant C and an exponent γ > 0 depending on ∇u L ∞ (R d ) . We treat the second righthand side term of (3.6.3) (that we denote by Rε,δ ) -the other terms are easier and can be treated similarly. Since for all k ,

|∇η k | δ -1 1 Q δ (k ) , we have k η k k ε(φ ξ k -φ ξ k , σ ξ k -σ ξ k )( • ε )∇η k 2 ( ε δ ) 2 k η k k 1 Q δ (k ) |(φ ξ k -φ ξ k , σ ξ k -σ ξ k )( • ε )| 2 .
Inserting this estimate in Rε,δ , and using the assumption ∇ū ∈ L ∞ (R d ), we obtain for all q ≥ 1 by Cauchy-Schwarz' inequality followed by Minkowski's inequality in probability, the support condition

Q δ (k) ∩ Q δ (k ) = ∅ ⇒ |k -k | < 2δ
, and the stationarity of ∇φ k ,

E ˆRd | Rε,δ | 2 q 2 1 q ε δ k k ∈Q 2δ (k) ˆQδ (k) E[|(φ ξ k -φ ξ k , σ ξ k -σ ξ k )( • ε )| 2q ] 1 q 1+ ∇ū 2(p-2) L ∞ (R d ) + k ∈Q 2δ (k) E[|∇φ ξ k | 2q(p-2) ] 1 q 1 2 .
By Theorem 15 and Corollary 16, and using that

µ d satisfies µ d (t 1 t 2 ) µ d (t 1 )µ d (t 2 ) and sup Q 4δ (k) µ d inf Q 4δ (k) µ d , this turns into E ˆRd | Rε,δ | 2 q 2 1 q ≤ Cq γ ( ε δ )µ d ( 1 ε ) k ( inf Q 4δ (k) µ d ) k ∈Q 2δ (k) |ξ k -ξ k | 2 |Q δ | 1 2 , (3.6.5)
for some constant C and an exponent γ > 0 depending on ∇u L ∞ (R d ) . It remains to reformulate the right-hand side sum. By Poincaré's inequality on Q 4δ (k), we have

k ∈Q 2δ (k) |ξ k -ξ k | 2 |Q δ | δ 2 ˆQ4δ (k) |∇ 2 ū| 2 ,
so that (3.6.4) follows from (3.6.5).

Step 4. Conclusion by monotonicity. We test (3.6.1) with u ε -ū2s ε,δ , and deduce by monotonicity of a ε that ˆRd

|∇(u ε -ū2s ε,δ )| 2 + |∇(u ε -ū2s ε,δ )| p ˆRd R ε,δ • ∇(u ε -ū2s ε,δ ).
By Young's inequality, we may absorb part of the right-hand side into the left-hand side, and obtain after taking the q-th moment of this inequality

E ˆRd |∇(u ε -ū2s ε,δ )| 2 + |∇(u ε -ū2s ε,δ )| p q 1 q E ˆRd |R ε,δ | 2 q 1 q .
This entails the claim in combination with (3.6.4) and the choice δ = ε. systems 3.7 Appendix

Deterministic PDE estimates and consequences

In this appendix, we recall mostly standard inequalities for nonlinear operators -∇ • a(•, ∇) and linear operators -∇ • a∇ (with unbounded coefficients) needed in the proofs of the paper. Based on these results we also prove the qualitative differentiability of correctors (with respect to ξ) when the equation is posed on a bounded domain, and we prove part of Theorem 14 for statistically isotropic operators.

Nonlinear systems: Caccioppoli, hole-filling, and Schauder

We start with Caccioppoli's inequality for nonlinear elliptic systems.

Lemma 23 (Caccioppoli's inequality). Let r > 0,

c 2 > 0, x ∈ R d and u ∈ W 1,p loc (R d ) be a weak solution of -∇ • a(•, ∇u) = 0 in B c 2 r (x). (3.7.1)
Then for all 0 < c 1 < c 2 ,

Bc 1 r (x) |∇u| 2 (1 + |∇u| p-2 ) c 1 ,c 2 inf c∈R d Bc 2 r (x)\Bc 1 r (x)
|u -c| r This implies the desired estimate (3.7.2) by Young's inequality, with exponents (2, 2) and (p, p p-1 ) for the first and second right-hand side terms, respectively, together with the identity 2ζ∇ζ = pη p-1 ∇η, and absorbing part of the right-hand side into the left-hand side.

The Widman hole-filling estimate for nonlinear systems follows from Lemma 23 by simple iteration (see e.g. [START_REF] Giaquinta | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs[END_REF]Section 4.4]).

Lemma 24 (Hole-filling estimate). There exists 0 < δ ≤ d such that if u ∈ W 1,p loc (B R ) is a weak solution of -∇ • a(•, ∇u) = 0 in the ball B R for some R > 0, then for all 0 < r ≤ R we have

Br |∇u| 2 (1 + |∇u| p-2 ) R r d-δ B R |∇u| 2 (1 + |∇u| p-2 ). (3.7.3)
We finally state regularity results for nonlinear equations, which are direct consequences of [START_REF] Uhlenbeck | Regularity for a class of non-linear elliptic systems[END_REF] and [START_REF] Kuusi | A nonlinear stein theorem[END_REF]Theorem 4] (for the uniform bound on the gradient).

Lemma 25. Let a be a monotone operator which has the form (3.1.9) and assume that A ∈ C α (B 4R ), for some R > 0 and α ∈ (0, 1). Let u ∈ W 1,p (B 4R ) be a distributional solution of

-∇ • a(•, ∇u) = 0.
Then, u ∈ C 1,α (B R ) and there exists a constant c depending on R and A C α (B 4R ) such that

∇u C α (B R ) ≤ c B 4R |∇u| p 1 p ,
where we recall that X C α = X L ∞ + X C 0,α . We state and prove Caccioppoli's inequality for linear elliptic systems with unbounded coefficients, from which we deduce Lemma 17 by optimizing the cut-off.

Lemma 26 (Caccioppoli's inequality for linear elliptic systems with unbounded coefficients). Let R > 0, a : B R → R d×d , and µ ∈ L 1 (B R ) be such that there exists a constant κ > 0 for which we have for all x ∈ B R and all

h ∈ R d h • a(x)h ≤ |h| 2 µ(x) ≤ κh • a(x)h. (3.7.4)
For all functions g and u related (in the weak sense) in B R via

-∇ • a∇u = ∇ • (g √ µ), (3.7.5)
we have for all

0 < ρ < σ ≤ R, ˆBρ |∇u| 2 µ κ J (ρ, σ, µ, u) + ˆBσ |g| 2 , (3.7.6)
where

J (ρ, σ, µ, u, g) := inf ˆBσ µ u - Bσ u 2 |∇η| 2 η ∈ C 1 c (B σ ), 0 ≤ η ≤ 1, η ≡ 1 in B ρ . (3.7.7)
Proof. Without loss of generality, we may assume that ffl Bσ u = 0. Let η ∈ C 1 c (B σ ) be such that η ≡ 1 in B ρ and 0 ≤ η ≤ 1. Testing the equation (3.7.5) with η 2 u and using the condition ( We then turn to the proof of Lemma 17.

Proof of Lemma 17. We split the proof into two steps.

Step 1. Proof that for all γ > 0

J (ρ, σ, µ, v) ≤ (σ -ρ) -1-1 γ ˆσ ρ ˆSr µ|v| 2 γ dr 1 γ , (3.7.9)
where S r := ∂B r . By scaling we may assume without loss of generality that ρ = 1 and σ = 2. Estimate (3.7.9) essentially follows by minimizing among radially symmetric cut-off functions. Indeed, for all ε > 0 we have This one-dimensional minimization problem can be solved explicitly. Set f (r) := ´Sr µ|v| 2 +ε. Using the competitor η(r) := 1 -´r 1 f -1 ´2 1 f -1 yields a control of this minimum by the harmonic average of f ,

J (1, 2, µ, v) ≤ inf ˆ2 1 η (r) 2 ˆSr µ|v| 2 + ε dr η ∈ C 1 (1, 2), 0 ≤ η ≤ 1, η(1) = 1, η(2) = 0 .
J (1, 2, µ, v) ≤ 2 1 ˆSr µ|v| 2 + ε -1 dr -1
.

By standard relations between quasi-arithmetic means, since γ > -1,

2 1 ˆSr µ|v| 2 + ε -1 dr -1 ≤ 2 1 ˆSr µ|v| 2 + ε γ dr 1 γ ,
and the claim (3.7.9) follows by letting ε ↓ 0.

Step 2. Proof of (3.3.16).

Let us first assume d ≥ 3 and note that q > d-1 2 implies q * ∈ [1, 2). Recall that for all s ∈ [1, d -1), r > 0 and φ ∈ W 1,s (S r ), Poincaré-Sobolev' inequality yields for s * = (3.7.10) By Hölder's inequality with exponents (q, q q-1 ), followed by (3.7.10) with s = q * and s * = 2q q-1 , (3.7.9) turns into

J (ρ, σ, µ, v) ≤ 1 (σ -ρ) 1+ 1 γ ˆσ ρ ˆSr µ q γ q
ˆSr |v| 2q q-1 (q-1)γ q dr 1 γ

(3.7.10) q 1 (σ -ρ)

1+ 1 γ ˆσ ρ ˆSr µ q γ q ˆSr |∇v| q * 2γ q * + r -2γ ˆSr |v| q * 2γ q * dr 1 γ .
We then choose γ = d-1 d+1 to the effect that γ q + 2γ q * = 1, so that by Hölder's inequality with exponents ( γ q , 2γ q * ) we obtain

J (ρ, σ, µ, v) q 1 (σ -ρ) 2d d-1 ˆBσ\Bρ µ q 1 q ˆBσ\Bρ |∇v| q * 1 q * + 1 ρ 2 ˆBσ\Bρ |v| q * 2 q * ,

Qualitative differentiability of correctors on bounded domains

In this section, we consider the approximation of correctors on bounded domains, both with Dirichlet and periodic boundary conditions. More precisely, let D be a smooth bounded domain of R d (resp. a cube Q L , L > 0), let A : D → M d (λ) be of class C α (resp. C α per (Q L )), and set a :

D × R d → R d , (x, ξ) → A(x)(1 + |ξ| p-2
)ξ for some p ≥ 2. We show that the periodic corrector gradients ξ → (∇φ ξ , ∇σ ξ ), where (φ ξ , σ ξ ) are solutions of (3.1.7) and (3.2.2) on D with homogeneous boundary conditions (resp. Q L -periodic with vanishing average), are Fréchet-differentiable and that their derivatives are given by the linearized corrector gradients. More precisely: Proof. We only give the arguments for ξ → ∇φ ξ . The differentiability of ξ → ∇σ ξ can be proved similarly. Let ξ, e ∈ R d , h ⊂ (0, 1) be a sequence that goes to 0, and set δ h φ ξ :=

φ ξ+he -φ ξ h
. We first show that we can extract a converging subsequence of (∇δ h φ ξ ) by local regularity and Arzela-Ascoli's theorem. Then, we show that this limit coincides with ∇ φξ,e . The starting point is the corrector equation (3.1.7) in the form -∇ • (a(x, ξ + ∇φ ξ+he ) -a(x, ξ + ∇φ ξ )) = ∇ • (a(x, ξ + ∇φ ξ+he ) -a(x, ξ + he + ∇φ ξ+he )), (3.7.11) that we rewrite, using the smoothness of ξ → a(•, ξ), as

-∇ • a (1) h ∇δ h φ ξ = ∇ • a (2) h e, (3.7.12)
where a

h := ˆ1 0 Da(•, ξ + ∇φ ξ + t(∇φ ξ+he -∇φ ξ ))dt and a

h ∈ C α (Q L ) and by using (3.7.13) for both ξ = ξ and ξ = ξ + he and arguing as in (3.5.25), there exists a constant c depending on |ξ|, D , A C α (D) such that

a (1) h C α (D) + a (2) h C α (D) ≤ c 1 .
(3.7.14)

On the one hand, by testing the equation (3.7.12) with δ h φ ξ and using that a

h is uniformly elliptic, we deduce Using the convergence of the gradient (3.7.17), we can pass to the limit as h ↓ 0 in (3.7.18), which implies that ψ solves (3.2.9). By uniqueness, ψ = φξ,e , and (3.7.17) holds without extracting a subsequence.

ˆD |∇δ h φ ξ | 2 ˆD |a (2) h | 2 (

Periodic setting: Proof of Theorem 13

In this paragraph, we show that if we have a good control of the critical set of the corrector of the leading order operator (an anisotropic p-Laplacian), then the homogenized operator ā belongs to M(p, 1, 2, c) for some c > 0. In what follows we set b : (x, ξ) → A(x)|ξ| p-2 ξ and c : (x, ξ) → A(x)ξ.

Since ā ∈ M(p, 1, p, c) by construction, it suffices to prove the improved monotonicity property:

For all ξ 1 , ξ 2 ∈ R d , (ā(ξ 1 ) -ā(ξ 2 ), ξ 1 -ξ 2 ) ≥ c|ξ 1 -ξ 2 | 2 (|ξ 1 | p-2 + |ξ 2 | p-2 ). (3.7.19)
We first introduce b :

R d → R d , ξ → ffl Q b(x, ξ + ∇ψ ξ (x))dx, where ψ ξ ∈ W 1,p per (Q) solves the corrector equation -∇ • b(x, ξ + ∇ψ ξ (x)) = 0.
By homogeneity, for all t > 0 and all ξ ∈ R d we have b(tξ) = t p-1 b(ξ).

Step where φξ,e ∈ H 1 per (Q) solves -∇ • a ξ (e + ∇ φξ,e ) = 0 and a ξ : x → Da(x, ξ + ∇φ ξ (x)). Hence, for all ξ 1 , ξ 2 ∈ R d we have On the one hand, by the corrector equations for φ ξ and ψ ξ we have

(ā(ξ 1 ) -ā(ξ 2 ), ξ 1 -ξ 2 ) = ˆ1 0 (ξ 1 -ξ 2 ) • Dā(ξ 1 + t(ξ 2 -ξ 1 )) • (ξ 1 -ξ 2 )dt. ( 3
-∇ • (a(ξ + ∇φ ξ ) -a(ξ + ∇ψ ξ )) = ∇ • c(x, ξ + ∇ψ ξ ),
so that, testing with ψ ξ -φ ξ and using the monotonicity of a, we obtain

ˆQ |∇(ψ ξ -φ ξ )| 2 (1 + |ξ + ∇ψ ξ | p-2 + |ξ + ∇φ ξ | p-2 ) ˆQ |∇(ψ ξ -φ ξ )||ξ + ∇ψ ξ | ˆQ |∇(ψ ξ -φ ξ )|(1 + |ξ + ∇ψ ξ |) p-2 2 (1 + |ξ + ∇ψ ξ |) 2-p 2 and therefore ˆQ |∇(ψ ξ -φ ξ )| 2 (1 + |ξ + ∇ψ ξ | p-2 + |ξ + ∇φ ξ | p-2 ) ˆQ(1 + |ξ + ∇ψ ξ |) 4-p .
Applied to ξ = sξ s this yields using that ψ sξs = sψ ξs ˆQ |∇( ) for some ω independent of s ≥ 1, {χ s } s≥1 is indeed bounded in C 1,α (Q) by [START_REF] Kuusi | Guide to nonlinear potential estimates[END_REF]Theorem 13]. The conclusion then follows using that 1 s p-2 a sξs = Dã s (x, ξ s + ∇χ s ) and that ξ → Dã s (x, ξ) is continuous (uniformly wrt s, x).

Substep 1.3. Proof of (3.7.21). We assume without loss of generality that ξ s → ξ ∞ . Since ξ → ∇ψ ξ is Lipschitz from R d to C α (Q), (3.7.22) can be upgraded to (along the subsequence giving the liminf) where a sym sξs is the symmetric part of a sξs . By assumption, there exists r > 0 such that which might not hold true in general due to the Lavrentieff phenomenon -see e.g. [START_REF] Zhikov | On the problem of the passage to the limit of nonuniformly elliptic equations in divergence form[END_REF] in a similar context.

lim s↑+∞ 1 s p-2 a sξs -b ξ∞ C α (Q) = 0. ( 3
R d \ T r (ξ ∞ ) is connected (where T r (ξ ∞ ) := {x + B r | x ∈ R d , |ξ ∞ + ∇ψ ξ∞ (x)| = 0}). Since [0, 1] d ∩ R d \ T r (ξ ∞ ) is closed
∈ R d , (ā(ξ 1 ) -ā(ξ 2 ), ξ 1 -ξ 2 ) ≥ c|ξ 1 -ξ 2 | p , from which (3.7.19) follows for |ξ 2 | 1, |ξ 1 | 1 using |ξ 1 -ξ 2 | p |ξ 1 -ξ 2 | 2 (|ξ 2 | p-2 -|ξ 1 | p-2 ) |ξ 1 -ξ 2 | 2 (1 + |ξ 2 | p-2 + |ξ 1 | p-2 ).

Statistically isotropic random setting: Proof of Theorem 14

In this subsection, we exhibit a class of random monotone operators a whose homogenized operator ā belongs to M(p, 1, 2, c) for some c > 0. If a is a p-Laplacian then ā is homogeneous of degree p -1.

The upcoming result relies on a perturbation of this property. To define this class we make both structural assumptions on (x, ξ) → a(x, ξ) and on the probability law P. We emphasize that our arguments are purely qualitative and do not require Hypothesis 3.1.2. We start with the structural assumption on the operator (which quantifies what we mean by perturbation of a p-Laplacian) Definition 3.7.1. We define a class A of nonlinear maps â :

[λ, 1] × R d → R d with quasi-diagonal structure, that is, such that for all (α, ξ) ∈ [λ, 1] × R d â(α, ξ) = ρ(α, |ξ|)ξ, (3.7.25)
where ρ : [λ, 1] × R + → R + is continuously differentiable, and such that the map ξ → â(α, ξ) is asymptotically of p-Laplacian type. More precisely, we assume that inf α ρ(α, t) ≥ λ(1 + t p-2 ) and that there exist two differentiable functions ρ

1 : [λ, 1] → R + and ρ 2 : [λ, 1] × R + → R such that for all (α, t) ∈ [λ, 1] × R + ρ(α, t) = ρ 1 (α)t p-2 + ρ 2 (α, t),
and that there exist a constant C and an exponent 0 ≤ β < p -2 such that

|ρ 2 (α, t)| + t|∂ t ρ 2 (α, t)| ≤ C(1 + t β ).
As a consequence, â is variational in the sense that â(α, ξ) = D Ŵ (α, ξ), where Ŵ is given by Ŵ (α, ξ) := ´|ξ| 0 sρ(α, s)ds ≥ λ(

: [λ, 1] × R d (α, ξ) → α(1 + |ξ| p-2 )ξ, with ρ 1 (α) =
α and ρ 2 (α, t) = α (and therefore C = 1 2 and β = 0).

The operator

â : [λ, 1] × R d (α, ξ) → 1+|ξ| p+q-2
1+α|ξ| q + 1 ξ, for any q ≥ 0, and with ρ 1 (α) := 1 α , ρ 2 (α, t) := α-t p-2 α(1+αt q ) + 1, C = and β = (p -2 -q) ∨ 0.

We are in position to state the main result of this section.

Theorem 25. Let â ∈ A and let A : R d → [λ, 1] be a stationary and ergodic random field (which is locally C α -this is convenient but not necessary) which is statistically isotropic in the sense that for all rotations R ∈ SO(d), A and A(R•) have the same (joint) distribution. Consider the random monotone operator a : R d × R d (x, ξ) → â(A(x), ξ). Then, the associated homogenized map ā belongs to M(p, 1, 2, c) for some c depending on λ, p and â. In particular, for all Then, there exists a constant c > 0 depending on c and p such that for all ξ 1 , ξ

ξ 1 , ξ 2 ∈ R d (ā(ξ 1 ) -ā(ξ 2 ), ξ 1 -ξ 2 ) ≥ 1 c (1 + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 , (3.7.26) and |ā(ξ 1 ) -ā(ξ 2 )| ≤ c(1 + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 |. ( 3
2 ∈ R d (a(ξ 1 ) -a(ξ 2 ), ξ 1 -ξ 2 ) ≥ c(1 + |ξ 1 | p-2 + |ξ 2 | p-2 )|ξ 1 -ξ 2 | 2 .
Proof. W.l.o.g we may assume that |ξ 1 | > |ξ 2 | > 0. We fix s > 0 and we define f : [s, +∞) → R as

f : t → tρ(t) -sρ(s) -c(1 + t p + s p ) p-2 p (t -s),
where c will be fixed later. Differentiating f and using the assumption (3.7.29), we obtain

f (t) = d dt (tρ(t)) -c((p -2)(1 + t p + s p ) -2 p t p-1 (t -s) + (1 + t p + s p ) p-2 p ) ≥ c(1 + t p ) p-2 p -c((p -2)(1 + t p + s p ) -2 p t p-1 (t -s) + (1 + t p + s p ) p-2 p ).
Since (t -s)t

= |ξ 1 | > t 2 = |ξ 2 |, (a • (ξ 1 ) -a • (ξ 2 ), ξ 1 -ξ 2 ) = t 2 1 ρ(t 1 ) + t 2 2 ρ(t 2 ) -(ξ 1 , ξ 2 )(ρ(t 1 ) + ρ(t 2 )) = (t 1 ρ(t 1 ) -t 2 ρ(t 2 ))(t 1 -t 2 ) + (t 1 t 2 -(ξ 1 , ξ 2 ))(ρ(t 1 ) + ρ(t 2 )) (3.7.30),(3.7.31) ≥ c(1 + t p 1 ) p-2 p (t 1 -t 2 ) 2 + c(t 1 t 2 -(ξ 1 , ξ 2 ))(1 + t p 1 ) p-2 p ≥ c 2 (1 + t p 1 ) p-2 p |ξ 1 -ξ 2 | 2 ,
and the claim follows by redefining c.

We now prove Theorem 25.

Proof of Theorem 25. Since (3.7.27) is a direct consequence of the monotonicity of a and the bound (3.1.8) on the corrector gradient, the subtle part is the proof of (3.7.26). We split the proof into two steps. In the first step we argue that it suffices to prove a version of (3.7.26) obtained by an approximation of the corrector on bounded domains, which we prove in the second step.

Step 1. Approximation.

By assumption there exists W : Hence, (3.7.26) follows if we prove that there exists a constant C > 0 independent of L such that for all ξ 1 , ξ 2 ∈ R d we have Step 2. Proof of (3.7.33).

R d × R d → R such that a(x, ξ) = DW (x, ξ).
(ā L (ξ 1 ) -āL (ξ 2 ), ξ 1 -ξ 2 ) ≥ 1 C (1 + |ξ 1 | + |ξ 2 |) p-2 |ξ 1 -ξ 2 | 2 . ( 3
If āL were the p-Laplacian, ζ L would satisfy (ζ L ) (t) = p t ζ L (t). The idea is to derive a similar ODE in our setting based on the identity āL (ξ) = D W L (ξ), from which we shall prove (3.7.33) by differentiating and using our structural assumptions on a.

Substep 2.1. Formula for ζ L via an ODE argument. On the one hand, by the weak formulation of (3.1.7) tested with φ L te , we have

(ζ L ) (t) = 1 t E B L a(x, te + ∇φ L te (x)) • (te + ∇φ L te (x))dx ,
which, in combination with the form of a, yields

t(ζ L ) (t) = E B L ρ(A(x), |te + ∇φ L te (x)|)|te + ∇φ L te (x)| 2 dx = E B L ρ 1 (A(x))|te + ∇φ L te (x)| p dx + E B L ρ 2 (A(x), |te + ∇φ L te (x)|)|te + ∇φ L te (x)| 2 dx . (3.7.34)
On the other hand, using that W (x, ξ) = ´|ξ| 0 sρ(A(x), s)ds and the decomposition of ρ, we also have

ζ L (t) = W L (te) = E B L ˆ|te+∇φ L te (x)| 0 sρ(A(x), s)dsdx = 1 p E B L ρ 1 (A(x))|te + ∇φ L te (x)| p dx + E B L ˆ|te+∇φ L te (x)| 0 sρ 2 (A(x), s)dsdx . ( 3
(ζ L ) (t) = p(p-1) ζ L (t ) t p + ˆt t s -1-p h(s)ds =: γ L (t) t p-2 + d 2 dt 2 t p ˆt t s -1-p h(s)ds -p(p -1)t p-2 ˆt t s -1-p h(s)ds =: R L (t) (3.7.36)
In the following two substeps we provide a bound from below for γ L (t) and a bound from above for R L (t).

Substep 2.2. Choice of t 0 and lower bound on γ L (t).

On the one hand, recall that ζ L (t) = W L (te) and that W (x, ξ) = ´|ξ| 0 sρ(A(x), s)ds ≥ λ(

1 2 |ξ| 2 + 1 p |ξ| p ), so that ζ L (t) = W L (te) = E L inf v∈W 1,p 0 (B L ) B L W (x, te + ∇v)dx ≥ λ p E L inf v∈W 1,p 0 (B L ) B L |te + ∇v| p = λ p t p .
On the other hand, the assumption on ρ 2 implies that there exists a constant c > 0 such that for all t ≥ 1 and α ∈

[λ, 1], |ρ 2 (α, t)| + t|∂ t ρ 2 (α, t)| ≤ c(1 + t β ). ( 3
(t)| ≤ cE L B L (1 + |te + ∇φ L te | β )|te + ∇φ L te ||e + ∇ φL te,e | ≤ cE L B L (1 + |te + ∇φ L te |) β+1-p-2 2 (1 + |te + ∇φ L te | p-2 ) 1 2 |e + ∇ φL te,e | ≤ cE L B L (1 + |te + ∇φ L te | p-2 )|e + ∇ φL te,e | 2 1 2 E L B L (1 + |te + ∇φ L te |) p-2(p-2-β) 1 2 .
For the second right-hand side factor, we use the (deterministic) energy estimate on ∇φ L ξ , which yields for t ≥ 1

E L B L (1 + |te + ∇φ L te |) p-2(p-2-β) 1 2 ≤ ct p 2 -(p-2-β) ,
whereas for the first right-hand side factor we use the (deterministic) energy estimate on ∇φ L ξ,e (in favor of which we shall argue below), which yields for t ≥ 1

E L B L (1 + |te + ∇φ L te | p-2 )|e + ∇ φL te,e | 2 1 2 ≤ ct p-2 2 . ( 3

Periodization in law and functional inequalities

Periodization in law of a

We start with the definition of the periodized ensemble P L . Definition 3.7.2. Let L ≥ 1. The probability P L is the stationary and centered Gaussian ensemble of scalar fields G defined by the covariance function c L : x ∈ R d → k∈Z d c(x + Lk). Clearly, the covariance function c L and thus the realizations G are Q L -periodic. We identify P L with its push forward under the map G → A := (x → B(G(x))), where B is defined in Hypothesis 3.1.2.

We now recall the qualitative convergence as L ↑ +∞ (see Lemma 42 of Chapter 5 for a proof). systems Lemma 29. There exists a family of coupling probabilities P Ω L indexed by L ≥ 1 between the probabilities P L and P, that generate Gaussian fields G L and G of covariance functions c L and c, respectively, and such that A := χ B(G) and A L := χ B(G L ) satisfy for all R > 0 and q ≥ 1

E Ω L sup Q R |A -A L | q -→ L↑+∞ 0.
(3.7.41)

Functional calculus

The ensemble P L satisfies the following logarithmic-Sobolev inequality (see for instance [START_REF] Duerinckx | Multiscale functional inequalities in probability: concentration properties[END_REF][START_REF] Clozeau | Bias in the representative volume element method: periodize the ensemble instead of its realizations[END_REF]).

Proposition 9. There exists ρ > 0 such that for all functional

F of A with E L [|F | 2 ] < +∞ E L [F 2 log(F )] -E L [F 2 ]E L [log(F )] ≤ 1 ρ E L ˆQL |∂ x F | 2 dx , (3.7.42)
where for all

x ∈ Q L ∂ x F (A) := sup δA lim sup h↓0 F (A + hδA) -F (A) h , (3.7.43)
and where the supremum runs over coefficient fields δA that are supported in B(x) and bounded by 1 in C α (B(x)). 5 The logarithmic Sobolev inequality (3.7.42) yields control of moments (see e.g. [START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF]).

Lemma 30. For all q ≥ 1 and all random variables F we have

E L |F -E[F ]| q 1 q √ q E L ˆQL |∂ x F | 2 dx q 2 1 q . (3.7.44)
The following standard result gives the link between algebraic moments and stretched exponential moments for non-negative random variables. Lemma 31. Let X be a non-negative random variable. We have the following equivalence:

∃C 1 > 0 such that E L [exp( 1 C 1 X)] ≤ 2 ⇔ ∃C 2 > 0 such that ∀q ≥ 1, E L [X q ] 1 q ≤ qC 2 . (3.7.45)
The last result of this subsection allows us to exchange supremum and expectation.

Lemma 32. Let X be a stationary random field. If there exists an exponent γ > 0 such that for all q

≥ 1 E L X q L ∞ (B) 1 q ≤ q γ , (3.7.46)
then we have for all ε > 0, R ≥ 1, and q ≥ 1

E L (R -ε X L ∞ (B R ) ) q 1 q d,ε q γ . ( 3
> 0. Consider N (R, d) R d points (x i ) i∈ 1,N ⊂ B R such that B R ⊂ N i=1 B(x i ). We then have X L ∞ (B R ) ≤ max i∈ 1,N X L ∞ (B(x i )) . (3.7.48)
By the discrete q -∞ estimate for all q ≥ 1, (3.7.48) turns into

X L ∞ (B R (x)) ≤ N i=1 X L ∞ (B(x i )) 1 q .
Therefore, by taking the the q-th moment, using the triangle inequality, the stationarity of X and the assumption (3.7.46), we get

E L X q L ∞ (B R ) 1 q ≤ max i∈ 1,N E L X q L ∞ (B(x i )) 1 q N 1 q (3.7.46) q γ R d q ,
which yields the desired estimate (3.7.47) provided q ≥ d ε (and therefore in the whole range of exponents by Hölder's inequality).

Convergence of the periodization in law of the correctors

In this subsection (and here only), we denote by (∇φ ξ , ∇σ ξ ) the nonlinear corrector gradients associated with the ensemble P, and by (∇φ L ξ , ∇σ L ξ ) the nonlinear corrector gradients associated with the periodized ensemble P L . Proposition 10. For all L ≥ 1, we consider the coupling ensemble

E Ω L [•] defined in Lemma 29. Let ξ ∈ R d . If (∇φ L ξ , ∇σ L ξ ) satisfies for all q ≥ 1 E L |(∇φ L ξ , ∇σ L ξ )| q 1 q q 1 (3.7.49)
(where the multiplicative constant does not depend on L), then for all R ≥ 1 and q ≥ 1, we have

sup Q R E Ω L |(∇φ ξ , ∇σ ξ ) -(∇φ L ξ , ∇σ L ξ )| q 1 q -→ L↑+∞ 0. (3.7.50)
As a consequence, for all x ∈ R d , we have

E Ω L ˆB(x) (φ ξ , σ ξ ) -(φ L ξ , σ L ξ ) + B (φ L ξ , σ L ξ ) 2 q 2 1 q -→ L↑+∞ 0. (3.7.51)
In addition, for all unit vectors e ∈ R d , the linearized correctors (∇ φξ,e , ∇σ ξ,e ) are well-defined, and if for all q ≥ 1

E L |(∇φ L ξ , ∇σ L ξ )| q 1 q q 1 (3.7.52)
then we have for all R ≥ 1 and q ≥ 1, ), and then we successively build the Gaussian fields G Ln in such a way that the conditional probabilities P Ω (G Ln |G) are independent of one another. Hereafter, for simplicity, we abusely use L instead of L n , and replace the limit n ↑ +∞ by L ↑ +∞. We split the proof into three steps.

sup Q R E Ω L |(∇ φξ,e , ∇σ ξ,e ) -(∇ φL ξ,e , ∇σ L ξ,e )| q 1 q -→ L↑+∞ 0. ( 3
Step 1. Proof of (3.7.50).

We start with the argument for ∇φ L ξ , and first show that we can extract a converging subsequence as L ↑ +∞. By Lemma 25 applied to (3.1.7) (with A replaced by A L ) combined with (3.7.49), we get for all R ≥ 1 and q ≥ 1 sup

L≥1 E Ω ξ + ∇φ L ξ q C α (Q R ) 1 q R sup L≥1 E Ω Q 2R |ξ + ∇φ L ξ | p q p 1 q R,|ξ|,q 1. (3.7.54)
By (3.7.54) and Arzela-Ascoli's theorem, there exists ψ ∈ C 1,α (Q R , L q (dP Ω )) such that (up to a subsequence that we do not relabel)

sup Q R E Ω |∇φ L ξ -∇ψ| q 1 q → L↑+∞ 0. ( 3
p p-1 (dP Ω ), χ ∈ C ∞ c (R d ) and define a L (x, ζ) = A L (x)(1 + |ζ| p-2 )ζ. By equation (3.1.7) (with the ensemble P L ) for φ L ξ , we have E Ω X ˆRd ∇χ(x) • a L (x, ξ + ∇φ L ξ (x))dx = 0.
Thus, using Lemma 29 and the convergence (3.7.55), we can pass to the limit when L ↑ +∞ to obtain

E Ω X ˆRd ∇χ(x) • a(x, ξ + ∇φ ξ (x))dx = 0,
which shows that ψ solves (3.1.7), and therefore that ∇ψ = ∇φ ξ and that (3.7.55) holds without taking a subsequence. We proceed the same way to prove the convergence (3.7.50) for (∇σ L ξ ), combining the equations (3.2.2) and (3.2.3) with the strong convergence

sup Q R E Ω L |a(•, ξ + ∇φ L ξ ) -a(•, ξ + ∇φ ξ )| q → L↑+∞ 0,
which follows from (3.7.50) for (∇φ L ξ ).

Step 2. Proof of (3.7.51). First we claim that for all R ≥ 1 and all functions ζ on Q R we have the Poincaré inequality .7.56) This simply follows by summation over dyadic scales of the following standard estimates for all

ˆQR ζ - B ζ 2 R 2    d = 1 : √ R d = 2 : log(R + 1) d > 2 : 1    ˆQR |∇ζ| 2 . ( 3
0 ≤ i ≤ log R ˆQ2 i+1 ζ - Q 2 i ζ 2 (2 i ) 2 ˆQ2 i+1 |∇ζ| 2 , Q 2 i ζ - Q 2 i-1 ζ 2 (2 i ) 2 Q 2 i |∇ζ| 2 .
ˆB(x) (φ ξ , σ ξ )-(φ L ξ , σ L ξ )+ B (φ L ξ , σ L ξ ) 2 R 2    d = 1 : √ R d = 2 : log(R + 1) d > 2 : 1    ˆQR |∇(φ ξ , σ ξ )-∇(φ L ξ , σ L ξ )| 2 ,
so that (3.7.51) follows from (3.7.50).

Step

C α (Q R ) ] R,|ξ| 1
for all q ≥ 1. We then extract a converging subsequence as before and identify the limit using the weak formulation of (3.2.9) together with the convergence Da

L (•, ξ + ∇φ L ξ ) → L↑+∞ Da(•, ξ + ∇φ ξ ) in C 0 (Q R , L q (dP Ω
)) (which follows from (3.7.50)).

Large-scale averages

We prove in this section estimates used to control large-scale averages, which are variations around [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. We fix L ≥ 1 and ξ ∈ R d , and use the short-hand notation r for r ,ξ,L .

Lemma 33. Let m ∈ (0, 1), r ≤ 3r (0) and f : R d → R + a measurable function. We have Indeed, if y ∈ B (x 0 ) and x ∈ B r 8 , we have

|y -x| ≤ |y -x 0 | + |x 0 -x| ≤ r ,ξ,L (x 0 ) + r + r 8 = r (x 0 ) + 9 8 r.
By the 1 16 -Lipschitz property of r , we have r (0) -1 16 r ≤ r (x 0 ) ≤ r (0) + 1 16 r. Together with the assumption r ≤ 2r (0), this entails 13 16 r (0) ≤ r (x 0 ) and r (x 0 ) +

x i | ≤ 1 16 r + r (x 0 ) - 1 4 r = r (x 0 ) - 3 16 r,
which concludes the proof of (3.7.61). We deduce from the sub-additive property of x ∈ R + → x m and the fact that for all i ∈ 1, N ,

x i + B 1 8 r ⊂ B 3 2 r B r 8 B r (x 0 )+ 9 8 r (x) f m dx N i=1 B r 8 B r (x 0 )-3 16 r (x+x i ) f m dx B 3 2 r B r (x 0 )-3 16 r (x) f m dx,
which concludes the proof of (3.7.58) since the 1 16 -Lipschitz property of r entails for all x 0 ∈ B r and This constant C (which only depends on d and our upper bound 1 16 on the Lipschitz constant of r ) will be used to define the best Lipschitz constant for r (cf. Definition 3.3.1).

x ∈ B 3 2 r r (x 0 ) - 3 16 r ≤ r (x) + 1 16 |x -x 0 | - 3 16 r ≤ r (x) + 1 16 ( 3 2 r + r) - 3
Proof. Without loss of generality, we may assume that x 0 = 0. Since r is 1 16 -Lipschitz, we have for all integrable non-negative functions g ˆRd We now discuss about the paring of two bilinear forms in this context, necessary to formulate the extension of Price's formula of the next section. We recall that under the assumption (4.1.3), the trace provides a pairing between a bounded symmetric bilinear form ċ on H ⊃ X * and a bounded symmetric bilinear form b on X ⊃ H. More precisely, the trace provides a pairing between the Riesz representations Ċ and B as bounded symmetric linear operators on H. Even more precisely, the pairing is given by the trace of the product ĊB, which requires ĊB to be of trace class The paring defined in (4.1.5) is canonical in the finite dimensional case, as a paring between a form b on X and a form ċ on X * (where in that case, we can obviously take X = H and A = Id). Suppose that X is a space of dimension N < +∞ and set (e where (e * n ) n∈ 1,N denotes the canonical dual basis of X * and µ nm = b(e n , e m ). We claim that in this case, the paring (4. We claim that this survives on X in form of lim

B (x) g dx ∼ ˆRd g, (3.7
N ↑∞ P N g -g 2 X L = 0 (4.2.6)
uniformly in L, where we made explicit the variable g ∈ X that • L integrates over. Indeed, we obtain from (4.1.3) that

P N g -g 2 X ≤ n>N m>N (h n , g) H (h m , g) H (Ah n , Ah m ) H ,
and thus by definition5 (4.1.1)

P N g -g 2 X L ≤ n>N m>N c L (h n , h m )(Ah n • Ah m ) H . ( 4
c L (h n , h m )(Ah n , Ah m ) H = m>N (Ah n , Ah m ) H h m , C L h n H = ((1 -P N )A * Ah n , C L h n ) H = (Ah n , A(1 -P N )C L h n ) H .
Hence, (4.2.6) follows by inserting the above identity in (4.2.7) and by using the fact that A(1 -P N )C L is Hilbert-Schmidt (as a product of continuous maps and an Hilbert-Schmidt operator) with |A(1 -P N )C L | HS ≤ |A| HS (using (4.1.2)):

P N g -g 2 X L ≤ n>N (Ah n , A(1 -P N )C L h n ) H ≤ n>N |Ah n | 2 H 1 2 n>N |A(1 -P N )C L h n | 2 H 1 2 ≤ |A| HS n>N |Ah n | 2 H 1 2
, which vanishes as N ↑ +∞ since A is Hilbert-Schmidt. In addition, by the same argument as for (4.2.6) it follows that P N g 2 X L is uniformly bounded in N and L so that by Fernique's theorem [33, Theorem 2.8.5], for all q < +∞ and uniformly in N and L P N g p X L

1.

(4.2.8)

We now define our approximation F N via pull-back under P N :

F N (g) = F (P N g). ( 4
F N L = F L , (4.2.10)
uniformly in L. We now claim that also the right-hand side term of (4.2.2) converges, that is

lim N ↑∞ d 2 F N (g). dc L dL L = d 2 F (g). dc L dL L , (4.2.11)
uniformly in L. Indeed, on the level of the second Fréchet derivative, the linear transformation (4.2.9) translates into

d 2 F N (g)(h, h ) = d 2 F (P N g)(P N h, P N h ).
From this we obtain as for (4.1.6) (where ĊL denotes the Riesz representation of the bilinear form

dc L dL in H) d 2 F N (g). dc L dL = +∞ n=0 d 2 F (P N g)(P N h n , P N ĊL h n ) = +∞ n=0 d 2 F (P N g)(h n , P N ĊL P N h n ), (4.2
.12)

d 2 F (g). dc L dL = +∞ n=0 d 2 F (g)(h n , ĊL h n ), (4.2
ĊL P N h n ) L = d 2 F (g)(h n , ĊL h n ) L , (4.2.14)
uniformly in L. In addition, using (4.2.1) and (4.1.3), one has

|d 2 F (P N g)(h n , P N ĊL P N h n )| ≤ (1 + P N g X ) p |Ah n | H |AP N ĊL P N h n | H ,
which gives with (4.2.8)

| d 2 F (P N g)(h n , P N ĊL P N h n ) L | ≤ C|Ah n ||AP N ĊL P N h n | H
for some generic constant C independent of L and N . Hence by Cauchy-Schwarz and the fact that

AP N ĊL P N is Hilbert-Schmidt, n>M | d 2 F (P N g)(h n , P N ĊL P N h n ) L | ≤ C n>M |Ah n | 2 1 2 |AP N ĊL P N | HS .
Then, as for (4.1.9), we use Step 2. Proof of (4.2.2) for finite rank functional. From (4.2.10) and (4.2.11) we learn that it is enough to establish (4.2.2) for F that are of the form (4.2.9), that is, of the form

|AP N ĊL P N | HS ≤ |A| HS |P N ĊL P N | ≤ |A| HS | ĊL | ≤ C (for some generic constant C independent of L), so that n>M | d 2 F (P N g)(h n , P N ĊL P N h n ) L | ≤ C n>M |Ah n |
F (g) = f ( 1 (g), • • • , N (g)) (4.2.15)
for ( n ) n∈ 1,N ⊂ X * and f is twice continuously differentiable of at most algebraic growth in its variables x n , • • • , x N , which we assemble in x ∈ R N . We first rewrite the paring d 2 F. dc L dL in this context. By the chain rule we obtain from (4.2.15)

d 2 F (g) = N n=1 N m=1 (∂ nm f )(x) n ⊗ m . ( 4

.2.16)

We consider the centered Gaussian measure • LN on R N defined by the push-forward of • L under (4.2.4), that is,

F (g) L =: f (x) LN , (4.2.17)
where we made the variable of integration explicit. With the standard identification

(R N ) * ∼ = R N , the covariance of • LN is given by c L,nm := c L ( n , m ), (4.2.18)
where a difficulty arises from the fact that the so defined symmetric N × N -matrix may not be definite but only semi-definite.

× N -matrix L → C L differentiable d dL exp(- 1 2 x • C -1 L x) det C -1 L = 1 2 x • C -1 L dC L dL C -1 L x -tr C -1 L dC L dL exp(- 1 2 x • C -1 L x) det C -1 L = 1 2 tr C -1 L x ⊗ C -1 L x -C -1 L dC L dL exp(- 1 2 x • C -1 L x) det C -1 L = 1 2 N n=1 N m=1 ∂ nm [exp(- 1 2 x • C -1 L x)] det C -1 L dc L,nm dL ,
Testing this identity with f we obtain, by two integration by parts (which are justified since f has polynomial growth)

d dL ˆRN dx f (x) exp(- 1 2 x • C -1 L x) √ detC -1 = 1 2 N n=1 N m=1 ˆRN dx (∂ nm f )(x) dc L,nm dL exp(- 1 2 x • C -1 L x) detC -1 L .
The goal is now to show that the computations done in the finite dimensional case to go from (4. We now show that we can exchange the two summations in (4.3.5). To do so, we check the summability of the summands: using that (λ n ) n∈N is bounded, the Cauchy-Schwarz' inequality and the fact that (f n ) n∈N is an orthonormal basis in X as well as (4.3.1), one obtains

+∞ k=0 +∞ n=0 |λ n n (h k ) n ( Ċh k )| ≤ sup n∈N |λ n | +∞ k=0 +∞ n=0 | n (h k )| 2 1 2 +∞ n=0 | n ( Ċh k )| 2 1 2 = sup n∈N |λ n | +∞ k=0 h k X Ċh k X (4.3.1) = sup n∈N |λ n | +∞ k=0 |Ah k | H |A Ċh k | H ,
which is finite since both A and A Ċ are Hilbert-Schmidt in H. Therefore, we can exchange the summations in (4.3.5) to obtain (using also the definition of n in (4.

3.3)) b. ċ = +∞ n=0 λ n +∞ k=0 n (h k ) n ( Ċh k ) (4.3.3) = +∞ n=0 λ n +∞ k=0 (h k , f n ) X ( Ċh k , f n ) X (4.3.1) = +∞ n=0 λ n +∞ k=0 (h k , A * Af n ) H ( Ċh k , A * Af n ) H = +∞ n=0 λ n Ċ +∞ k=0 (h k , A * Af n ) H h k , A * Af n H = +∞ n=0 λ n ċ(A * Af n , A * Af n ),
which gives (4.

Application for the Chapter 5

We now specify the spaces X and H that we will use for our application in the next chapter and we rewrite (4.3.2) in a very specific form that we will use in the next chapter.

(g) = ˆhn (x)g(x)dµ(x). (4.3.12)
We now prove that b admits a kernel in the sense of (4.3.9). For all g, g ∈ X, we have from (4. is well defined in L 2 dµ⊗dµ (R d × R d ). Indeed, using the fact that (h n ) n∈N is orthonormal in X, we have for all n ≥ 0 and p ≥ 0

ˆˆ n+p k=n λ k h n (x)h n (y) 2 dµ(x)dµ(y) = n+p k=n λ 2 k ≤ +∞ k=n λ 2 k , which vanishes as n ↑ +∞ since (λ n ) n∈N is squared integrable. Therefore k ∈ L 2 dµ⊗dµ (R d × R d
) and (4.3.13) turns into b(g, g ) = ˆˆk(x, y)g(x)g(y)dµ(x)dµ(y), and thus k is the8 kernel of b in the sense of (4.3.9). We now prove (4.3.10). We first reinterpret, for all n ∈ N, the linear form n given by (4.3.12) as the Bochner integral Corollary 27. We assume that the assumptions of Theorem 26 are satisfied and consider X and H as in (4.3.6) and (4.3.7). We assume in addition that d 2 F is Hilbert-Schmidt in X and we denote by

n = ˆδx h n (x)dµ(x), (4.3
∂ 2 F ∈ L 2 dµ⊗dµ (R d × R d ) its kernel.
We have the following version of Price's formula

d dL F L = 1 2 ˆˆ ∂ 2 F (x, y) L dc L dL (δ x , δ y )dµ(x) dµ(y). (4.3.17)
Chapter 5

Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations

We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble • and the corresponding linear elliptic operator -∇ • a∇. In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called "representative" volume element, i. e. a large box with, say, periodic boundary conditions. The main message of this chapter is: Periodize the ensemble instead of its realizations.

By this we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble • . We make this point by investigating the bias (or systematic error), i. e. the difference between a hom and the expected value of the RVE method, in terms of its scaling w. r. t. the lateral size L of the box. In case of a suitable periodization of • , we rigorously show that it is O(L -d ). In fact, we give a characterization of the leading-order error term for the strategy of periodizing • . For the strategy of periodizing the realizations, the behaviour is expected to be O(L -1 ) (see the numerical experiments [103]) and formal argument will be added in the work in preparation in [START_REF] Clozeau | Bias in the representative volume element method: periodize the ensemble instead of its realizations[END_REF])

We carry out the rigorous analysis in the convenient setting of ensembles • of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price's theorem (see Chapter 4) and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of secondorder correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green's function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green's function.

This Chapter is based on joint work in preparation with Felix Otto, Marc Josien and Qiang Xu.

The Chapter is organized as follows. In Section 5.1, we recall the assumptions of the chapter as well as our main result. In Section 5.2, we present a refinement of our main theorem and the main ideas. In Section 5.3 we present the structure of the proof as well as the auxiliary results.

Representation formula

We start with an informal, but detailed, derivation of the representation formula, see (5.2.9), which might be the most conceptual piece of our work.

Let us fix two vectors ξ and ξ * and focus on the component ξ * • āξ; we denote by φ the solution of (5.1.2) with e i replaced by ξ.2 By stationarity3 of ∇φ and of • L , we have where the two minus signs in the denominator are for later convenience. 4 Here ∂ 2 F ∂g(x)∂g(y) denotes the kernel representing the second Fréchet derivative of F , seen as a bilinear form on the space of functions on R d . As a derivative w. r. t. the noise g, it can be seen as a Malliavin derivative.

ξ * • āξ L = F L where F := (ξ * • a(∇φ + ξ))(0). ( 5
We define F by (5.2.1). By the change of variables z

x-y, which capitalizes on the translation invariance of the covariance, and (more directly) by the stationarity of • L in conjunction with the stationarity of ∇φ that leads to , a](∇φ (1) + ξ);

indeed, the first identity (formally) follows from applying ∂ 2 ∂g(0)∂g(z) to (5.1.2) and then testing with φ * (1) , whereas the second identify follows from testing (5.2.4) with ∂ 2 φ (1) ∂g(0)∂g(z) . Resolving the commutator [ This representation separates a first factor Γ hom , which only depends on the type of the periodic lattice (here cubic) and the homogenized coefficient a hom , from a second factor that only depends on the whole-space ensemble • , via its covariance function c and covariances involving its first-and second-order correctors.

Small contrast regime and non-degeneracy

In this subsection, we (formally) identify the leading order (5.2.26) of the r. h. s. of (5.2.25) in the small-contrast regime. We then argue that this leading-order error term typically does not vanish, even in the high-symmetry case of an isotropic ensemble.

We start with the derivation of (5.2.26): To leading order in a small ellipticity contrast 1 -λ, the quantity ∇φ Restricting to the case of scalar A for convenience, the expression further simplifies to Q ijm (z) ≈ -z m a (0)a (z) e i ⊗ e j .

Restricting ourselves w. l. o. g. to ensembles • with c(0) = g 2 (0) = g 2 (z) = 1, we see that a (0)a (z) depends on the Gaussian ensemble • only through c(z). We thus write a (0)a (z) = A (c(z)) for some function A, so that by the chain rule to leading order in the contrast.

It remains to argue that the two factors in (5.2.26) typically do not vanish. The second factor in (5.2.26) does not vanish in the typical case of A > 0 and c ≥ 0. Indeed, by definition of A, we then have A > 0 and thus A(c) > 0 for c > 0, so that ´Rd dzA(c(z)) > 0 because of c(0) = 1.

For the first factor in (5.2.26), we restrict ourselves to an isotropic ensemble, namely the case where c is radially symmetric, in addition to A being scalar. In line with this, we show that the trace of the first factor in (5.2.26) does not vanish:

lim T ↑∞ k∈Z d k m ∂ m G T,hom (k) = 0.
(5.2.27)

For our isotropic ensemble, the contravariant two-form a is invariant in law under orthogonal transformations, and so is a hom , which thus is a multiple of the identity, so that is a multiple of ∇ • a hom ∇. Hence by definition of G T,hom , (5.

Structure of the proof of Theorem 30

In this section, we formulate the main intermediate results that lead to Theorem 30: In Subsection 5.3.1, we introduce the massive approximation in order to rigorously derive the analogue of the representation formula (5.2.28) from Subsection 5.2.1, see Proposition 11. In Subsection 5.3.2 we argue, following Subsection 5.2.1, that a re-summation allows for removing the massive approximation in the representation formula, see Proposition 12. It relies on second-order homogenization, as introduced in Subsection 5.2.2. In Subsection 5.3.3 we sketch how to pass from the representation given by Proposition 12 to the asymptotics stated in Theorem 30. This essentially relies on corrector estimates and the estimate of the homogenization error, see Subsections 5.3.4 and 5.3.5. In Subsection 5.3.4, we formulate the uniform stochastic estimates on first and second-order correctors needed to capture the asymptotics L ↑ ∞, see Proposition 13. In Subsection 5.3.5, we formulate the stochastic second-order estimate of the homogenization error, applied to the Green's function, see Proposition 14.

Massive approximation

As became apparent in Subsection 5.2.1, there is divergence in the sum over the periodic cells, see (5.2.7). We avoid it by replacing the operator -∇ • a∇ by 1 T -∇ • a∇ where T < ∞ will eventually tend to infinity. This has the desired effect that the corresponding Green's function G T (a, x, y) and its derivatives now decay exponentially in |y-x| √ T , more precisely: and thus C in (5.3.1) is well controlled. We refer to Appendix 5.5.1 for a proof. The language of "massive" approximation arises from field theory where such a zero-order term is often introduced to suppress an infrared divergence, like here. Assimilating m 2 to the inverse of a time scale T however makes the connection to stochastic processes, since 1 T -∇ • a∇ is the generator of a diffusiondesorption process where T is the time scale of desorption, and ultimately to parabolic intuition. As a collateral of the massive approximation, we have to replace the definitions (5.1.2) and (5.1.3) by

1 T φ (1)
T i -∇ • a(∇φ

T i + e i ) = 0, āT e i :=

[0,L) d a(∇φ (1)
T i + e i );

(

with analogous definitions for the transposed medium a * . From Schauder's theory and (5.3.2), (φ

T , ∇φ

T ) is Hölder continuous, more precisely: for any p < +∞ and α < α φ

T , ∇φ where we recall that φ

T = i ξ i φ (1)
T i .

The z-integral on the r. h. s. of (5.3.5) converges absolutely for |z| ↑ ∞ since the exponential decay of ∇∇G T (0, z) dominates the linear growth of ∂c L ∂L (z), cf. (5.2.8). The singularity at z = 0 is to be interpreted by duality, using that the other factors are locally smooth in z.

Let Q (1) and Q (2) be defined as in (5.2.19) and (5. Periodic homogenization theory suffices to establish Proposition 12 and in particular to ensure that all five expressions on the r. h. s. of (5.3.10) are well-defined, including the third one. Indeed, it helps to momentarily think of having rescaled length by the fixed L. This puts us into the context of a 1-periodic coefficient field a, which in addition is Hölder continuous. By and rescaling back, we obtain ( 5.3.11) This estimate yields the absolute convergence of the third term on the r. h. s. of (5.3.10), since the decay (5.3.11) over-compensates the linear growth of ∂c L ∂L . Finally, periodic homogenization and (5.3.2) ensure the following convergences of the massive quantities: for any p < +∞ and any x, y, the following convergences holds in L p • (φ

(1) T , ∇φ (1)
T ) → T ↑+∞ (φ (1) , ∇φ (1) ) and ∇∇G T (x, y) → T ↑+∞ ∇∇G(x, y) as well as (∇ 3 G T (x), ∇ 2 G T (x)) → T ↑+∞ (∇ 3 G(x), ∇ 2 G(x)).

(5.3.12)

We refer to Appendix 5.5.2 for a proof of (5.3.12).

From representation to asymptotics

In order to pass from the representation in Proposition 12 to the asymptotics in Theorem 30, we have to show that the first r. h. s. term of (5.3.10), up to the factor L d+1 , converges to the r. h. s. term of (5.2.24), and that the remaining terms are o(L -(d+1) ). Note that from (5.1.4), the fifth term of (5.3.10) is immediately of order L -d-1-2α . Below, we discuss the first fourth terms. ensemble instead of its realizations Lemma 41 is an easy consequence of the pathwise Lipschitz estimate [80, Theorem 1]. More precisely, we refer to [80, (16)], which takes the form of

B 1 |∇u| 2 1 2 d,λ r d 2 * B R |∇u| 2 1 2 ,
with the random radius r * defined in [80, (12)]. It easily follows from the estimates on (φ

i , σ

i ) in Proposition 13 that r p * Corollary 32 amounts to an inner regularity estimate for a-harmonic functions u, in terms of the norms L ∞

x L p • and W -2,1

x L p • on the level of the gradient ∇u. We call this estimate an annealed9 estimate, since now on both sides of (5.3.44), the probabilistic norm is inside. where we recall that the massive corrector φ

Proofs

T is defined in (5. , which directly provides the fifth term of (5.4.17). Thus, we only have to study the r. h. s. integral in (5.3.5). As explained in Section 5.3.2, we split the integral into the near-field contribution ´dzη L (z) (which gives the fourth r. h. s term of (5.4.17)) and the far-field contribution ´dz(1 -η L )(z) (that we denote by Π L,T ). On the far-field part, we appeal to the two-scale expansion of ∇∇G T and, by the definition of the homogenization error E T , the tensors Q

T and Q

T naturally appear in (∇φ * (1) T + ξ * )(0) • a (0)∇∇G T (0, z)a (z)(∇φ

T + ξ)(z) =∂ ij G T (z)ξ * • Q (1) T ij (z)ξ + ∂ ijm G T (z)ξ * • Q (2) (1)
T ijm (z)ξ + (∇φ * (1) T + ξ * )(0) • a (0)E T (0, z)a (z)(∇φ

T + ξ)(z), (5.4.18) which we may insert in Π L,T to the effect of

Π L,T = -ˆdz(1 -η L (z)) ∂ ij G T (z)ξ * • Q (1) T ij (z)ξ L ∂c L ∂L (z) =: Π (1) L,T + ˆdz(1 -η L (z)) ∂ ijm G T (z)ξ * • Q (2) T ijm (z)ξ L ∂c L ∂L (z) =: Π (2) L,T + ˆ(1 -η L (z)) (∇φ * (1) T + ξ * )(0) • E T (0, z)(∇φ (1)
T + ξ)(0)

L ∂c L ∂L (z)dz, (5.4.19
) where we identify the third above r. h. s. term as the third r. h. s. term of (5.4.17). Then, we shall treat separately Π L,T . On the one hand, using (5.2.8), we may expand ∂c L ∂L as in (5.4.19), in such a way that the sum on k = 0 may be transfered to the Green function. Namely, we obtain (5.4.20) where we used the change of variables z z + kL and the fact that Q

Π (1) L,T = k =0 ˆdz(1 -η L (z)) ∂ ij G(z)ξ * • Q (1) T ij (z)ξ L k n ∂ n c(z -kL) = ˆdz ξ * • Q (1) T ij (z)ξ k =0 ((1 -η L )∂ ij G T )(z + kL)k n L ∂ n c(z),
T is L-periodic. Since ∂ ij G T is an even function, the following identity holds: (5.4.22)

k =0 ∂ ij G T (kL)k n = 0.
On the other hand, by (5.2.8) and by a change of variables, we similarly rewrite Π

L,T as Π

L,T = -ˆdz ξ • Q

(2)

T ijm (z)ξ k =0 ((1 -η L )∂ ijm G T )(z + kL)k n L ∂ n c(z) = -ˆdz ξ * • Q (2) T ijm (z)ξ k =0 ∂ ijm G T (kL)k n + (2)
T Lijmn L

∂ n c(z).

(5.4.23)

We finally remark that, in (5. Asymptotic analysis as T ↑ ∞. We now show that each term in (5.4.17) pass to the limit as T ↑ ∞ and converge to their massless counterpart. Thus, we need to establish that this limit makes sense for each of the five r. h. s. terms of (5.4.17); in this task, the dominated convergence theorem is our main tool. Note that the estimates (5.3.11), (5.3.13) and (5.3.14) hold, uniformly in T , at the level of the massive quantities E T ,

T Lijmn and

(1)

T Lijn (thanks to (5.3.1) for the two latters). Therefore, combined with the convergences of the massive quantities (5.3.12), the second, third, fourth and fifth r. h. s term of (5.4.17) converge to their massive less counterpart as T ↑ +∞. Therefore, the subtle part is in the first r. h. s term of (5.4.17), that we treat in details.

In the sequel, L ≥ 1 is fixed. We prove that lim

T ↑∞ ˆdz Γ T /L 2 ijmn ξ * • Q (2) T ijm (z)ξ -z m ξ * • Q (1) T ij (z)ξ L ∂ n c(z) = ˆdz Γ ijmn ξ * • Q (2) ijm (z)ξ -z m ξ * • Q (1) ij (z)ξ L ∂ n c(z).
(5.4. [START_REF] Bella | Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations[END_REF] We claim that the only additional needed ingradient is the well-posedeness of Γ ijmn given by definition (5. (5.4.26)

Indeed, thanks to the assumption (5.1.4) on c, (5.3.4) and (5.4.26), the integrand is bounded (uniformly in T ) by (1 + |z|) -d-2α . We conclude using the convergences (5.3.12) and the Lebesgue convergence theorem.

Here comes the argument for (5.4.26). We fix a smooth compactly supported η with η = 1 on the 5.4. Proofs 259 unit cube (-1 2 , 1 2 ) d ; we use it to split the lattice, which we interpret as a Riemann sum, as follows:

k =0 k n ∂ ijm G T (k) = ˆRd \(-1 2 , 1 2 ) d dxη(x)x n ∂ ijm G T (x) + ˆdx(1 -η)(x)x n ∂ ijm G T (x) + k =0 k n ∂ ijm G T (k) - ˆk+(-1 2 , 1 2 ) d
dxx n ∂ ijm G T (x) .

(5.4.27)

The first r. h. s. integral effectively extends over a compact subset of R d \{0} and thus obviously converges for T ↑ ∞, thanks to (5.3.12). On the second r. h. s. integral in (5.4.27) we perform two integrations by parts:

ˆdx(1 -η)(x)x n ∂ ijm G T (x) = ˆdx -∂ j η(x)δ mn ∂ i G T (x) + ∂ m η(x)x n ∂ ij G T (x) .
Again, the r. h. s. integral effectively extends over a compact subset of R d \{0} and converges for T ↑ ∞, thanks to (5.3.12). We finally turn to the last contribution in (5.4.27); clearly each summand has a limit T ↑ ∞. This extends to the sum because of dominated convergence: Each summand is dominated, in absolute value, by the Lipschitz norm of x → x n ∂ ijm G T (x) on the translated cube k + (-1 2 , 1 2 ) d , which by the uniform-in-T decay of G T is |k| -d-1 , an expression that is summable in k ∈ Z d \{0}.

Proof of Lemma 37: Fluctuation estimates

As announced above, we show only (5.3.29) by closely following [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]. The only difference is that we appeal not only to the annealed Calderón-Zygmund estimates as in [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF], but also to the annealed weighted estimates contained in Lemma 38. For a deterministic and periodic vector field f , we consider the random variable of zero average

F := ˆ[0,L) d f • ∇φ (2) ij .
We employ on it the spectral gap inequality (cf. [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Lem. 3.1]), which, combined with the Bochner estimate (assuming that p ≥ 2), reads

|F | p 1 p ˆ[0,L) d ∂F ∂g p 2 p 1 2 .
(5.4.28)

We split the proof into three steps. First, we establish that the Fréchet derivative of F is given by

∂F ∂g = ∇v • a (∇φ (2) ij + φ (1)
i e j ) -(∇w j + ve j ) • a (∇φ

i + e i ), (5.4.29) where v and w j are defined through (5.4.34) . Last, we get (5.4.43) by recalling that u is periodic of vanishing average in [- 1 2 , 1 2 ) d , so that Proof. Let R ≥ 1 and p < +∞. We also set L 1 and a discretization parameter N ∈ N * , the value of which will be fixed at the end of the proof. By (5.3.2) (which also holds for g), there holds for a fixed α g p C 0,α (B R ) (5.4.50)

ˆ[-1 2 , 1 2 ) d |u| 2p 1 2p = ˆ[-1 2 , 1 2 ) d u - ˆ[-1 2 , 1 2 ) d u 2p 1 2p ≤ ˆ[-1 2 , 1 2 ) d dx ˆ[-
1 p + g p C 0,α (B R )
14 Notice that the set of discretization points {xi, i ∈ {-N, • • • , N } d } becomes denser and spreader in R d when N ↑ ∞. ensemble instead of its realizations

Step 2: On account of (5.4.56), there exists a subsequence of (u Ln ) n (that we do not relabel) and a function u ∞ ∈ L 2 (Ω; H 1 loc (R d )) such that, for any R ≥ 1, the following weak convergences hold We show that ∇u ∞ = ∇u, almost-surely. Hereafter, for simplicity, we replace L n by the lighter notation L and the limit n ↑ ∞ by L ↑ ∞.

We verify that u ∞ solves the equation (5.4.55) in the sense of distributions. To do so, let X ∈ L q (Ω) for q > 2 be a random variable and ζ ∈ C ∞ c (R d ) be a smooth function. By (5.4.55), there holds

X ˆ∇ζ • a L ∇u L + f L Ω = 0.
(5.4.63)

Thus, it follows from (5.4.61), (5.4.62) and (5.4.59) that, in the limit L ↑ ∞, we have

X ˆ∇ζ • a∇u ∞ + f Ω = 0,
which shows that u ∞ satisfies (5.4.55) in the sense of distributions, • Ω -almost-surely. Also, by means of the weak lower semi-continuity of the L 2 norm, we obtain

B R |u ∞ | 2 1 2 Ω ≤ lim inf L↑∞ B R |u L | 2 1 2 Ω R 1 2
for all R ≥ 1.

This together with the Markov inequality gives As a consequence, we obtain (5.4.64), which establishes the strong convergence (5.4.60).

Equipped with Lemma 44, we proceed with the:

Proof of Lemma 43. Let • Ω L be the coupling ensembles between • and • L that are built in Lemma 42. Notice that we have already established in (5.4.61) that a and a L satisfy (5.4.54). The proof falls in four steps. First, we show that is sufficient to establish a version of (5.4.54) where p = 2 and where the spatial L ∞ norm is replaced by the L 2 norm. Then in Step 2, Step 3 and Step 4, we successively show that φ (1) , ∇φ (1) , σ (1) , ∇σ (1) and ∇φ (2) satisfy the convergence (5.4.54) by appealing to Lemma 44.

Step 1. From Proposition 13, (5.3.2) and Schauder theory applied to (5.1.2), (5.3.18) and (5.3.19), one has for any p < +∞ and α < α ∇φ

L , φ

L , ∇σ

L , σ

L , ∇φ Hence, by Arzela-Ascolie's theorem, it is sufficient to establish ∇φ (1) , φ (1) , ∇σ (1) , σ (1) , ∇φ (2) -∇φ

L , φ

L , ∇σ

L , σ

L , ∇φ Step 2. We first consider f := a and u := φ (1) . Indeed, by Proposition 13, u and f satisfy (5.4.56) ((5.4.57) and (5.4.58) being consequences of the homogenization theory). Moreover, by Lemma 42 and 5.1.6, the condition (5.4.59) is also satisfied. As a consequence of (5.4.60) there holds ∇φ (1) -∇φ for any R ≥ 0. Interpolating as described in Step 1, this implies that ∇φ (1) -∇φ

L satisfies (5.4.54). Next, by (5.3.24) in Proposition 13 (where a similar estimate holds for • ) applied with g = ∇v and -∆v = R -d 1 B R , there holds (5.4.75)

B R φ (1) 2 1 2 L + B R φ (1)
We construct w with help of the Riesz representation theorem, so that we automatically have ˆuw = ˆ( n w) 2 ∼ ˆ|∇ 2n w| 2 ˆw2 , (5.4.76)

where we used higher-order L 2 -regularity and a higher-order Poincaré estimate. We obtain from inserting (5.4.75) into (5.4.74) ˆ(η 4n u) 2 ˆ|η 4n+1 ∇u| 2 4n 4n+1 ˆw2 1 4n+1 + ˆw2 .

Combining this with Caccioppoli's estimate ´|η 4n+1 ∇u| 2

´(η 4n u) 2 and Young's inequality, we obtain by the choice of η ˆB1 |∇u| 2 ≤ ˆ|η 4n+1 ∇u| 2 ˆw2 .

(5.4.77)

It remains to post-process this inner regularity estimate for an a-harmonic function u.

In route to an annealed estimate, we express the r. h. s. of (5.4.77) in terms of u, which is conveniently done in terms of the complete orthonormal system of eigenfunctions {w k } k∈N ⊂ H 2n 0 (B 2 ) and eigenvalues {λ k } k ⊂ (0, ∞) of the Dirichlet-2n , which is a positive operator with compact inverse:

ˆw2 = k 1 λ 2 k ˆuw k 2 = k 1 λ k ´uw k 2
´( n w k ) 2

(5.4.76)

k 1 λ k ´uw k 2 ´|∇ 2n w k | 2 .
We insert this into (5.4.77)

ˆB1 |∇u| 2 k 1 λ k ´uw k 2 ´|∇ 2n w k | 2
and apply (•) .

that the use of the second order correctors would allow to deal with artificial boundary conditions for approximating the homogenized matrix in dimension 3. In the latter application, we have to appeal to massive term approximation, defined by adding a massive term 1 T φ

(2) ij in (6.1.1). Optimal estimates on the massive term correctors are then necessary and they can be easily deduced from the optimal time decay estimates on u (2) (as we did in Chapter 2, see Corollary 5).

At first sight, in dimension 3 and under the assumption MSG (1.1.63), the gradient of second order correctors can be constructed with help of u (2) only for β > 2, which corresponds to the regime where φ can be constructed stationary itself (see Corollary 3). Indeed, this is easily inferred in the small contrast regime, that we perform hereafter. We assume that a δ = (1 + δA)Id, for a small δ 1 and for the scalar Gaussian model A = A(m ξ) ∈ R for |m(x)| ≤ (1 + |x|) -1 2 (d+β) . By linearizing around δ ↓ 0, the first order approximation of u (2) is given by u

(2) ij = δu (2) + o(δ) where ∂ τ u (2) -∇ • a∇u (2) = 0 in R d × (0, +∞), u (2) (0) = ∇ • (φ e i -σ i )e j ),

with

-∆φ e i = ∇ • Ae i and -∆σ ijk = ∂ j (e k • ∇φ e i ) -∂ k (e j • ∇φ e i ).

In the following, we only consider the part of u (2) corresponding to the initial value ∇ • (φ e i e j ) (the flux part is treated the same way) and for the ease of the reading we keep the notation u (2) . Recall that since d = 3 and β > 2, φ e i can be expressed with help of the first order semigroup u e i (see (1.2.4) with e = e i ): . Therefore, u (2) is explicitly given by: for all (T, x) ∈ (0, +∞) × R d We now apply MSG (1.1.63) to the effect of: for all q < +∞ E[(u (2) ) q ] We now consider two regimes in .

φ e i = ˆ+∞
6.1. On linear elliptic systems 6.1.2 Representative volume element method: beyond the Gaussian setting with short-range correlations and linear equations

In Chapter 5, we prove optimal estimates for the representative volume element method for approximating the homogenized coefficient. The present work is however limited to Gaussian distributions with integrable covariance, essentially for two reasons:

• Because of the way we compute the derivative d dL a L , using Price's formula of Chapter 4, which is restricted to Gaussian fields.

• Because of the way we periodize the law: we strongly use the integrability of the covariance function allowing for a straightforward periodization (see (5.1.7)).

A natural extension consists of considering other type of random distributions, based for instance on point processes or Gaussian distributions with slowly decaying correlations. A first step would be to understand the periodization in law. Such a periodization in law is immediate, for instance, in the case of Poisson point processes or i.i.d coefficients (see [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]), whereas the case of slowly decaying correlations does not allow a straightforward periodization. Once we have define the periodization in law • L , a second step would be to understand the derivative with respect to L. An alternative of the Price's formula of Chapter 4 has to be found in the non Gaussian setting.

An other possible work is to extend the results of Chapter 5 in the nonlinear setting (with the same Gaussianity of the coefficient). Such work in this direction has been done recently in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] for quadratic nonlinearity (that is for the model (1.1.4) with p = 2). In [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF], however, the proof is more in the vein of the work of Gloria, Neukamm and Otto in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF] where the authors introduce a massive term into the corrector equation in order to screen the resulting boundary layer, which leads to a logarithmically worse estimate. A possible way to avoid this loss is to adapt the strategy of Chapter 5. This would require several results, where one of them are already established in Chapter 3. More precisely:

• One needs stochastic estimates on the correctors for the periodized ensemble • L . The stochastic estimates for the first order correctors has been already established in Chapter 3, but we will also need stochastic estimates on the second order correctors (also for the linearized ones). Such estimates may be established with help of the perturbative large-scale regularity developed in Chapter 3.

• One needs to define the Green's function of the linearized operator -∇ • Da(•, ξ + ∇φ)∇ and to prove its second order two-scale expansion. To do so, the perturbative large-scale regularity theory of Chapter 3 will not be enough. However, since we prove good control of the correctors in Chapter 3, we have access to the non-perturbative regularity theory for the linearized operator, using the result in [START_REF] Bella | A Liouville theorem for elliptic systems with degenerate ergodic coefficients[END_REF], but in the regime p < 2d d-2 in dimension 3. This may allow to prove the existence of the Green's function as well as (by combining the regularity theory for the linearized operator and the stochastic control of the first and second order correctors) its second order two-scale expansion.

On nonlinear elliptic systems

Nonlinear systems with monotone coefficients

In Chapter 3, we developed a quantitative homogenization theory for the nonlinear system (1.1.4) in the full range p ≥ 2 for d = 3 and under LSI (1.1.57). Some questions and extensions, that we list hereafter, are still open.

• Does such a quantitative homogenization theory hold with other mixing conditions as finite range of dependence (1.1.56) (as addressed in [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF]5,[START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF] for p = 2) or MLSI (1.1.62)? The quenched Meyers' estimates of Theorem 18 (and its weighted version of Theorem 19) do hold for general stationary ergodic coefficients -and therefore in the setting of finite range of dependence or MLSI. They are however useless without a good control of the Meyers minimal radius (provided by Theorem 21 for Gaussian coefficients with integrable correlation). As a direct consequence of the control of the Meyers minimal radius, we obtain a control of moments of the corrector gradient (cf. Lemma 15). For finite range of dependence [START_REF] Armstrong | Higher-order linearization and regularity in nonlinear homogenization[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF], such estimates are obtained by combining a rate of convergence (any would do) for the Dirichlet problem with a Campanato argument based on C 1,α -regularity for the homogenized operator. On the one hand, it is not completely clear how to adapt the duality arguments (the Fenchel transform of a map with p-growth has p p-1 -growth -which we can also seen to some extent at the level of the stochastic integrability of ∇σ ξ in Definition 3.2.1) to prove convergence rates. On the other hand, without the a priori knowledge that ā satisfies the strong monotonicity (1.1.5), one cannot easily turn this convergence rate into a moment bound on the corrector gradient. For MLSI, the difficulty is that it does not provide the CLT scaling (see for instance (1.1.65) for β < d), which was crucial to buckle in the proof of Theorem 21. A way would be to use the powerful approach developed in the linear setting based on large-scale regularity, this time for the nonlinear equation (1.1.4). A possible first investigation may be to consider the example of Theorem 14. A possible second investigation is to consider the case of scalar equations with 2 ≤ p < 2(d-1) d-3 for which we now a C 1,α regularity theory for the homogenized operator (see the discussion in Section 3.1.1).

• In Chapter 3, we investigate the oscillations of ∇u ε that we reconstruct with help of ∇u in form of the two-scale expansion (1.1.31). An other natural question is to understand the random fluctuations of the macroscopic observables ´g • ∇u ε that qualitatively converges to ´g •∇u. In the linear setting, this question has been successfully studied by Gloria, Duerinckx, Otto, Mourrat and Gu in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF][START_REF] Gu | Scaling limit of fluctuations in stochastic homogenization[END_REF][START_REF] Duerinckx | Scaling limit of fluctuations in stochastic homogenization[END_REF] where they proved that under LSI or finite range of dependence (more general results for the Gaussian setting with non-integrable covariance has been established in [START_REF] Duerinckx | Scaling limit of fluctuations in stochastic homogenization[END_REF]) the centred rescaled observables ε -d 2 ´g • (∇u ε -E[∇u ε ]) converge in law to a Gaussian. We can even look for a finer descriptions of this convergence by means of a two-scale expansion. It was observed in [START_REF] Gu | Scaling limit of fluctuations in stochastic homogenization[END_REF] that the naive two-scale expansion which constits to replace ε -d 2 ´g • ∇u ε by ε -d 2 ´g • (1 + εφ e i ∂ x i )u does not work. The two-scale expansion has to be done in the level of the so-called homogenization commutator. The homogenization commutator, defined by Ξ ε = (a(

1 . 1

 11 Introduction to the homogenization theory . . . . . . . . . . . . . . . . . 1.1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Qualitative assumptions for a qualitative homogenization theory . . . . . . 1.1.3 The formal two-scale expansion . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Two examples of explicit homogenization . . . . . . . . . . . . . . . . . . . Example 1: A one-dimensional example . . . . . . . . . . . . . . . . . . . . Example 2: A two-dimensional example of a layered material . . . . . . . . 1.1.5 Quantitative ergodic assumptions for a quantitative homogenization theory The small contrast regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . The general framework of quantitative homogenization . . . . . . . . . . . . 1.2 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Questions arising in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 List of Preprints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields (Chapter 2) . . . . . . . . . . . . . . . . . . . . . 1.2.4 Quantitative homogenization theory for nonlinear elliptic equations and systems (Chapter 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Price's formula for infinite dimensional Gaussian measures (Chapter 4) . . . 1.2.6 Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations (Chapter 5) . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 3 :

 13 Figure 1.3: One-dimensional heterogeneous material model by resistors in series.

  d , F (• + z, a) = F (•, a(• + z, •)), we have P-almost surely 7 Br F (x, a)dx → r↑+∞ E[F ].

.1. 10 )Figure 1 . 4 :

 1014 Figure 1.4: A piece of a sample of a random checkerboard. The conductivity matrix is equal to a 0 in the white region and a 1 in the black region. We can define a random map (x, ξ) → a(x, ξ) satisfying (1.1.5) and (1.1.6) with a law satisfying (1.1.8) and (1.1.9) by setting for all z ∈ Z d and x ∈ z + [0, 1) d + O a(x, ξ) = a b(z) (s + |ξ| p-2 )ξ.(1.1.11)

20 Chapter 1

 201 . Introduction• Do we have the convergence of the sequence (u ε , ∇u ε ) ? In which topology ?

Figure 1 . 6 :Figure 1 . 7 :

 1617 Figure 1.6: A diagram of the homogenization process of (1.1.4).

  .28) which yields, together with the bound (1.1.6) of a and Cauchy-Schwarz' inequality as well as (1.1.25) (which holds in general for • ) |a(ξ 1 ) -a(ξ 2 )| (1.1.6)

.1. 42 )

 42 Chapter 1. Introduction which in turn can be upgraded to εφ ξ ( x ε ) → ε↓0 0 in L 3 ([0, 1]) and almost surely by dominated convergence. Thus u 2sc ε -u ε → ε↓0 0 in L 3 ([0, 1]) and almost surely. Second, by differentiating (1.1.41) and using (1.1.39), we have

- 1 C

 1 d (for some C > 0) and with the oscillation derivative (1.1.60) replaced by

r -d 1

 1 Br . We plug the derivatives in (1.1.62) and we use Jensen's inequality ´B (x) f (A)1 Br 2 d ´B (x) |f (A)| 2 1 Br in the regime ≤ r and Minkowski's 29 More generally, if for all x ∈ R d , |c(x)| γ(|x|) holds for some non-increasing Lipschitz functions γ : R+ → R+, then A = A(m W) satisfies MLSI/MSG (1.1.62) and (1.1.63) with weight π( ) ∼ |γ ( )|. See [60, Section 3.1].

1 1 2

 12 +2 d+1 β∧d there exists some constant C < +∞ depending on d, λ, β and α such that

Theorem 7 (

 7 Main result of Chapter 3-3). Assume that a(x, ξ) = A(x)(s + |ξ| p-2 )ξ with A(x) = b(x)Id for some scalar-valued function b and that for all R ∈ SO(d), b(R•) and b have the same (joint) distribution (in which case A is statistically isotropic). Then there exists c > 0 such that ā satisfies the strong monotonicity conditions (1.1.5) and (1.1.6) with λ = c.

Definition 1 . 2 . 1 .

 121 For given L < ∞, let • L be the stationary and centered Gaussian ensemble of scalar fields g defined by the covariance function c L (x) := k∈Z d c(x + Lk).

F

  .1.4) where for all x ∈ R d and ∈ [1, +∞)∂ fct x, F := sup lim sup h→0 (a + hδa) -F (a) h , sup B (x) |δa| ≤ 1, δa = 0 outside B (x) ,(2.1.5)and where the weight π satisfies π( ) = |γ ( )|.

Corollary 1 ( 2 ≤

 12 Decay of the semigroup).There exists a constant c < +∞ depending on λ and d such that for all T ≥ 1, R ≥ √ T and unit vector e ∈ R d ˆηR ( C ,d,λ,β (T )η β (T ),(2.1.23)

( 2 . 1 . 24 ) 1 1 2

 212412 and for some random variable T ∈ R + * → C ,d,λ,β (T ) which depends on d, λ, β and satisfies: for all α < +2 d+1 β∧d there exists some constant C < +∞ depending on d, λ, β and α such that

  sup

.1. 25 ) 1 .

 251 Remark Let us briefly comment on the results of Theorem 10 and Corollary 1.

2

 2 y)dy, and thus we have, after an integration in time and combined with a Young's inequality that sup t∈(-r 2 ,0) Br |∇v(t, y)| 2 dy ≤ Cr |∇v(s, y)| 2 dy ds + 2r 2 Cr |∇v(s, y) • ∇∂ τ v(s, y)|dy ds ≤ Cr |∇v(s, y)| 2 dy ds + r 4 Cr |∇∂ τ v(s, y)| 2 dy ds. Combining this with (2.3.2) applied with ρ = r and R = 2r yields sup t∈(-r 2 ,0) Br |∇v(t, y)| 2 dy Cr |∇v(s, y)| 2 dy ds + r 2 C 2r |∂ τ v(s, y)| 2 dy ds.

.3. 15 ) 0 =

 150 For all τ ∈ (-∞, s), we have, by testing(2.3.15) with w r ˆRd w r (τ, y)∂ τ k s (τ, y)dy -ˆRd ∇k s (τ, y) • a(y)∇w r (τ, y)dy = 0, and (2.3.8) with k s ˆRd k s (τ, y)∂ τ w r (τ, y)dy + ˆRd ∇k s (τ, y) • a(y)∇w r (τ, y)dy = 0. By summing the two identities above, integrating in time over τ ∈ [0, s) and noticing that from the initial conditions of w r and k s we have ˆRd ˆs 0 w r (τ, y)∂ τ k s (τ, y)dτ dy + ˆRd ˆs 0 k s (τ, y)∂ τ w r (τ, y)dτ dy = ˆwr (τ, y)k s (τ, y)dy s ˆ∇k s (0, y) • a(y)f r (y)edy -ˆ∇w r (s, y) • h(y)dy, we get ˆ∇w r (t, y) • h(y)dy = ˆ∇k s (0, y) • a(y)f r (y)e dy. Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields It follows by integrating over s ∈ [0, -t], using the formula (2.3.7) combined with the Cauchy-Schwarz inequality and (2.3.5) that ˆRd h(y) • ∇v r (t, y)dy ≤ Br

  .3.21) for which v r (s, •) = v r (s, •) as long as s ≤ 0. It then follows from Lemma 2 applied to the equation (2.3.21) and the large-scale C 0,1 estimate (2.4.6) (since from the inclusion B R (x) ⊂ R d \B R and the assuption

  2.43). It remains to show that C (r, x) := 1+r d 2 * (0)+ ´Rd \B 1 (x) η 1 (y)r d 2 * (ry+x)dy has the desired stretched exponential moment. From the moment bound (2.4.3) of r * and the equivalence between algebraic moments and exponential moments (see Lemma 9), we have for all p ∈ [1, +∞) and all y ∈ R d r dp * (y)

2 by r d 2 *

 22 3.43) which clearly holds by replacing g r with v T (1, •) and we bound d (0). Regime < √ T . We use the estimate (2.3.43) which clearly holds by replacing g r with v T (1, •).

  (2.1.18) of µ β . (ii) In the regime ≥ √ T we use (2.3.75), the change of variables → √ T and the fact that r ≤ √ T in the last line:

1 -

 1 .3.103) Second, we have by (2.3.102), a polar change of coordinates and the change of variables ρ → ρ r ˆ+∞

  3.129) by the definition (2.1.18) of µ β . (ii) In the regime ≥ √ T we use (2.3.123), the Minkowski inequality in L p • (Ω) and the fact that r ≤ √ T in the last line:

Substep 1 . 1 .

 11 Proof that for all ∈ [1, r) y)dt |f r (y)|dy 2 dx d M ,1 (r, ), (2.3.142) and ˆT 2

- 1

 1 3.154) by replacing ε(1 + 2 d β∧d ) by ε. We now turn to the proof of (2.3.155). First, using the moment bound (2.3.102) on r * we have ˆ+∞ r

( 2 j 2 η

 22 iii) Finally, for β ≤ 2 and d = 2, we combine (2.3.156),(2.3.157) and (2.3.158) to obtain β (1 + log 2 (r + 1)) + 1 + log(r + 1).To conclude, using the same decomposition as before and in addition the moment bound (2.3.108) of D we get ˆ+∞ r +∞ j=0 ε,β (2 j ) K p j, ,ε (r, ) 1 p p d+2 β∧d +ε(1+2 d β∧d ) χ 2 d,β (r). for correlated coefficient fields Proof of Corollary 1: Decay of the semigroup.

1 2

 1 η β (T ) = 0, which yields almost surely lim T →+∞ ˆ∇ψ(y) • a ˆT 0 ∇u(s, y)ds dy = ˆ∇ψ(y) • a ˆ+∞ 0 ∇u(s, y)ds dy.To conclude, we can pass to the limit as T ↑ +∞ in (2.3.161) and obtain almost surely ˆ∇ψ(y) • a(y)e dy + ˆ∇ψ(y) • a(y) ˆ+∞ 0 ∇u(s, y)ds dy = 0. (2.3.163) Since ´+∞ 0 ∇u(t, •)dt is curl free and belongs to L 2 (Ω × R d ), there exists a potential ζ ∈ L 2 (Ω × R d ) such that ´+∞ 0 ∇u(t, •)dt = ∇ζ and (2.3.163) takes the form ˆ∇ψ(y) • a(y) edy + ˆ∇ψ(y) • a(y)∇ζ = 0, which means that -∇ • a(∇ζ + e) = 0 in the distributional sense in R d . Since ´+∞ 0 ∇u(t, •)dt has finite second moment, it is well known that ζ own the sub-linear property. By the uniqueness of ∇φ defined by (2.1.8), this concludes the argument for (2.3.159).

  and p ∈ [1, +∞). On the one hand, by the triangle inequality combined with the Poincaré inequality in the space L 2 (R d , g 1 dx) and the stationarity property of ∇φ, we have for all

1 r

 1 2α Exc(∇u + g, r) + sup r∈[r * ,θR]1 r 2α Exc(∇u + g, r), 2.4. Appendix 139 we may absorb the second r.h.s term of the previous inequality into the l.h.s of (2.4.13), which yields sup r∈[r * ,θR] 1 r 2α Exc(∇u+g, r) sup r∈[θR,R] 1 r 2α Exc(∇u+g, r)+ sup r∈[r * ,R]

Hypothesis 3 . 1 . 1 .

 311 Let p ≥ 2, and consider the strongly monotone and non-degenerate operator a(x, ξ) := A(x)(1 + |ξ| p-2 )ξ, (3.1.9)

  λ|ξ| 2 and |A(x)ξ| ≤ |ξ|. Under Hypothesis 3.1.1, the monotone map a almost surely satisfies a(•, 0) ≡ 0, (3.1.1), and (3.1.3) (and, incidently, also (3.1.4)) for some C depending only on p and λ, so that the qualitative homogenization result of Theorem 2 applies.

Hypothesis 3 . 1 . 2 .

 312 On top Hypothesis 3.1.1, assume that A(y) = χ * B(G(y)), (3.1.10) where B : R → M d is a Lipschitz map and G is a stationary centered random Gaussian field on R d (that is, E[G] = 0) characterized by its covariance function C : R d → R, x → C(x) := E[G(x)G(0)],

Theorem 12 .

 12 Assume Hypothesis 3.1.2 and let f ∈ L p (R d ) d . Let the weight µ d : R k → R + (for k = 1 and d) be given by µ d (z) = For all ε > 0 we denote by u ε ∈ Ẇ 1,p (R d )/R the unique weak solution of (3.1.5), by ū ∈ Ẇ 1,p (R d )/R the unique weak solution of the homogenized equation (3.1.6), and by ū2s ε the two-scale expansion (3.1.11) for the choice δ = ε. If the homogenized solution ū satisfies ∇ū ∈ L

  is a connected set. Then there exists C > 0 such that ā satisfies (3.1.3) and (3.1.4). Remark 3.1.1. The assumptions of Theorem 13 are quite strong. They are satisfied in dimension d = 2 by

Theorem 14 .

 14 On top of Hypothesis 3.1.1, assume that A(x) = b(x)Id for some scalar-valued function b and that for all R ∈ SO(d), b(R•) and b have the same (joint) distribution (in which case A is statistically isotropic). Then there exists C > 0 such that ā satisfies (3.1.3) and (3.1.4).

2 . 5 )

 25 (see the proof of Theorem 12 for the precise statement). Note that a ξ φ ξ and σ ξ both scale like (|ξ| 2 + |ξ| p )

154 Chapter 3 .

 1543 Quantitative homogenization theory for nonlinear elliptic equations and systems Corollary 16 (Control of nonlinear corrector differences). Under Hypothesis 3.1.2, for all K > 0 and all ξ 1 , ξ 2 ∈ R d with |ξ 1 |, |ξ 2 | ≤ K, we have for all x ∈ R d ,

Lemma 13 .

 13 Under Hypothesis 3.1.2, for all ξ ∈ R d set a ξ := Da(•, ξ + ∇φ ξ ). For all e ∈ R d there exists a unique random field φξ,e that solves almost surely in the distributional sense in R d the linearized corrector equation -∇ • a ξ (e + ∇ φξ,e ) = 0, (3.2.9)

. 10 ) 1 ( 1 + 1 ,

 10111 which is anchored via ´B σξ,e = 0 almost surely, whose gradient ∇σ ξ,e is stationary and is bounded in the senseE |∇σ ξ,e | p p-|ξ| p-2 )and which satisfies the property ∇ • σξ,e = a ξ (e + ∇ φξ,e ) -āξ e,

3. 2 . 155 Theorem 17 .

 215517 Strategy of the proof, extensions, and limitations Under Hypothesis 3.1.2, for all ξ, e ∈ R d with |e| = 1, the stationary extended linearized corrector gradient ∇( φξ,e , σξ,e ) satisfies for some exponent α ξ > 0 and some constant c ξ > 0 depending on λ, p, d, and |ξ|, E exp(c ξ |∇( φξ,e , σξ,e )| α ξ ) ≤ 2.(3.2.11)For all g ∈ L 2 (R d ), averages of (∇ φξ,e , ∇σ ξ,e ) display the CLT scaling in the form ˆg(∇ φξ,e , ∇σ ξ,e ) ≤ C ξ,g ˆ|g| 2

so that the diagram of Figure 1 . 1 .

 11 2 takes the equivalent form of Figure 3.1.

Figure 3 . 1 :

 31 Figure 3.1: Reformulation of the commutative diagram

Lemma 14 (

 14 Well posedness of r ,ξ,L ). Let (x, ξ) ∈ R d × R d and L ≥ 1. There exist two constants c 1 , c 2 > 0 depending on p and d such that, P L -almost surely, r ,ξ,L satisfies r ,ξ,L (x, c 2 ) ≤ r ,ξ,L (x, c 1 ) ≤ r ,ξ,L (x, c 1 ), (3.3.3) and r ,ξ,L (x, c 1 ) ≤ L. (3.3.4)

√ d 2 .

 2 By (3.7.62),(3.7.63), the Hölder inequality with exponents ( m, m m-1 ) (with m as in Theorem 18) and the unweighted Meyers estimate (3.3.10), we have with β

  and the hole filling estimate (3.3.25) with exponent δ > 0 yields in combination with the plain energy estimate ´QL |∇v| 2 µ ξ ´QL |h| 2 and the assumption ´QL |h| 2 = 1

.3. 30 )

 30 The claim(3.3.26) now follows from (3.3.27), (3.3.29) and (3.3.30).

2 .

 2 The claim (3.3.32) then follows from (3.3.33), (3.3.34), (3.3.36), and (3.3.37), and the identity |Q k | = (2 j-3 R) d .

m 2

 2 dx. (3.3.47) Controlling the last right-hand side of (3.3.42) the same way, (3.3.23) follows from (3.3.43) and (3.3.47).

.3. 55 ) 1 R 2 p + α 1 + 2 p

 551212 Indeed, by Jensen's inequality, with the short-hand notation α := inf η∈R ffl B R p |φ ξ (x) -η| p , (3.3.54) yields α |ξ| p so that α 1, which implies α α , whence the reformulation(3.3.55).

.3. 65 )

 65 Then lim h↓0 δ h F = δF = ´g • ∇δφ ξ , which we now rewrite by duality. Testing (3.3.60) with δφ ξ and then(3.3.65) with u, we obtainδF = ˆ∇u • δAa • (ξ + ∇φ ξ ),and the claim (3.3.59) follows by taking the supremum over δA. It remains to argue in favor of the convergence of δ h φ ξ to δφ ξ , which actually holds in C 1,α (Q L ). First, recall that {φ ξ (A + hδA)} h is a bounded set in C 1,α (Q L ) by Lemma 25. By testing (3.3.63) with hδ h φ ξ , we obtain by monotonicity

.3. 66 )

 66 Since E L [∇φ ξ ] = 0, by (3.7.44) and (3.3.59), we have for all q ≥ 1

m- 2 = d + 1 .

 21 3.50), and set m := 2 + 2ε d+1 , to the effect that m ≤ m and 2ε Since r is 1 16 -Lipschitz and r ≥ 1, we have ˆQL r (x) d-δ B(x) |∇u| 2 µ ξ dx ˆQL B(x) r d-δ |∇u| 2 µ ξ dx, so that by (3.7.65) combined with the estimate r ≤ L, with periodicity, and using again the Lipschitz-continuity of r , we obtain ˆQL r (x) d-δ B(x)

  .3.71) The claim (3.3.49) then follows from (3.3.68), (3.3.69) and (3.3.71).

Theorem 22 (

 22 Annealed Meyers' estimate). Under Hypothesis 3.1.2, for all ξ ∈ R d , with κ := ( m-2)∧1 8

180 Chapter 3 .

 3 Quantitative homogenization theory for nonlinear elliptic equations and systems By the 1 16 -Lipschitz property of r , we have the implication for all z ∈ B (y) |y -x| ≥ 2r D =⇒ |z -x| ≥ |y -x| -r (y) ≥ |y -x| -(r (x) + 1 16 |y -x|) ≥ 15 16 |y -x| -r (x) ≥ ( 15 8 -1 2 )r D ≥ r D =⇒ 1 D (z) = 0, so that (3.3.74) follows for r D ≥ 2r (x) in the stronger form ˆD E L B (y)

  Perturbative regularity theory for the linearized operator 181 from which (3.3.75) follows.

3. 4 . 15 183

 415 Control of correctors: Proof of Theorem Proof of Theorem 15. By (3.4.2) and Theorem 21, assumption (3.7.49) in Proposition 10 is satisfied for ∇φ ξ . Let us show that this also yields assumption (3.7.49) for ∇σ ξ . Indeed, by maximal regularity for the Laplacian applied to equation (3.2.2) we have for all q > 1, ´QL |∇σ ξ | q 1 q

Substep 2 . 1 .

 21 Sensitivity calculus.Recall the defining equation for σ ξ,ij

and where u 1

 1 , u 2 ∈ H 1 per (Q L ) are the unique weak solutions of -∇ • a * ξ ∇u 1 = ∇ • g (3.5.8)and (with an implicit sum over the repeated index k)-∇ • a * ξ ∇u 2 = ∂ k (D 2 a(ξ + ∇φ)(e + ∇ φ)e k • ∇u 1 ). (3.5.9) (These equations are well-posed since the Q L -periodic maps ∇φ and a * ξ are bounded almost surely.) Let us give the quick argument. Following Step 1 of the proof of Proposition 5, we let δA be an increment of A localized in B(x) and consider for h small enough

  .10) with b ξ := D 2 a(ξ + ∇φ).We now proceed by duality. First, we test (3.5.10) with u 1 and (3.5.8) with δ φ to obtainδF 1 = lim h↓0 δ h F 1 = ˆQL ∇u 1 • δADa • (ξ + ∇φ)(e + ∇ φ) + ˆQL ∇u 1 • b ξ ∇δφ(e + ∇ φ). (3.5.11) Second, we test (3.3.65) with u 2 and (3.5.9) with δφ to get ˆQL ∇u 1 • b ξ ∇δφ(e + ∇ φ) = ˆQL ∇u 2 • δAa • (ξ + ∇φ).(3.5.12)

3. 5 .

 5 Control of corrector differences: Proof of Theorem 17 191

3. 5 .

 5 Control of corrector differences: Proof of Theorem 17 193 the definition of η 1 and η 2 to the effect that ˆQL E L sup B(x)

  B R |∇ φξ,e | 2 µ ξ by the growth of ffl B 2R µ ξ . Definition 3.5.1 (Linear minimal scale). Let ξ ∈ R d , L ≥ 1, |e| = 1 and C > 0. For all x ∈ Q L , we define the linear minimal scale r ,ξ,e,L (x, C) via r ,ξ,e,L (x, C) := inf r=2 N ,N ∈N ∀R ≥ r : B R |∇ φξ,e | 2 µ ξ ≤ C B 2R µ ξ . (3.5.19) As for the Meyers minimal radius, r ,ξ,e,L (•, C) is bounded by L as soon as C is large enough, due to periodicity and to the plain energy estimates for φξ,e in form of ´QL |∇ φξ,e | 2 µ ξ ´QL µ ξ . In what follows we fix such a constant C, fix e and ξ, and use the short-hand notation r for r ,ξ,e,L (•, C), r for r ,ξ,L , φ for φ ξ , and φ for φξ,e . The upcoming lemma uses local regularity and hole-filling to control sup B |∇ φ + e| 2 µ ξ by r and r . Lemma 22 (Quenched bounds on the linearized correctors). For all ξ ∈ R d , there exist two exponents 0 < β ≤ d (the linear hole-filling exponent of Lemma 15) and γ > 0 and a non-negative stationary random field χ (depending on r , A C 0,α (R d ) and |ξ|) with the following properties: For all x ∈ R d sup B(x) |e + ∇ φ| 2 µ ξ ≤ χ(x)(r (x)) d-β , (3.5.20)

.5. 21 ) 194 Chapter 3 .

 211943 Quantitative homogenization theory for nonlinear elliptic equations and systems

2 |ξ + ∇φ| p 1 p ( 3 . 3 . 5 ) 23 )

 2133523 .5.22) for some constant C > 0 depending on d, p, A C 0,α (R d ) , and |ξ|, where 0 < δ ≤ d is the nonlinear hole-filling exponent of Lemma 15. (We recall thatX C α (B) = X L ∞ (B) + X C 0,α (B) .)On the one hand, by Lemma 25 applied to the equation (3.1.7) combined with the estimate (3.3.5), we haveξ + ∇φ C α (B) A C 0,α (R d ) B ≤ A C 0,α (R d ) (1 + |ξ|)rOn the other hand, recall thata ξ = ADa • (ξ + ∇φ) with a • : ζ ∈ R d → (1 + |ζ| p-2 )ζ,and thus for all ζ ∈ R d |Da • (ζ)| 1 + |ζ| p-2 and |D 2 a • (ζ)| 1 + |ζ| p-3 . (3.5.24) Therefore, by (3.5.23) and (3.5.24),

,

  which yields(3.5.20) for χ := Cr 2(γ+1)(d-δ) p-2 p +β (for some constant C > 0 depending on d, p, |ξ| and A C 0,α (R d ) ). The claimed moment bounds on χ follow from Theorem 21 (for a suitable γ > 0). systems

3. 6 .

 6 Quantitative two-scale expansion: Proof of Theorem 12 197

  Without loss of generality, we may assume thatx = 0. Let η ∈ C ∞ c (R d ) be a standard cut-off for B c 1 r in B c 2 rand set ζ 2 = η p to the effect that 2ζ∇ζ = pη p-1 ∇η. By testing the equation (3.7.1) with ζ 2 (u -c) and by making use of the monotonicity (3.1.4) of a and the property a(•, 0) = 0 in form of ´ζ2 ∇u • a(•, ∇u) ´ζ2 |∇u| 2 (1 + |∇u| p-2 ), we have ˆζ2 |∇u| 2 (1 + |∇u| p-2 ) ˆ|ζ||∇ζ||u -c||∇u| + ˆ|ζ||∇ζ||u -c||∇u| p-1 .

3

 3 Linear elliptic systems: Caccioppoli and Lemma 17

202 Chapter 3 .

 3 Quantitative homogenization theory for nonlinear elliptic equations and systems

27 (

 27 Differentiability of correctors). For all D the corrector gradients ξ → (∇σ ξ , ∇φ ξ ) are Fréchet-differentiable in C 0 (D) and for all directions e ∈ R d we have(∂ ξ ∇φ ξ • e, ∂ ξ ∇σ ξ • e) = (∇ φξ,e , ∇σ ξ,e ),where ( φξ,e , σξ,e ) solve (3.2.9) and (3.2.10) with homogeneous Dirichlet boundary conditions (resp. Q L -periodic with vanishing average).

D 1 +

 1 Da(•, ξ + ∇φ ξ+he + (1 -t)he)dt. Next, for all ξ ∈ R d , using local regularity (this time up to the boundary) in form of Lemma 25 (applied to the equation (3.1.7)) and an energy estimate on D, we have ξ + ∇φ ξ C α (D) D, A C α (D) ˆD | ξ + ∇φ ξ | p 1 p | ξ| p . (3.7.13) Therefore, a

1 . 1 .

 11 Proof of (3.7.19) for |ξ 1 |, |ξ 2 | Substep 1.1. Reformulation. By Lemma 27, correctors are differentiable and we thus have for all ξ, e ∈ R d e • Dā(ξ)e = Q (e + ∇ φξ,e ) • a ξ (e + ∇ φξ,e ),

.7. 20 ) 1 . 1 s

 2011 Since Dā is non-negative, the claim (3.7.19) follows for |ξ 1 |, |ξ 2 | 1 provided we prove that e • Dā(ξ)e ≥ c|ξ| p-2 for all |ξ| 1 and all e ∈ R d with |e| = Fix such a direction e. For all s > 0 let ξ s ∈ R d be such that |ξ s | = 1 and e • Dā(sξ s )e = inf |ξ|=1 e • Dā(ξ)e, which exists since ξ → Dā(ξ) is continuous. In the following two substeps we prove the needed estimate in form of lim inf s↑∞ e • 1 s p-2 Dā(sξ s )e > 0. (3.7.21) Substep 1.2. Proof of lim s↑∞ p-2 a sξs -b ξs C α (Q) = 0, (3.7.22) 3.7. Appendix 205 where b ξ : x → Db(x, ξ + ∇ψ ξ (x)).

  7.23) by Arzela-Ascoli's theorem. To this end we rewrite the corrector equation for χ s := 1 s φ sξs as -∇ • ãs (x, ξ s + ∇χ s ) = 0 with ãs (x, ξ) := b(x, ξ) + 1 s p-1 c(x, sξ). Since by assumption, ãs satisfies (3.2.19

Remark 5 .

 5 The strong assumption on the critical set of ψ ξ is solely used to ensure that lim inf s↑+∞ ˆQ(e + ∇ φsξs,e ) • 1 s p-2 a sym sξs (e + ∇ φsξs,e ) ≥ inf φ∈H 1 per (Q) ˆQ(e + ∇ φ) • b sym ξ∞ (e + ∇ φ),

  .7.27) Note that the example of Theorem 14 satisfies the assumptions of Theorem 25. The proof of Theorem 25 relies on an approximation argument, an ODE argument, and the following property. Lemma 28. Let a • : R d → R d be such that for all ξ ∈ R d a • (ξ) = ρ(|ξ|)ξ, (3.7.28)where ρ : R + → R + is a differentiable function that satisfies for some constant constant c > 0 and for all t > 0 d dt (tρ(t)) ≥ c(1 + t p )

  Likewise, there exists W : R d → R such that ā(ξ) = D W (ξ). For all L ≥ 1, we denote by φ L ξ the unique weak solution in W 1,p 0 (B L ) of (3.1.7), and definea L (ξ) := B L a(x, ξ + ∇φ L ξ (x))dx, W L (ξ) := B L W (x, ξ + ∇φ L ξ (x))dx,which satisfy a L (ξ) = DW L (ξ). As a direct consequence of the homogenization result we have almost-surely lim L↑+∞ (a L (ξ), W L (ξ)) = (ā(ξ), W (ξ)). Set āL (ξ) := E a L (ξ) and W L (ξ) := E W L (ξ) . Since we have the uniform almost sure bound |a L (ξ)| + |W L (ξ)| ≤ C(1 + |ξ| p ), we obtain on the one hand that āL (ξ) = D W L (ξ) and on the other hand (by dominated convergence) that lim L↑+∞ (ā L (ξ), W L (ξ)) = (ā(ξ), W (ξ)).

.7. 47 ) 5

 475 The fact one can assume δA C α (B(x)) ≤ 1 comes from the convolution in (3.1.10).

3. 7 .

 7 Appendix 213 Proof. Let R ≥ 1 and ε

  7.56) to ζ = (φ ξ , σ ξ ) -(φ L ξ , σ L ξ ) and using the anchoring condition ffl B (φ ξ , σ ξ ) = 0, for all x ∈ R d we have with R = 2(|x| + 1)

58 ) 8 B

 588 The estimate (3.7.57) follows from sup x 0 ∈Br B (x 0 ) Let x 0 ∈ B r be fixed. Since r ,ξ,L is1 16 -Lipschitz, we have for all x ∈ B r (x 0 ) ⊂ B r (x 0 )+ 9 8 r (x).(3.7.59)

  i ) i∈ 1,N its canonical basis. Then, b admits the representation

µ

  nm ċ(e * m , e * n ). (4.1.11)

  1.6) to (4.1.11) hold here. To do so, we fix an orthonormal basis (h k ) k∈N of H. By definition (4.1.6), b. ċ = +∞ k=0 b(h k , Ċh k ), which turns into, using (4.3.4), b. ċ = +∞ k=0 +∞ n=0 λ n n (h k ) n ( Ċh k ). (4.3.5)

3 . 2 )

 32 by noticing that A * Af n is the Riesz representation of n in H by the assumption (4.3.1).

λ

  n ˆˆh n (x)h n (y)g(x)g(y)dµ(x)dµ(y). (4.3.13) In addition, the function k : (x, y) → +∞ n=0 λ n h n (x)h n (y) (4.3.14)

∂ 2 F 3 )

 23 ∂g(-x)∂g(z-x) L = ∂ 2 [ξ * •(a(∇φ (1) +ξ)(x)]With help of the corrector for the (pointwise) dual 5 coefficient field a * in direction ξ * , i. e. the periodic solution φ *(1) of∇ • a * (∇φ * (1) + ξ * ) = 0,(5.2.4)the inner integral can be rewritten more symmetrically asˆRd ξ * • ∂ 2 [a(∇φ (1) + ξ)] ∂g(0)∂g(z) = ˆRd (∇φ * (1) + ξ * ) • ∂ 2 [a(∇φ (1) + ξ)] ∂g(0)∂g(z) = ˆRd (∇φ * (1) + ξ * ) • [ ∂ 2 ∂g(0)∂g(z)

∂ 2 ∂g

 2 (0)∂g(z) , a] by Leibniz' rule we obtain ˆRd ξ * • ∂ 2 [a(∇φ(1) + ξ)] ∂g(0)∂g(z)= ˆRd (∇φ * (1) + ξ * ) • ∂ 2 a ∂g(0)∂g(z)(∇φ(1) + ξ)+ ˆRd (∇φ * (1) + ξ * ) • ∂a ∂g(0)∇ ∂φ(1) ∂g(z) + term with z, 0 exchanged.Denoting a := A (g) and a := A (g), we remark that by (5.1.5) we have ∂a(x) ∂g(z) = a (z)δ(x -z). Applying operator ∂ ∂g(z) on (5.1.2), we thus obtain the representation∂∇φ (1) (x) ∂g(z) = -∇∇G(x, z)a (z)(∇φ + e)(z) (5.2.5)in terms of the mixed derivatives of the non-periodic Green function 6 G = G(a, x, y) associated with the operator -∇ • a∇. Hence the above turns intoˆRd ξ * • ∂ 2 a(∇φ (1) + ξ) ∂g(0)∂g(z) = δ(z) (∇φ * (1) + ξ * ) • a (∇φ (1) + ξ) (0) -(∇φ * (1) + ξ * ) • a (0) • ∇∇G(0, z) a (∇φ (1) + ξ) (z) -(∇φ * (1) + ξ * ) • a (z) • ∇∇G(z, 0) a (∇φ (1) + ξ) (0).Applying • L we obtain by stationarity ˆRd ξ * • ∂ 2 a(∇φ(1) + ξ) ∂g(0)∂g(z)L = δ(z) (∇φ * (1) + ξ * ) • a (∇φ (1) + ξ) L -(∇φ * (1) + ξ * ) • a (0) • ∇∇G(0, z) a (∇φ (1) + ξ) (z) L -(∇φ * (1) + ξ * ) • a (0) • ∇∇G(0, -z) a (∇φ (1) + ξ) (-z) L .(5.2.6)Inserting this into (5.2.3), and noting that since ∂c L ∂L , as derivative of a covariance function is even, the two last terms have the same contribution, we obtaind dL ξ * • āξ L = -ˆRd dz (∇φ * (1) + ξ * ) • a (0)• ∇∇G(0, z) a (∇φ(1) + ξ) (z) * (1) + ξ * ) • a (∇φ(1) + ξ) L ∂c L ∂L (0).

( 5 . 2 . 7 ) 30 .

 52730 Theorem Let d > 2 and A be symmetric. Suppose • satisfies Assumption 28 and let a hom denote the homogenized coefficient . For all L, let • L be as in Definition 5.1.1, ā be defined by (5.1.3), Γ hom,T defined by(5.2.18), and Q L be as in Definition 5.2.1. Then the following limits exist:Γ hom,ijmn := lim T ↑∞ Γ hom,T ijmn , (5.2.22) Q ijm (z) := lim L↑∞ Q Lijm (z) pointwise, uniformly bounded in z,(5.2.23)and the latter only depends on • (and not the lattice). Moreover, we havelim L↑∞ L d+1 d ā L dL = Γ hom,ijmn ˆRd dzQ ijm (z)∂ n c(z). (5.2.24)With the tools of this paper, the asymptotics of d ā L dL could be characterized up to order O(L -d-d 2 ). Let us comment on the representation of the leading error term arising from (5.2.24), namely d lim L↑0 L d a hom -ā L = Γ hom,ijmn ˆRd dzQ ijm (z)∂ n c(z). (5.2.25)

  may be neglected w. r. t. e i ; likewise φ(1) i e m + ∇φ (2)im may be neglected w. r. t. e i . Hence to leading order, (5.2.21) reduces toξ * • Q ijm (z)ξ ≈ -z m ξ * • a (0)e i e j • a (z)ξ .

Q

  ijm (z)∂ n c(z) ≈ -z m ∂ n A(c(z)) e i ⊗ e j .Normalizing A such that A(0) = 0, we obtain by integration by parts ˆRd dzQ ijm (z)∂ n c(z) ≈ δ mn ˆRd dzA(c(z)) e i ⊗ e j . ensemble instead of its realizations Hence the r. h. s. of (5.2.25) is given by limT ↑∞ k∈Z d k m ∂ m ∇ 2 G T,hom(k)ˆRd dzA(c(z)) (5.2.26)

  m ∂ m G T,hom (k) = 0. (5.2.28) By scaling, we have G T,hom (k) = 1 √ T d-2 G 1,hom ( k √ T ).Hence we see that the sum in (5.2.28) can be interpreted as a Riemann sum that in the limit T ↑ ∞ converges to the integralˆRd dkk m ∂ m G 1,hom (k) = -d ˆRd dkG 1,hom (k),which indeed does not vanish.

  |G T (a, x, y)| + |x -y||G T (a, x, y)| + |x -y| 2 |∇∇G T (a, x, y)| ≤ C(a)|x -y| -d+2 e -|x-y| √ T , (5.3.1) where the constant C depends at most polynomially on the Hölder constant [a] α ,B 1 (x) (next to depending on d and λ). In view of Assumption 28 (and Definition 5.1.1), which for α < α ensures Hölder continuous realizations g and thus a, any stochastic moment of [a] α ,B 1 (x) is finite, that is for all p < +∞ [a] p α ,B 1 (x) L 1, (5.3.2)

T

  ) p C 0,α ([0,L) d ) L p,L 1,(5.3.4)where• C 0,α ([0,L) d ) = • L ∞ ([0,L) d ) + [•] α ,[0,L) d .We now can rigorously establish the massive version of formula (5.2.7):Proposition 11. It holdsd dL ξ * • āT ξ L = ˆRd dz (∇φ * (1) T + ξ * )(0) • a (0)× ∇∇G T (0, z)a (z)(∇φ

  2.20). Then we haved dL ξ * • āξ L = L -(d+1) ˆRd dz Γ ijmn ξ * • Q (2) ijm (z)ξ -z m ξ * • Q (1) ij (z)ξ L ∂ n c(z) ij (z)ξ L ∂ n c(z) + ˆRd dz(1 -η L )(z) (∇φ * (1) + ξ * )(0) • a (0) × E(0, z)a (z)(∇φ (1) + ξ)(z) L ∂c L ∂L (z) + ˆRd dzη L (z) (∇φ * (1) + ξ * )(0) • a (0)× ∇∇G(0, z)a (z)(∇φ(1) + ξ)(z) L ∂c L ∂L (* (1) + ξ * ) • a (∇φ(1) + ξ) L ∂c L ∂L (0).(5.3.10)

  |E(x, y)| ≤ C(a)L 2 |y -x| -d-2 , where the constant C depends at most polynomially on the non-dimensionalized Hölder constant L α [a] α (next to depending on d and λ). In particular we have, form (5.3.2) sup x,y |y -x| d+2 |E(x, y)| L < ∞.

1 1 |∇u| 2 1 2 ,Corollary 32 .R 2

 112322 for all p < ∞. On the other hand, by standard Schauder theory in C α we have |∇u(0)| ≤ C(a) B where C depends at most polynomially on the local Hölder norm [a] α ,B 1 . Since α < α, we obtain from Assumption 28 that [a] p α ,B 1 1 p L < ∞ for all p < ∞. Noe (5.3.43) follows from combining both estimates; note that the loss in stochastic integrability is unavoidable, since it compensates the fact that both r * and [a] α are not uniformly bounded. As mentioned, we use Lemma 40 only in its form combined with Lemma 41 Let d > 2 and • satisfy Assumptions 28; let • L be defined as in Definition 5.1.1. Let the random function u be a-harmonic in the ball B R of radius R. Then we have for all p , p < ∞ |∇u(0)| p sup |∇ 2 g| provided p < p. (5.3.44) Here p,p has the same meaning as in Proposition 13.

5. 4 . 1

 41 Proof of Proposition 11: Derivative in LWe prove(5.3.5) by an approximation argument on the flux a(∇φ T + e) to which we apply the version of Price's formula (4.3.17) of Corollary 27 of Chapter 4. We finally let the approximation parameter goes to +∞.

Fix

  z ∈ R d and define F (•, z) : a ∈ X → ξ * • (a(∇φ

( 1 +

 1 3.3) and where X := a [a] γ,α := sup x∈R d |x| 2 ) -γ sup y:|y-x|≤1 |a(y) -a(x)| |y -x| α + |a(x)| < +∞ , Formula (5.3.5) established in Proposition 11

  1 -η L )∂ ij G T )(z + kL) -∂ ij G T (kL))k n L ∂ n c(z). (5.4.21)Next, we reorganize the above r. h. s. term into brackets by means of a Taylor-like expansionk =0 (((1 -η L )∂ ij G T )(z + kL) -∂ ij G T (kL))k n = z m k =0 ∂ ijm G T (kL)k n + (1)T Lijmn (z), ensemble instead of its realizations (which defines[START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] T Lijmn ) so that (5.4.21) readsΠ (1) L,T = ˆdz ξ * • Q (1) T ij (z)ξ z m k =0 ∂ ijm G T (kL)k n + (1)T Lijmn (z) L ∂ n c(z).

  4.22) and(5.4.23), we may use a scaling argument and replace∂ ijm G T (kL) = L -d-1 ∂ ijm G T /L 2 (k).(5.4.24)As a conclusion, combining (5.3.5),(5.4.19),(5.4.22), (5.4.23) and (5.4.24), yields the far-field contribution and therefore (5.4.17).

  3.7) and ( •L -almost-surely) Γ ijmn ≤ sup T ≥1 |Γ T ijmn | 1.

5. 4 . 4 Lemma 42 . 1 ,

 44421 Proof of Corollary 31: Limit L ↑ ∞The proof of Corollary 31 rest on a coupling procedure, which jointly builds Gaussian fields of covariance functions c and c L that become closer when L ↑ ∞. More precisely: Under the assumption 28, there exists a family of coupling ensembles • Ω L indexed by L ∈ [1, +∞) between the ensembles • and • L , that generate Gaussian fields g L and g of covariance functions c L and c, respectively, and such that the following convergence holds, for any R ≥

48 ) 2 N d √ C -C L 2 N

 4822 we recall that • C 0,α = sup| • | + [•] α . We introduce the discretization points x i := i/ √ N for i ∈ Z d , the Gaussian vectors G N , and the Gaussian field g N built from interpolating linearly w. r. t. the points x i with 14G N i := g(x i ) fori ∈ {-N, • • • , N } d and g N (x i ) := G N i for i ∈ {-N, • • • , N } d , 0 fori ∈ Z d \{-N, • • • , N } d .(5.4.47)It is sufficient to build the coupling on the level of the Gaussian vectors G N for N sufficiently large, since(5.4.46) implies that, if √We denote by C and C L the covariance matrices associated with G N under • and • L defined by:C ij := c (x i -x j ) and (C L ) ij := c L (x i -x j ) for i, j ∈ {-N, • • • , N } d . Now,by [53, Th.], there exists an ensemble • ΩL generating two Gaussian vectors G N and G N L of covariance matrices C and C L , respectively, such that the euclidean distance G N -G N L 2 is minimal in the following sense: norm • and the fact that the function √ • is 1/2-Hölder continuous, we easily get the following string of inequalities trace √ C -C L 2 ≤ (2N + 1) d √ C -C L d C -C L N 2d sup i,j{-N,••• ,N } d C ij -(C L ) ij .

u

  Ln n↑∞ u ∞ weakly in L 2 (Ω × B R ), ∇u Ln n↑∞ ∇u ∞ weakly in L 2 (Ω × B R ).(5.4.62)

Step 3 :

 3 in m ∈ N. Hence, it follows from the Borel-Cantelli lemma that for any ν > 0, so that u ∞ satisfies(5.4.58).Recall that the solution to (5.4.55) with the condition (5.4.59) is unique up to the addition of a (random) constant (see the Liouville theorem [80, Cor. 1]). Whence ∇u ∞ = ∇u. Since we already know that ∇u L ∇u in L 2 (Ω × B R ), then by ellipticity of a, it suffices to show the conservation of the norm lim sup L↑∞ ˆBR ∇u L • a∇u L Ω ≤ ˆBR ∇u • a∇u Ω (5.4.64) to get the strong convergence ∇u L → ∇u in L 2 (Ω × B R ). By the strong convergence (5.4.61) of a L and by (5.4.56), we first obtain that lim sup L↑∞ ˆBR ∇u L • a∇u L Ω ≤ lim sup L↑∞ ˆBR ∇u L • a L ∇u L Ω . 5.4. Proofs 267 Then, making use of the coupling properties • Ω and (5.4.55) yields ∇u L • a L ∇u L Ω = ∇u • a∇u L (5.4.55)= -∇u • f L = -∇u L • f L Ω .(The third above equality is due to the stationarity of f and u under • L .) Whencelim sup L↑∞ ˆBR ∇u L • a∇u L Ω ≤ lim sup L↑∞ -ˆBR ∇u L • f L Ω = -ˆBR ∇u • f Ω ,where we appealed to the strong convergence of f L given by (5.4.59) and the weak convergence of ∇u L . Using once more the coupling properties of • Ω and (5.4.55) yields-∇u • f Ω = -∇u • f = ∇u • a∇u = ∇u • a∇u Ω .

5 . 4 . 5

 545 L ≥ R. Since, by a Poincaré inequality combined with a triangle inequality, there holdsφ (1) -φ (1) L 2 L 2 (B R ) L d R 2 ∇φ (1) -∇φ (1) L 2 L 2 (B R ) Ω L + Proof ofLemma 40: Improved Caccioppoli inequality By scaling, it is enough to consider R = 1; we fix a smooth cut-off function η for B 1 in B 2 . Starting point is the following localized version of a standard L 2 -based interpolation estimate, with n ∈ N to be fixed later, which we take from the proof of [29, Lemma 4]: ˆ(η 4n 2n w) 2 ˆ|η 4n+1 ∇ 2n w| 2 4n 4n+1 ˆw2 1 4n+1 + ˆw2 . (5.4.74) We apply it to w ∈ H 2n 0 (B 2 ) (as usual, H 2n 0 (B 2 ) denotes the closure of C ∞ 0 (B 2 ) w. r. t. to the H 2n (B 2 )-norm) that (weakly) solves 2n w = u in B 2 .

p 2 .´|∇ 2n w k | 2 p 2

 22 By Hölder's inequality in k we obtain ˆB1 |∇u|

0u

  e i (s, •)ds = -ˆ+∞ 0 ˆ∇Γ(s, • -y) • A(y)e i dy ds, where Γ(s, x) = (4πs) -d 2 e -|x| 2 4s

u ( 2 ) 1 π d 2 Γ 0 ∂ 0 ∂

 21200 (T, x) = -ˆ∇Γ(T, x -y) • φ e i e j dy = -ˆ+∞ 0 ˆˆ∂ j Γ(T, x -y)∂ i Γ(s, y -y )A(z)dy dy ds.By differentiating with respect to the randomness (in the sense of (1.1.61)) and using the semigroup property of Gaussian kernels in form of Γ(T,•) Γ(s, •) = (T + s, •), we have for all (z, ) ∈ R d × (1, +∞) ∂ fct z, u (2) (T, x) i ∂ j Γ(T, •) Γ(s, •)(x -y)ds dy i ∂ j Γ(T + s, x -y)ds dy.

0 ∂ 2 dz d 1 2 .

 022 i ∂ j Γ(T + s, x -y)ds dy (6.1.2)

  For the far-field contribution, one has to make use of stochastic cancellations, and this is here that our quantitative assumption (1.1.50) plays a important role. Using (1.1.50), we may express the second moment of φ f -φ f (0) as:26

1.1.52)) and E[|g(0)| 2 ] = 1, to obtain: for all x ∈ R d 32 Chapter 1. Introduction

  2 can be upgraded to Cq if MLSI (1.1.62) is satisfied. Multiscale functional inequalities allow us to consider long-range correlations, compared to standard functional inequalities of Definition 2. For instance, the Gaussian based model (1.1.13) with β ≤ d satisfies MLSI (1.1.62) with weight π( ) = -1-β and where the functional derivative (1.1.61) has to be replaced by

  1. N. Clozeau. Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields, Arxiv Preprint, arXiv:2102.07452, 2021. 2. N. Clozeau and A. Gloria. Quantitative nonlinear homogenization: control of oscillations, Arxiv Preprint, arXiv:2104.04263v2, 2021.

3. N. Clozeau, M. Josien, F. Otto and Q. Xu. Bias in the representative volume element method: periodize the ensemble instead of its realizations, In preparation, 2021. 1.2.3 Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields (Chapter 2)

  Let d > 2 and A be symmetric. Under the Assumption 8 on • , for all L, and with the Definition 1.2.1 of • L we have for the expectation ā L of ā defined in(5.1.3)

	1.2.19)
	Clearly, the covariance function c L and thus the realizations g are L-periodic. As in Assumption 8
	we identify • L with its push forward under (1.2.18).
	Equipped with the ensemble • L , we can state our main result.
	Theorem 9.

  On the other hand, in the case where β = 2 and d ≥ 2, we split the variance into the far-field contribution ´Rd \Br dx and the near-field contribution ´Br dx where, for each contributions, we bound x → ´γm (y)(1 + |x -y|) -2 dy. For the far-field contribution, we use ˆγm (y)(1 + |x -y|) -2 dy =

	ˆB |x|	(x)	γ m (y)(1 + |x -y|) -2 dy +	ˆRd \B |x|	(x)	γ m (y)(1 + |x -y|) -2 dy
	2 (2.1.29)	1 (|x| + 1) d-1	ˆB |x|	2 (|y| + 1) -2 dy + (|x| + 1) -2 ˆ|γ m |
							2
	(2.1.29)
							x)e k ds dτ
			ˆr2	ˆr2
	×		∂ ijm Γ(s + τ , y)e m ds dτ (1 + |x -y|) -β dx dy.
				1	0
							(2.1.28)
	In what follows, we use the short-hand notation γ m := note that a direct computation gives	´r2 1	´r2 0 ∂ ijm Γ(s + τ , •)e m ds dτ and
	ˆ|γ m | r and |γ m (y)|		r (|y| + 1) d ∧	1 (|y| + 1) d-1 .	(2.1.29)
	On the one hand, in the case where β > 2 and d = 2, x → (1 + |x|) -β is integrable and thus
	by the Cauchy-Schwarz inequality the variance (2.1.28) is of order of
	ˆˆ|γ k (y)| 2 (1 + |x -y|) -β dx dy	(2.1.29)	ˆBr	1 (|y| + 1) 2 dy + r 2	ˆRd \Br	1 (|y| + 1) 4 dy
					∼ log(r + 2).

  Corollary 2 (Bounds on the flux and the gradient of correctors). We have for all r ≥ 1, T ≥ 1 and unit vector e ∈ R d|(q e ) r -(q e ) r | + |(q e,T ) r -(q e,T ) r | + |∇(φ e ) r | + |∇σ r | + |∇(σ T ) r | + |∇(φ e,T ) r | ≤ C ,d,λ,β (r)π

	Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with
					for correlated coefficient fields
	to the decay (2.1.23), the idea of the proof is clear: since lim T →+∞	u e (T, •) = 0 and	´+∞ 0	∇u e (t, •)dt is
	well defined in L 2 loc (R d , L 2 • (Ω)), we have by integrating the equation (2.1.10) in time
				ˆ+∞
				-∇ • ae -∇ • a	∇u e (t, •)dt = 0,
				0
	and we then recognize the corrector equation (2.1.8). By uniqueness, we then conclude that
				ˆ+∞
				∇φ e (•) =	∇u e (t, •)dt.	(2.1.31)
				0
	Formula (2.1.31) combined with (2.1.23) then allow us to prove bounds on the gradient of correctors
	as well as on the flux.
					-1 2 (r),
					(2.1.32)
	with some random variable r ∈ [1, +∞) → C ,d,λ,β (r) which depends on d, λ, β and satisfies: for all
	α <	1 2 +	1 β∧d 5 2 d+2	there exists some constant C < +∞ depending on d, λ, β and α such that
					. Thanks

  For the contribution in [1, T ], we once again use the Minkowski inequality in L 2 (R

											2
											dx
		ˆˆ1						2
	d	d		|∇u(t, y)||∇v T (t, y)|dt dy
	≤ d	ˆ	0 ˆ1 0 Br(z)	|∇u(t, y)| 2 dy dt	1 2		ˆ1 0 Br(z)	|∇v T (t, y)| 2 dy dt	1 2	dz	2
	(2.2.7),(2.2.20)	d	ˆ	ˆηr (y -z)g 2 r (y)dy	1 2	dz	d .	(2.2.21)
							2	dx			ˆT 1	s -1	ˆ	ˆB	(x)	2	dx	1 2	ds	2
									(2.2.19)

2 (R d ) combined the identity ´= ´fflBr(z) dz, the Cauchy-Schwarz inequality, (2.2.7) applied with R = r and T = 1 and [92, Lemma 3] applied to the equation (2.2.11) in form of ˆ1 0 Br(z) |∇v T (t, y)| 2 dy dt ˆηr (y -z)g 2 r (y)dy, (2.2.20) Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields to get ˆ ˆB (x) ˆ1 0 ∇u(t, y) ⊗ ∇v T (t, y)dt dy d ) combined with (2.2.5) and (2.2.19) to get for all ≥ 1 ˆ ˆB (x) ˆT 1 ∇u(t, y) ⊗ ∇v T (t, y)dt dy |∇v T (t, y)|dy

  the pointwise bound (2.2.40) for local average of ∇v r 2 and the decay (2.2.26) of averages of ∇u.

  2. Proof of (2.2.40). We use the same type of decomposition as in Substep 1.2 : we have ∇v r = +∞ k=0 ∇v r,k where this time, for all k ≥ 1, B k := B 2 k+1 \B 2 k and B 0 := B 1 . We then split the proof into two steps. Substep 2.1. We argue in favor of the first alternative in (2.2.40), that is when the r.h.s is equal to

	r * (0)r log(1+|x|) d 2
	(|x|+1)

d . From the assumption (2.2.38) used in form of |f r (x)| + (|x| + 1)|∇f r (x)| r (|x|+1) d , we note that 1 r f r,k satisfies (2.3.5) with radius 2 k . Thus by (2.3.6), we have for all k ≥ 0 and √ -t ≥ 2r * (x)

  and the domination of the Gaussian kernel by the exponential kernel |q r (T )|

		(2.1.13) 1 + ˆηr (y) ˆgr (y)dy + ˆgr (y) ˆr2	ˆT 0	∇u(s, y)ds dy
											2	1 2
											dy	1.
	For the second r.h.s term of (2.3.28), we write with	´=	´fflBr(x) and the Cauchy-Schwarz inequality
	ˆηr (y)	ˆT r 2	∇u(s, y)ds dy	ˆ	Br(x)	η 2 r (y)dy	1 2	Br(x) ˆT r 2	∇u(s, y)ds	2	dy	1 2	dx. (2.3.29)
	Using the energy estimate (2.2.32) applied to the equation (2.1.10) in form of, for all T > 0

0 ∇u(s, y)ds dy + ˆηr (y) ˆT r 2 ∇u(s, y)ds dy 1 + ˆηr (y) ˆr2 0 ∇u(s, y)ds 2 dy 1 2 + ˆηr (y) ˆT r 2 ∇u(s, y)ds dy. (2.3.28) For the first r.h.s term of (2.3.28), we use the energy estimate (2.2.32) in form of ˆηr (y) ˆr2 0 ∇u(s, y)ds

  For this, we decompose R d into B r and R d \B r to the effect of The first r.h.s term of (2.3.31) is controlled by, using the1 8 -Lipschitz regularity of r * in form of sup x∈Br r * (x) r * (0) + r r * (0) + 1 and the Cauchy-Schwarz inequality

	2.3. Proofs															91
	It remains to show that
			ˆr d 2 * (x)		Br(x)	η 2 r (y)dy	1 2	dx 1 + r * (0) + d 2	ˆRd \B 1	d 2 * (rx)dx. η 1 (x)r	(2.3.30)
	ˆr d 2 * (x)	Br(x)	η 2 r (y)dy	1 2	dx =	ˆBr	r * (x) d 2	Br(x)	η 2 r (y)dy	1 2	dx+ ˆRd \Br	r * (x) d 2	Br(x)	η 2 r (y)dy	1 2	dx.
																	(2.3.31)
		ˆBr	r * (x) d 2		r (y)dy η 2	1 2
				Br(x)
											√ r T	)	ˆr d 2 * (x)	Br(x)	η 2 r (y)dy	1 2	dx.

(2.3.29)

turns into ˆηr (y) ˆT r 2 ∇u(s, y)ds dy r -d 2 log(

  Second, by the localized energy estimate (2.2.32) applied to the equation (2.1.10) with a replaced by a + hδa and for R = ≥ 1, we obtain

	2.3. Proofs											93
	We will prove that			I 1 + I 2	d 2		ˆe-|x-z| 2c	|g r (y)| 2 dy dz	1 2	.	(2.3.38)
																B (z)
	Since the arguments are similar we only give the details for I 2 . By the Cauchy-Schwarz inequality,
	we have
	I 2 ≤	ˆRd \B (x)	e	|x-z| 2c	B (z) ˆ1 0	∇δ h u(s, y)ds	2	dy dz	1 2	ˆRd \B (x)	e -|x-z| 2c	B (z)	|g r (y)| 2 dy dz	1 2	.
																(2.3.39)
	It remains to estimate the first r.h.s factor of (2.3.39). First, by the localized energy estimate (2.2.34)
	applied to the equation (2.3.34) and	´B	(x) |δa(y)e| 2 dy	1 2	d 2 , we obtain
	ˆRd \B (x)	e	|x-z| 2c	B (z) ˆ1 0	∇δ h u(s, y)ds	2	dy dz	1 2	d 2 +	ˆB	(x) ˆ1 0	δa(y)∇u(a + hδa, s, y)ds	2	dy
																+	ˆ1 0	1 1 -t ˆ1 t	ˆB	(x)	|δa(y)∇u(a + hδa, s, y)| 2 dy	ds dt.
																(2.3.40)
										ˆη	(y -x)	ˆ1 0	∇u(a + hδa, t, y)dt	2	dy	1 2	d 2 ,	(2.3.41)
	and
	ˆ1 0	1 1 -t ˆ1 t	ˆη	(y-x)|∇u(a+hδa, s, y)| 2 dy	1 2
	Regime have ˆgr (y)e 1 • a(y) < √ T . Using that ˆ1 0 ∇δ h u(t, y)dt dy ´=	´fflB (z) and by splitting R d into B (x) and R d \B (x) we ˆ ∇δ h u(t, y)dt g r (y)dy dz B (z) ˆ1 0 = ˆB (x) B (z) ˆ1 0 ∇δ h u(t, y)dt g r (y)dy dz (=: I 1 )
																(2.3.37)

.

[START_REF] Braides | Homogenization of multiple integrals[END_REF]

where T x, (η r ) is defined in (2.2.48).

+

ˆRd \B (x) B (z) ˆ1 0 ∇δ h u(t, y)dt g r (y)dy dz. (=: I 2 )

  Now, since δa is supported in B (x), the localized energy estimates (2.2.33) and (2.2.34) applied to the equation (2.3.34) with R = combined with (2.3.41) and (2.3.42) yield

	2.3.42) Finally the combination of (2.3.39), (2.3.40), (2.3.41) and (2.3.42) yields (2.3.38). It then follows from (2.3.37) that ˆgr (y)e 1 • a(y) ˆ1 0 ∇δ h u(t, y)dt dy d 2 ˆe-|x-z| 2c B (z) |g r (y)| 2 dy dz 1 2 . (2.3.43) Regime ≥ √ T . Using the Cauchy-Schwarz inequality and by dominating the Gaussian kernel by the exponential kernel, we have ˆgr (y)e 1 • a(y) ˆ1 0 ∇δ h u(t, y)dt dy ≤ ˆηr (y) ˆ1 0 ∇δ h u(t, y)dt 2 dy 1 2 , (2.3.44) with, by applying the localized energy estimates (2.2.32) and (2.2.33) to (2.3.34) for R = r ˆηr (y) ˆ1 0 ∇δ h u(t, y)dt 2 dy 1 2 ˆηr (y)|δa(y)| 2 dy 1 2 + ˆηr (y)|δa(y)| 2 ˆ1 0 ∇u(a + hδa, t, y)dt + ˆ1 0 1 ∇u(a+hδa, t, y)-∇u(a, t, y)dt 2 dy+t 2 ( √ t ∨1) -d B (x) |∇u(a+hδa, t, y)-∇u(a, t, y)| 2 dy h 2 d , 2 dy (2.3.46) which allow us to pass to the limit when h ↓ 0 in (2.3.44) and (2.3.45), and obtain lim sup h→0 ˆgr (y)e 1 • a(y) ˆ1 0 ∇δ h u(t, y)dt dy T x, (η r )(0). It concludes the argument for (2.3.36). Step 2. Second r.h.s term of (2.3.35). We prove that 1 -t ˆ1 B (x) ˆ1 0 lim sup	1 2

t ˆηr (y)|δa(y)| 2 |∇u(a + hδa, s, y)| 2 dy 1 2 ds dt . (2.3.45) for correlated coefficient fields h→0 ˆT 1 ˆgr (y)e 1 • a(y)∇δ h u(t, y)dy dt ˆB (x) ˆT 1 |∇u(t, y)||∇v T (t, y)|dt dy + ˆB (x) |∇v T (1, y)| 1 + ˆ1 0 ∇u(s, y)ds dy

  and obtain the first r.h.s term of(2.3.47). It remains to control the first r.h.s term of (2.3.51). To this aim, we integrate in time the equation(2.3.34) between 0 and 1:

					(2.3.49)
	and also, by testing (2.3.34) with v T
		ˆT 1	ˆvT (t, y)∂ (2.3.50)
	Consequently, by summing (2.3.49) and (2.3.50), using an integration by part in time and the fact
	that v T (T, •) ≡ 0, we get
	ˆT 1	ˆgr (y)e 1 •a(y)∇δ h u(t, y)dy dt = ˆδh u(1, y)v T (1, y)dy-	ˆT 1	ˆ∇v T (t, y)•δa(y)∇u(a+hδa, t, y)dy dt. (2.3.51)
	Moreover, from (2.3.46) we can pass to the limit when h ↓ 0 in the second r.h.s term of (2.3.51),
	namely
		lim h→0 ˆT 1	ˆ∇v

.

3.48)

We reformulate the l.h.s of

(2.3.47)

with help of the dual system (2.3.48). We have by testing the equation (2.3.48) with δ h u and integrating in time ˆT 1 ˆδh u(t, y)∂ t v T (t, y)dy dt-ˆT 1 ˆ∇δ h u(t, y)•a * (y)∇v T (t, y)dy dt = -ˆT 1 ˆgr (y)e 1 •a(y)∇δ h u(t, y)dy dt, t δ h u(t, y)dy dt + ˆT 1 ˆ∇v T (t, y) • a(y)∇δ h u(t, y)dy dt = -ˆT 1 ˆ∇v T (t, y) • δa(y)∇u(a + hδa, t, y)dy dt. T (t, y) • δa(y)∇u(a + hδa, t, y)dy dt = ˆT 1 ˆ∇v T (t, y) • δa(y)∇u(t, y)dy dt,

  y)dt dy -ˆ∇v T (1, y) • δa(y) e dy.

	(2.3.52)

The first r.h.s term of (2.3.52) combined with the third one and

(2.3.46)

give the second r.h.s term of

(2.3.47)

. The second r.h.s term of (2.3.52) is then dominated in two ways.

Regime ≥ √ T . For the generic case ≥ r * (0), we use the identity ´= ´fflB (y) and we split the integral into the two contributions ´Rd \B 4 and ´B4 in form of

  (0) + |y| 8 + 5 ≤ 6 + |y| 8 and thus |y| ≤ 16 3 ≤ 7 . Therefore, using the property (2.4.20) combined with the Cauchy-Schwarz inequality, we get

.3.53) Indeed, from the 1 8 -Lipschitz regularity property of r * , if there exists z ∈ B r * (y) (y) ∩ B 5 then |y| ≤ |y -z| + |z| ≤ r * (y) + 5 ≤ r *

  ˆfr (y)e 1 • δa(y)edy + ˆfr (y)e 1 • δa(y)

						ˆr2
						∇u(t, y)dt dy
				ˆr2	0
			+
	and (2.3.45) applied with r =	√	s to get	´r2 1 s -1 2 g √	s (y)ds
	lim sup h→0	ˆfr (y)e 1 • a(y)	ˆ1 0	∇δ h u(t, y)dt dy	lim sup h→0

0 ˆfr (y)e 1 • a(y)∇δ h u(t, y)dy dt.

(2.3.54)

The two first r.h.s terms of (2.3.54) give directly the two first r.h.s terms of (2.2.50), respectively. As in (2.3.35), we make the decomposition ˆr2 0 ˆfr (y)e 1 • a(y)∇δ h u(t, y)dy dt = ˆfr (y)e 1 • a(y) ˆ1 0 ∇δ h u(t, y)dt dy + ˆr2 1 ˆfr (y)e 1 • a(y)∇δ h u(t, y)dy dt. (2.3.55) Control of the second r.h.s term of (2.3.55). This term is controlled the same way as we did in Step 2 of the proof of Lemma 7 and provides the third, the fifth and the sixth r.h.s term of (2.2.50). Control of the first r.h.s term of (2.3.55). It remains to argue that the first r.h.s term of (2.3.55) is dominated by the fourth r.h.s term of (2.2.50). We distinguish the two regimes ≥ r and ≤ r. Regime ≥ r. We use the assumption (2.2.49) in form of, for all y ∈ R d , |f r (y)|

  We then split the integral into the far-field contribution |x| ≥ 2 and the near-field contribution |x| < 2 . For the near-field contribution, we make use of a dyadic decomposition and (2.3.56) in form of |f r (y)| (|y| + 1) -d+1 to get

	2.3. Proofs					97
	ˆB2 B (y)	|f r (z)|	ˆ1 0	∇δ h u(t, z)dt dz dy	ˆB3 ˆB1 ˆ1 0 |f r (y)| ∇δ h u(t, y)dt dy ∇δ h u(t, y)dt dy ˆ1 0
				(2.3.56)	+ ˆB1 ˆ1 0 log 2 (3 ) n=0 ∇δ h u(t, y)dt dy ˆB2 n+1 \B 2 n |f r (y)|	ˆ1 0	∇δ h u(t, y)dt dy

1 • a(y) ˆ1 0 ∇δ h u(t, y)dt dy ˆ B (y) |f r (z)| ˆ1 0 ∇δ h u(t, z)dt dz dy.

  .[START_REF] Duerinckx | Stochastic homogenization of nonconvex unbounded integral functionals with convex growth[END_REF] where we recall that v T is defined in(2.2.45). This estimate is a consequence of the localized energy estimate (2.2.32) and the large-scale regularity estimate (2.2.39). Indeed, we first notice that v T = (ṽ k (• -T, •)) k∈ 1,d with ṽk is the weak solution of (2.2.36) for e = e k and f r = g r . In addition, from the identity (2.3.7) we get for all (t, x) ∈ (-∞, T ] × R d

	ˆT -t
	∇v T (t, x) =	∇w(s, x)ds,	(2.3.59)
	0

with w = (w k ) k∈ 1,d and w k solves (2.3.8) with f r = g r and e = e k . Therefore, in the regime √ T -t ≥ 2r * (x), we use (2.2.39) (where we bound

  We split this step into two parts. The first part is devoted to the control of the first fourth r.h.s terms of (2.2.44) and the second part is devoted to the control of the fifth term. ´T r 2 and we apply the triangle inequality followed by the Jensen inequality, the identity ´fflB (x) = ´as well as the Cauchy-Schwarz inequality to the effect of

	Substep 1.1. Proof that for all ∈ [1,	√	T )
	ˆ		ˆB	(x) ˆT 0	∇u(t, y)dt g r (y)dy	2	dx	1 2	d r -d 2	1 + log(	√ r T	) ˆrd * (rx)g 2 1 (x)dx	1 2	,
														(2.3.62)
	and
	-d	ˆ	ˆB	(x)	g r (y)dy	2	+ -d + ˆˆe -|x-z| ˆB (x) |∇v T (1, y)| 1 + 2c |g r (y)| 2 + |∇v T (1, y)| 2 dy dz dx ∇u(s, y)ds dy 2 ˆ1 0	dx	d r -d . (2.3.63)
														B (z)
	Argument for (2.3.62). Since r ≤ 0 + ˆ √ T , we split ´T 0 into ´r2 ˆB 2 dx 1 2 d ˆ ˆr2 ∇u(t, y)dt	2	r (y)dy g 2	1 2
														0
														+	ˆˆB (x) ˆT r 2	∇u(t, y)dt	2	dy	ˆB	(x)	g 2 r (y)dy dx	1 2	.
														(2.3.64)
	ˆ ˆr2	∇u(t, y)dt	2	g 2 r (y)dy	1 2	r -d 2	ˆηr (y)	ˆr2	∇u(t, y)dt	2	dy	1 2 (2.2.32)
			0											0
														1
														2
														.	(2.3.61)

.60) with C (r, ) := r d * (0) ˆrd * (rx) log 2 (|x| + 1) (|x| + 1) 2d dx + ˆr2d * (x) (|x| + 1) 2d dx + ˆrd * (rx)g 2 2 (x)dx (x) ˆT 0 ∇u(t, y)dt g r (y)dy For the first r.h.s term of (2.3.64), we dominate the Gaussian kernel by the exponential kernel and we use the localized energy estimate (2.2.32) applied to (2.1.10) in form of r -d 2 . (2.3.65) For the second r.h.s term of (2.3.64), since < √ T , we have by the Minkowski inequality in L 2 (B (x)) and the decay estimate (2.3.57)

  Estimate(2.3.62) then follows in combination with the previous estimate, (2.3.64) and (2.3.65) as well as the estimate r * (rx) ∨ 1 r * (rx). for (2.3.63). On the one hand, we have from the Jensen inequality, the identity ´fflB (x) = ´and the Cauchy-Schwarz inequality as well as ´g2 r (y)dy r -d On the other hand, by noticing that from the Fubini-Tonelli theorem, we have for all measurable functions f : R d → R +

	Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with
	100 2.3. Proofs							for correlated coefficient fields 101
	(2.3.57)
	Argument ˆ ˆ	ˆB 2d r -d + (x) g r (y)dy ˆˆB (x) 2 |∇v T (1, y)| 2 dy + ˆB (x) |∇v T (1, y)| 1 + ˆB (x) 1 + ˆ1 0 ˆ1 0 ∇u(s, y)ds ∇u(s, y)ds dy 2 dx. (x) ˆT 1 ˆB dx |∇u(t, y)||∇v T (t, y)|dy dt 1 2 2	2	dx	(2.3.66)
	(x) |∇v ˆˆe -|x-z| 1+ ˆ1 0 ∇u(s, y)ds 2 dx (2.2.32) d ˆˆB (x) 2c
				B (z)
	Substep 1.2. Proof that for all ∈ [1,	√	T )
	ˆ	ˆT 1	ˆB	(x)	|∇u(t, y)||∇v T (t, y)|dy dt	2	dx	1 2
		d r -d 2 log(T ) r d * (0) ˆrd * (rx)	log 2 (|x| + 1) (|x| + 1) 2d dx +	ˆr2d * (x) (|x| + 1) 2d dx + ˆrd * (rx)g 2 2 (x)dx	1 2
									1
									2
									,

Then, using the localized energy estimate (2.2.32) applied to (2.1.10) with R = and the plain energy estimate ´|∇v T (1, y)| 2 dy r -d (the proof is identical as for (2.3.9)) we get

ˆˆB (x) |∇v T (1, y)| 2 dy ˆB T (1, y)| 2 dy dx 2d r -d .

(2.3.67)

f (y)dy dz dx = ˆf (y)

ˆ -d 1 B (y) (z) ˆe-

|x-z| 2c dx dz dy d ˆf (y)dy, we get ˆˆe -|x-z| 2c B (z) g 2 r (y) + |∇v T (1, y)| 2 dy dz dx d ˆg2 r (y) + |∇v T (1, y)| 2 dy d r -d . (2.3.68) The combination of (2.3.66), (2.3.67) and (2.3.68) give the desired (2.3.63). . (2.3.69) Let N := log 2 (T ) . We start by decomposing the time interval [1, T ] into dyadic intervals [2 j , 2 j+1 ] for j ∈ 1, N -1 . By the triangle inequality, the Cauchy-Schwarz inequality and the decay estimate

  We split this step into two parts. The first one is devoted to the control of the first fourth r.h.s terms of (2.2.44) and the second one is devoted to the control of the fifth r.h.s term.

											.3.76)
	and			D ,2 (r) =	Br	r d * (x)dx +	ˆRd \B 1	r d * (rx)g 1 (x)dx.	(2.3.77)
	Substep 2.1. Proof that for all ∈ [ √	T , +∞)
		ˆ	ˆB	(x) ˆT 0	∇u(t, y)dt g r (y)dy	2	dx	d D ,2 (r) log 2 (	√ r T	),	(2.3.78)
	and
	ˆ	ˆB
		(x)

  Argument for(2.3.79). We start with the first l.h.s term. We distinguish between the generic case ≥ r * (0) and the non-generic case < r * (0).

	2.3. Proofs											103
	and the Minkowski inequality in L 2 (B r (x)) as well as the decay estimate (2.3.57) (where we use that,
	since r ≥ 1, r * (x) r ∨ 1 ≤ 2r * (0)), we have
	ˆ	ˆT r 2	∇u(t, y)dt	2	g r (y)dy			Br	ˆT r 2	Br(x)	|∇u(t, y)| 2 dy	1 2	dt	2	dx
										(2.3.57)	+	ˆRd \Br Br r d * (x)dx + g r (x)	ˆT r 2 ˆRd \B 1	|∇u(t, y)| 2 dy * (rx)g 1 (x)dx log 2 ( 1 2 Br(x) r d √ dt T r	2 ). dx	(2.3.81)
	Regime ≥ r (x)	d	´Rd \B 3 as well as the property (2.4.20)
	ˆRd \B 4	ˆB	(x)	|∇v T (1, y)| 1 +	ˆ1 0	∇u(s, y)ds dy	2 (2.2.32) dx ≤	ˆB (x) ˆRd \B 4 × ˆB 1 + (x) d ˆB ˆRd \B 4 (x) |∇v T (1, y)| 2 dy ∇u(s, y)dy ˆ1 0	2	dx
	Since r ≤ (2.2.32) applied to (2.1.10) in form of √ T , we split the integral ´T 0 into ´ ´r2 0 ∇u(t, y)dt ´r2 0 + ´T r 2 and we use the localized energy estimate 2 g r (y)dy 1, to obtain
	ˆ	ˆT 0	∇u(t, y)dt	2	g r (y)dy	ˆ ˆr2 0 (2.2.32) 1 +	∇u(t, y)dt ˆ ˆT r 2 ∇u(t, y)dt 2 g r (y)dy + 2 g r (y)dy. ˆ ˆT r 2	∇u(t, y)dt	2	g r (y)dy (2.3.80)
														2
														dy dx,

y)dt g r (y)dy 2 dx ˆˆB (x) ˆT 0 ∇u(t, y)dt 2 g r (y)dy dx d ˆ ˆT 0 ∇u(t, y)dt 2 g r (y)dy. Finally, using that ´= ´fflBr(x) dx and for all x ∈ R d \B r , sup y∈Br(x) g r (y) g r (x), in form of ˆ ˆT r 2 ∇u(t, y)dt 2 g r (y)dy Br Br(x) ˆT r 2 ∇u(t, y)dt 2 dy dx+ ˆRd \Br g r (x) Br(x) ˆT r 2 ∇u(t, y)dt * (0). We split the integral into the far-field contribution |x| ≥ 4 and the near-field contribution |x| < 4 . For the far-field contribution, we write using the Cauchy-Schwarz inequality, the localized energy estimate (2.2.32) applied to (2.1.10) with R = and T = 1, the identity ´Rd \B 4 ´B |∇v T (1, y)| 2 dy (2.3.82) 2d ˆRd \B 3 |∇v T (1, y)| 2 dy (2.4.20) 2d

  energy estimates (2.2.32) as well as (2.2.33) applied to (2.1.10), the Minkowski inequality in L 2 (R d ) and the estimate ´´B (x)d ´, we have

	2.3. Proofs									105
	and the localized ˆT 2 x, (η r )(0)dx	d 1 + ˆηr (y)	ˆ1 0	∇u(t, y)dt	2	dy +	ˆ1 0	1 1 -t ˆ1 t	ˆηr (y)|∇u(s, y)| 2 dy	1 2	ds dt	2
	(2.2.32),(2.2.33)
											1
											2
											ds dt,

  dy.

				(2.3.90)
	Next, we have	ˆTx, (η )(y)dy	d .	(2.3.91)
	Indeed, we first split the integral into two contributions
	ˆTx, (η )(y)dy =	ˆB2 (x)

T x, (η )(y)dy + ˆRd \B 2 (x) T x, (η )(y)dy. On the one hand, since from the localized energy estimate (2.2.32) applied to (2.1.10) with R = we have sup y∈R d T x, (η )(y) 1 + ´1 0 -log(t) 1-t dt 1 (we bound the integral ´B (x) by ´in the definition (2.2.48) of T x, (η )(y)), we get ˆB2 (x) T x, (η )(y)dy d . On the other hand, for all y ∈ R d \B 2 (x) and z ∈ B (x) we have |z -y| ≥ |y -x| -≥ |y-x| 2 and thus η (z -y) η 2 (y -x). Therefore, the localized energy estimate (2.2.32) applied to (2.1.10) with R = yields

  for correlated coefficient fields and it concludes the argument for (2.3.91). The combination of (2.3.90), (2.3.91) and the estimate sup y∈R d ´Tx, (η )(y)dx d (which can be proved with the same computation as (2.3.91) by exchanging the role of x and y) leads to ˆ ˆRd \B 4 (1, z)| 2 dz dy dx, and we then proceed as we did from (2.3.82) to conclude. For the near-field contribution, we make use of the Minkowski inequality in L 2 (R d ) and the estimate sup y∈R d ´T 2

		|∇v T (1, z)| 2 dz	1 2	T x, (η )(y)dy	2	dx
		B (y) ˆˆR d \B 4 B (y) ˆRd \B 4 B (y) (2.3.90),(2.3.91) d 2d |∇v T x, (η r * (y) )(y)dx |∇v T (1, z)| 2 dz T x, (η )(y)dy dx	d (obtained
	the same way as (2.3.89)) to obtain
	ˆ	ˆB7

B r * (y) (y)

  We split the integral in the r.h.s of (2.3.94) into the far-field contribution |x| ≥ 4 and the near-field contribution |x| < 4 . For the far-field contribution, we write for all j ∈ 1, N -1 , using the Cauchy-Schwarz inequality, the decay estimate (2.3.57) applied for ≥ √ T ,

	Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with
	2.3. Proofs 108									107 for correlated coefficient fields
	and we conclude since ´B7 g 2r (x)dx	1, η	1 2 4r * (x) (x)	r * d 2 +1	(x)(|x| + 1) -d-1 and summing over
	j ∈ 0, N -1 .
	ˆRd \B 4 Regime < r * (0). We use the estimate (2.3.69) which holds for ≥ ˆ2j+1 2 j ˆB (x) |∇u(t, y)||∇v T (t, y)|dy dt 2 dx (2.3.57) ≤ d 2 -j ˆ2j+1 2 j and r d * (x) in each integral in the r.h.s (by using (2.3.57) for ≥ √ T ) and we estimate d by r d ˆB |∇v T (t, y)| 2 dy dx dt. √ T by removing r d * (rx) * (0) as ˆRd \B 4 (x) well as r -
	We then argue as from (2.3.82) to (2.3.83), (2.3.84), (2.3.85) and (2.3.86) (noticing that the evaluation
	at time 1 plays no role in the estimates) and finally obtain
	ˆRd \B 4	ˆ2j+1 2 j ˆB4 ˆB (x) |∇u(t, y)||∇v T (t, y)|dy dt 2 dx 2d r d * (0) ˆRd \B + ˆRd \B r d ˆ2j+1 ˆB 1 2 2	log 2 (1 + |x| r ) (|x| + r) 2d dx
										dx
						2 j	(x)
			ˆ2j+1					1 2
		≤								dy dt
				2 j ˆB5	ˆ2j+1
				d
				2
		2 (2.4.20) d ˆ		ˆ2j+1
					2
		(2.3.53) ≤	d 2	2 j ˆ2j+1 2 j ˆB7 B r * (x) (x) B r * (x) (x)	|∇u(t, y)||∇v T (t, y)|dt dy dx.	(2.3.95)
	Then, from the decay (2.3.57), we have
						ˆ2j+1		|∇u(t, y)| 2 dy dt	1 2	2 -j 2 ,	(2.3.96)
						2 j	B r * (x) (x)
	consequently, by the Cauchy-Schwarz inequality and (2.3.73), (2.3.95) turns into
		ˆB4	ˆ2j+1	ˆB	|∇u(t, y)||∇v T (t, y)|dy dt	2	dx	1 2
				2 j		(x)
	2 -j 2 (2.3.73) d d 2 2	N -1 j=0 2 1 ˆ ˆ2j+1 2 j ˆ2j+1 2 dx * (0) ˆB7 r d 2 ˆB7 log(1 + |x| ˆB7 1 r ) (|x| + r) d dx + η 2 4r * (x) (x)dx + ˆB ˆB7 (x)	g 2r (x)dx ,	(2.3.94)	2	dx	1 2	2	.

As in Substep 1.2, we decompose the time interval [1, T ] into dyadic intervals [2 j , 2 j+1 ] for j ∈ 1, N -1 and N := log 2 (T ) and we write by the triangle inequality ˆ ˆT 1 ˆB (x) |∇u(t, y)||∇v T (t, y)|dt dy 2 dx ≤ |∇u(t, y)||∇v T (t, y)|dy dt * (x)|x| -2d dx + -d , and we conclude by summing over j ∈ 1, N -1 , which give the two first r.h.s terms of (2.3.93). For the near-field contribution |x| ≤ 4 we write, using the Minkowski inequality in L 2 (B 4 ), the Fubini-Tonnelli theorem, the property (2.4.20) and the assumption ≥ r * (0) in form of (2.3.53) |∇u(t, y)||∇v T (t, y)|dy dt ˆ ˆB4 |∇u(t, y)| 2 |∇v T (t, y)| 2 1 B (x) (y)dx j |∇u(t, y)||∇v T (t, y)|dt dy |∇u(t, y)||∇v T (t, y)|1 B 5 (y)dt dy dx j B r * (x) (x) |∇v T (t, y)| 2 dy dt

  We treat separately the bound of the two lines in the definition (2.3.76) of D ,1 . First, we have by the Minkowski inequality in L p • (Ω), the stationarity property of r * and (2.3.102)

	2.3. Proofs												109
	It remains to show that	sup r≥1	D	p 2 ,2 (r)	≥1 p + sup 1	C p (r, )	1 2p	p	d β∧d ,	(2.3.100)
	and	ˆ+∞ 1	-1-β D p ,1 (r,	√	T )	1 p d	1 2	p	d+2 β∧d (1 + log 2 (	√ r T	)).	(2.3.101)
	We only provide the argument for (2.3.101), (2.3.100) is proved the same way. From the moment
	bound (2.4.3) on r * and the equivalence between algebraic moments and exponential moments (see
	Lemma 9), we have for all p ≥ 1
						r dp * (0)	1 p		p	d β∧d .	(2.3.102)
													.3.99)

  Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields

.3.105) The combination of (2.3.103), (2.3.104) and (2.3.105) gives (2.3.101). To conclude, the estimates (2.3.97), (2.3.98), (2.3.99), (2.3.100) and (2.3.101) give the desired bound (2.2.53).

  .111) where η β is defined in (2.1.24).

The combination of (2.3.109), (2.3.110) and (2.3.111) yields that for all x ∈ R d and R ≥ √ T B R (x)

  The estimate(2.3.63) is unchanged and gives the first contribution in(2.3.114). We improve the estimates (2.3.62) and (2.3.69). On the one hand, noticing that from the Minkwoski inequality in L 2 (R d ) and (2.3.106) as well as r ≥ 1 2 , we have for all x ∈ R d ˆB

	Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with
													for correlated coefficient fields
	where	E ,ε (r, ) := ˆrd * (rx)	ˆ+∞ 16 1	D ε (t, , rx)η ε,β (t)dt	2	B	r	(x)	g 2 1 (y)dy dx,	(2.3.115)
	and
	F j, ,ε (r, ) := ˆrd * (x)		ˆ2j+1 2 j	D 2ε (t, , x)dt	B (x)	r d * (0)r d log 2 (1 + |y| r ) (|y| + r) 2d dy +	r d * (y) (|y| + 1) 2d + r d g 2 2r (y)dy
			+ r d+ε * (x) ε	ˆRd \B (x)	|y -x| -d-ε r d * (0)r d log 2 (1 + |y| r ) (|y| + r) 2d dy +	r d * (y) (|y| + 1) 2d + r d g 2 2r (y) dy dx	1 2	.
													(2.3.116)
	(x) ˆT r 2	∇u(t, y)dt	2	dy ≤	ˆT r 2	ˆB	(x)	|∇u(t, y)| 2 dy 1 D 2 1 2 dt ˆ+∞ 2 (2.3.106) d r d * (x) 1 16 2
													dx
			(2.3.64),(2.3.65) ≤		d r -d 2 +	ˆ		ˆB	(x) ˆT r 2	∇u(t, y)dt	2	dy	ˆB	(x)	r (y)dy dx g 2	1 2
			(2.3.106)	r -d 2 +	d 2	ˆrd * (x)J ,ε (x)	ˆB	g 2 r (y)dy dx	1 2	,	(2.3.117)
													(x)
													2
													,	(2.3.114)

ε (t, , x)η ε,β (t)dt 2 , thus we deduce from (2.3.64) and (2.3.65) that ˆ ˆB (x) ˆT 0 ∇u(t, y)dt g r (y)dy

.

  We then split the argument into the far-field regime |x| ≥ 4 and the near-field regime |x| ≤ 4 . For the far-field regime, we use directly the Cauchy-Schwarz inequality combined with (2.3.2 dy dt using the arguments in (2.3.121) and (2.3.122). For the near-field regime, we use (2.3.106) and (2.3.118) in form of ˆ2j+1

																2.3.94)
	ˆ	ˆT 1	ˆB		(x)	|∇u(t, y)||∇v T (t, y)|dy dt	2	dx ≤	j=0 +∞	ˆ	2 j ˆ2j+1	ˆB	(x)	|∇u(t, y)||∇v T (t, y)|dy dt	2	dx	1 2	2
	for ≥	√	T ) and (2.3.118) to the effect of	106) (applied
		ˆRd \B 4 ˆ	ˆ2j+1 2 j ˆ2j+1 ˆB ˆB	(x)	|∇u(t, y)||∇v T (t, y)|dy dt ˆ2j+1 ˆB	2	dx
		≤										|∇u(t, y)| 2 dy dt	|∇v T (t, y)| 2 dy dt dx
		2 j (2.3.106),(2.3.118)	(x) d η 2 ε,β (2 j )	ˆRd \B 4	2 j ˆ2j+1 2 j	(x) D 2ε (t, , x)dt	ˆ2j+1 2 j	ˆB	(x)
										|∇u(t, y)| 2 dy dt	1 2 (2.3.106),(2.3.118)	η ε,β (2 j )	ˆ2j+1	D 2ε (t, , x)dt	1 2	, (2.3.126)
		2 j				B r * (x) (x)								2 j
	which has the effect of, combined with (2.3.95) and the Cauchy-Schwarz inequality
		ˆB4			ˆ2j+1	ˆB									2	1 2
																		dx
		(2.3.95) d 2	2 j ˆB7 B r * (x) (x) (x)	ˆ2j+1 2 j	|∇u(t, y)||∇v T (t, y)|dt dy dx
		(2.3.126) d 2 η ε,β (2 j )	ˆB7		ˆ2j+1	D 2ε (t, , x)dt	1 2	ˆ2j+1	|∇v T (t, y)|dt dy	1 2	dx, (2.3.127)
													2 j					B r * (x) (x)	2 j
	and yields the third term of (2.3.124) by using (2.3.73) (where we use η 4r * (x) (x) r d * (x)(|x| + 1) -2d ).
	Regime < r * (0). For the non-generic case < r * (0), we use (2.3.119), (2.3.121), (2.3.122)
	and we bound one d by r d * (0) which gives the second term of (2.3.124).

|∇v T (t, y)| 2 dy dt dx. (2.3.125) It then gives the first term of (2.3.124) by dominating ´2j+1 2 j ´B (x) |∇v T (t, y)| |∇u(t, y)||∇v T (t, y)|dy dt

  in form of (2.3.56), that we restate here: for all y ∈ R d

					|f r (y)|	r (|y| + 1) d ∧	1 (|y| + 1) d-1 .	(2.3.133)
	Second, we note that from (2.3.133), we have
	ˆ|f r (y)| 2 dy	ˆBr	1 (|y| + 1) 2(d-1) dy + r 2	ˆRd \Br	1 (|y| + 1) 2d dy	1 log(r + 1) for d = 2, for d ≥ 3,
										(2.3.134)
	and
	ˆB	|f r (y)|dy 1 <r	ˆB	1 (|y| + 1) d-1 dy + 1 ≥r	ˆBr	1 (|y| + 1) d-1 dy + r	ˆB	\Br	1 (|y| + 1) d dy
										(2.3.135)
			1

<r + r(1 + log( r + 1))1 ≥r . (2.3.136) Therefore, by arguing the same way as for (2.3.58), using the system (2.2.51) and the estimate (2.2.40) instead of (2.2.39) (since from (2.3.133), (2.2.38) holds) as well as the estimate (2.3.134), we have the following large-scale regularity estimate: for all t ∈ (-∞, r 2 ) and x ∈ R d B r * (x) (x) |∇v r 2 (t, x)| 2 dy r d * (0)

  .143) Proof of(2.3.142). We only give the argument for the second l.h.s term, the first one is dominated the same way. We split the argument into the far-field regime |x| ≥ 4 and the near-field regime |x| < 4 . For the near-field regime, we make use of a dyadic decomposition and the estimate(2.3.133) in form of |f r (y)| (|y| + 1) -d+1 to get

	Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with
											for correlated coefficient fields
	as well as
	ˆ	ˆB	(x)	|∇v r 2 (1, y)| 1+	ˆ1 0	∇u(s, y)ds dy	2	dx+ ˆG2 r, (x)dx	d (r d * (0)(1+log(r+1)1 d=2 )+M ,2 (r, )).
											(2.3.144)
	ˆB2	ˆB		ˆr2			2		ˆB3	ˆr2	2
						∇u(t, y)dt |f r (y)|dy	dx d	∇u(t, y)dt |f r (y)|dy
			(x)	0						ˆB1	0 ˆr2	2
											d	∇u(t, y)dt dy
											0
									(2.3.133)	+ d	log 2 (3 ) n=0 ˆB1 ˆr2 ˆB2 n+1 \B 2 n ∇u(t, y)dt dy ˆr2	2	2
											0
											log 2 (3 )	ˆr2
										+	2 n
											n=0	B 2 n+1
											n+1 )
	in form of
		ˆB1		ˆr2 0	∇u(t, y)dt dy	2	ˆB1 ˆ1 0	∇u(t, y)dt dy	2	+	ˆB1	ˆr2

0 ∇u(t, y)dt |f r (y)|dy 0 ∇u(t, y)dt dy 2 , which gives the second and third r.h.s terms of (2.3.139) by applying the localized energy estimate (2.2.32) to the equation (2.1.10) and the estimate (2.3.106) (applied for both = 1 and = 2 1 ∇u(t, y)dt dy 2

  .3.145) For the far-field contribution, we first make use of the Jensen inequality combined with the inequality ´Rd \B 2 ´B (x) dx d ´Rd \B and the decomposition ´Rd \B = ´Br\B + ´Rd \Br to write For the first r.h.s term of (2.3.146), we make use of a dyadic decomposition and the estimate (2.3.133) in form of |f r (y)| (|y| + 1) -d+1 to obtain fourth r.h.s term of (2.3.139) using (2.3.145). For the second r.h.s term of (2.3.146), we make use of a dyadic decomposition and (2.3.133) in form of |f r (y)| r(|y| + 1) -d to get which provides the first r.h.s contribution in (2.3.143). For the third l.h.s term, we can use previous estimates. To this aim, we use the Jensen inequality with measure T x, (η )(y)dy and (2.3.91) as well as ´Tx, (η )(y)dx d (which may be obtained by changing the role of x and y in the proof of (2.3.91)) combined with the inequality ´Rd \B 2 ffl B (y) dy ´Rd \B to obtain

	2.3. Proofs									119
		ˆBr\B	ˆr2 0	∇u(t, y)dt	2	|f r (y)| 2 dy =	log 2 (r) log 2 ( ) ˆB2 n+1 \B 2 n	ˆr2 0	∇u(t, y)dt	2	|f r (y)| 2 dy
										(2.3.133) log 2 (r)	2 -n(d-2)	ˆr2	∇u(t, y)dt	2	dy,
											log 2 ( )	B 2 n+1	0
	which gives the ˆRd \Br ˆr2 0	∇u(t, y)dt	2	|f r (y)| 2 dy =	+∞ n= log 2 (r) ˆB2 n+1 \B 2 n	ˆr2 0	∇u(t, y)dt	2	|f r (y)| 2 dy
										(2.3.133)	r 2	+∞	2 -nd	ˆr2	∇u(t, y)dt
											n= log 2 (r)	B 2 n+1	0
	ˆ	ˆRd \B 2	B (y)	|f r (z)| 2 dz		1 2	T x, (η )(y)dy	2	dx	(2.3.91) ˆˆR 2d d ˆRd \B |f r (y)| 2 dy.
	We then get (2.3.143) using (2.3.133) in form of
		ˆRd \B 2	ˆB ˆRd \B (x)	ˆr2 0 |f r (y)| 2 dy ∇u(t, y)dt |f r (y)|dy ˆBr\B (|y| + 1) -2(d-1) dy + r 2 2 dx 2d ˆRd \B ˆBr\B ˆRd \Br ˆr2 0 ∇u(t, y)dt (|y| + 1) -2d dy 2 |f r (y)| 2 dy 2 ˆr2 2-d (1 + log( r + 1)1 d=2 ), (2.3.147)
											= 2d	∇u(t, y)dt	|f r (y)| 2 dy
											+	ˆRd \Br	0 ˆr2
											(2.3.146)

0 ∇u(t, y)dt 2 |f r (y)| 2 dy . 2 dy, which finally gives fifth r.h.s of (2.3.139) using once more (2.3.145). Proof of (2.3.143). The estimate of the two first l.h.s terms is an immediate consequence of the Minkowski inequality in L 2 (R d ) and the estimate (2.3.89) applied with r = √ s, d \B 2 B (y) |f r (z)| 2 dz T x, (η )(y)dy dx 2d ˆRd \B 2 B (y) |f r (z)| 2 dz dy

  3.85) Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields (applied with T = r 2 ) combined with (2.3.137) to obtain ˆRd \B 4 ˆB

	(x) ˆRd \B |∇v r 2 (1, y)| 1 + r 2 log 2 (1 + |x|) ˆ1 0 (|x| + 1) 2d ∧ + (1 + log(r + 1)1 d=2 ) 2d r d * (0) ˆRd \B η 4r * (x) (x)dx + ∇u(s, y)ds dy 1 (|x| + 1) 2(d-1) dx ˆRd \B f 2 2r (x)dx , 2 dx
	which gives the first term in (2.3.140) using that η 4r * (x) (x) r d * (x)(|x| + 1) -2d . For the near-field
	contribution, we make use of the estimate (2.3.88) combined with (2.3.137) to obtain
	ˆB4	ˆB	(x)	|∇v r 2 (1, y)| 1 +	ˆ1 0	∇u(s, y)ds dy	2	dx	1 2

  identical as the one for (2.3.9)) combined with (2.3.139) ˆ|∇v r 2 (1, y)| 2 dy ˆ|f r (y)| 2 dy 1 + log(r + 1)1 d=2 , We split the argument between the far-field regime |x| ≥ 4 and the nearfield regime |x| < 4 . For the far-field contribution, we make use of (2.3.94),(2.3.125) and (2.3.121) as well as (2.3.122) combined with (2.3.137) to obtain

	2.3. Proofs						121
	ˆB Regime ≥ r ˆRd \B 4	(x)	ˆr2 1	|∇u(t, y)||∇v r 2 (t, y)|dt dy	2	dx	+∞ j=0
					2.32) to get
			ˆ	ˆB ˆˆB (x) (x) |∇v r 2 (1, y)| 1 + |∇v r 2 (1, y)| 2 dy ˆB ˆ1 0 (x)	∇u(s, y)ds dy 1 + ˆ1 0 ∇u(s, y)ds 2 dx	2	dy dx
	which gives the first contribution in (2.3.138). For the second l.h.s term of (2.3.144), we argue as in
	(2.3.90) and (2.3.92).
	Substep 1.2. Proof that for all ∈ [1, r)
	ˆ	ˆB	ˆr2
		(x)

2d ˆ|∇v r 2 (1, y)| 2 dy d r d * (0) ˆ|∇v r 2 (1, y)| 2 dy, and we conclude with the plain energy estimate in the equation (2.2.51) (for which a proof is 1 |∇u(t, y)||∇v r 2 (t, y)|dt dy 2 dx d +∞ j=0 2 j 2 η ε,β (2 j )K j, ,ε (r, ) 2 . (2.3.148)

We argue differently with the generic case ≥ r * (0) and the non-generic case < r * (0) and we use several previous estimates. * (0).

  First,(2.3.150) follows from the Minkowski inequality in L 2 (R d ) and (2.3.89) applied with r = √ s (noticing that the evaluation at 0 plays no role in the estimate). Secondly, using the Minkowski inequality in L 2 (R d ) and the assumption (2.1.19), we have

			ˆ	ˆB				2			ˆ	ˆB	ˆr2	2
						|f r (y)|dy	dx +	∇u(s, y)ds |f r (y)|dy	dx
			d	(x) ˆr2	s -1 ˆ|y|g √	s (y) 1 +	(x) ˆr2 ∇u(t, y)dt dy 0	2	,
					1						0
	which gives (2.3.151) using that |y|g √	s (y)	s	1 2 g √	2s (y), the estimates (2.3.80) and (2.3.81) with
	(T, r) replaced by (r 2 , s) to get
		ˆr2 1	s -1 ˆ|y|g √	s (y)	ˆr2 0	∇u(t, y)dt dy	ˆr2 1	s -1 2 ˆg√	2s (y)	ˆr2 0	∇u(t, y)dt dy
											.3.150)
	and
	ˆ	ˆB			2		ˆ	ˆB	ˆr2
			|f r (y)|dy		dx +
			(x)								(x)

0 ∇u(s, y)ds |f r (y)|dy 2 dx M ,3 (r).

(2.3.151) for correlated coefficient fields

  d=2 ≤ χ d,β (r) (where we recall that χ d,β (r) is defined in (2.1.21)):

	2.3. Proofs	123
	1 + log(r + 1)1
	1
	2
		χ d,β (r) and

2.3.138) combined with the Minkowski inequality in L p • (Ω) and the moment bound (2.3.102) of r * as well as ´r 1 1-β (1 + log( r + 1)1 d=2 )d

  .3.155) We start with (2.3.154). First, using the moment bound (2.3.108) of D and the definition of η ε,β

	in (2.3.107) as well as (2.3.102), we have for all ρ > 0, N p ,ε (r, , ρ)	1 p	p	d β∧d +ε(1+2 d+2 β∧d )

  We first note that the r.h.s of (2.3.159) is well defined as a random variable in L 2 • (Ω), since from (2.1.25) and (2.2.32) we have for all x ∈ R d We only provide the argument for (2.3.159), (2.3.160) follows the same way. To this aim, we prove that there exists a potential ζ ∈ L 2 (Ω × R d ), sub-linear at infinity, such that ´+∞ 0 ∇u(t, •)dt = ∇ζ and solving -∇ • a(∇ζ + e) = 0 in R d in the distributional sense. By uniqueness of ∇φ defined by (2.1.8), it shall imply (2.3.159). Let ψ ∈ C ∞ c (R d ) be supported in B R for some R > 0 and let √ T > R. We have by testing (2.1.10) with ψ and integrating in time from 0 to T ˆu(T, y)ψ(y)dy + ˆ∇ψ(y) • a(y)e dy + ˆ∇ψ(y) • a(y) ˆT 0 ∇u(s, y)ds dy = 0. (2.3.161) We now check that each term of (2.3.161) pass to the limit, almost surely, as T ↑ +∞. For the first l.h.s term of (2.3.161), we use the triangle inequality followed by the Poincaré inequality, (2.1.23) and (2.1.25):

	2.3. Proofs											127
	ˆ+∞ 0	∇u(t, x)dt	2 1 2	≤	ˆ1 0	∇u(t, x)dt	2 1 2	+	1 ˆ+∞	|∇u(t, x)| 2	1 2	dt 1.
						ˆ+∞
						e -t T ∇u(t, •)dt.			(2.3.160)
						0

.159) and for all T ≥ 1 ∇φ T (•) =

  x)dtIt remains to control the space integral of the above inequality. We apply Lemma 5 using (2.1.23), f ≡ 1 and g : t ∈ R + → η β (t) as well as the moment bound (2.4.3) of r * to obtain, for all r ≥ 1

												p 1 p
			≤	ˆ+∞		ˆgr (x)|∇u(t, y)|dy	p 1 p	dt
			≤	r 2 ˆ+∞		ˆηr (x)|∇u(t, x)| 2 dx	p 2	1 p	dt.	(2.3.164)
							r 2
								p	1
			|∇u(t, x)| 2 dx	2	p	p	1 2	d β∧d + 1 α t -1 2 η β (t).	(2.3.165)
		Br
	Consequently, for all t ≥ r 2
	p 2	1 p	≤	Br	|∇u(t, x)| 2 dx	p 2	1 p	+	ˆRd \Br	η p 2	1 p
			(2.3.165)	p	1 2	d β∧d + 1 α t -1 2 η β (t) +	ˆRd \Br	η r (x)|∇u(t, x)| 2 dx	p 2	1 p	.
												(2.3.166)

ˆηr (x)|∇u(t, x)| 2 dx r (x)|∇u(t, x)| 2 dx

For the second r.h.s term of the previous estimate, we decompose R d \B r into the family of annuli (B (n+1)r \B nr ) n≥1 to obtain, with

(2.3.165)

  We consider the regimes β = 2, d > 2 and β > 2, d = 2. Our main tools here are the fluctuation estimate (2.1.20) and the decay (2.1.23) of ∇u. We claim that φ

	2.3. Proofs	133
	where we recall that ξ d,β is defined in (2.1.34). The combination of (2.3.173), (2.3.174),
	(2.3.175), (2.3.177) and Lemma 9 gives the desired estimate (2.1.33).
	(ii)

),

(2.3.177)

  of the corrector equation (2.1.8) for both φ * e i and φ e i yields (∇φ e i -∇φ n e i ,T ) • a * (∇φ * e j + e j ) = (∇φ * e j -∇φ * ,n e j ,T ) • a(∇φ e i + e i ) ,

	2. Proof of (2.1.38). This estimate is a direct consequence of (2.1.37). Indeed by the definition (2.1.36) of a n T , we have 135 the weak formulation and we conclude that |e j •(a n T -a hom )e i | = | (∇φ * ,n e j ,T -∇φ * e j )•a(∇φ n e i ,T -∇φ e i ) | ≤ |∇φ * ,n e j ,T -∇φ * e j | 2 1 2 |∇φ n e i ,T -∇φ e i | 2 1 2 , so that the claim follows from (2.1.37), used for both a * and a. Proof of Corollary 6: Spectral resolution Let 0 < µ ≤ 1. The starting point is the use of the spectral theorem which allows one to rewrite the e 2.3. Proofs definition of (φ n e,µ -1 )

j • (a n T -a hom )e i = (∇φ * ,n e j ,T -∇φ * e j ) • a(∇φ n e i ,T + e i ) -(∇φ * e i + e j ) • a(∇φ e i -∇φ n e i ,T ) .

Since we have

(∇φ * e,j + e j ) • a(∇φ e i -∇φ n e i ,T ) = (∇φ e i -∇φ n e i ,T ) • a * (∇φ * e j + e j ) , n∈N , defined in Corollary 5, in the form, for all n ∈ N φ n e,T = g n (L, µ -1 )Θ, where g 0

  Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields with some constant C 1 which depends on λ and d.

	and if R = +∞ sup r≥r * r 2α Exc(∇u + g, r) d,λ,α sup 1 r≥r * Let α = 1+α 2 and r Set ξ := ffl Cρ g and let w	1 r 2α	Cρ	|h -	1 r 2α Cρ h| 2 + |g -Cr |h -Cρ	Cr g| 2 . h| 2 + |g -	Cr (2.4.7) g| 2 , (2.4.8)

* ≤ r ≤ ρ ≤ R. We prove that Exc(∇u + g, r) ≤ C 1 r ρ 2α Exc(∇u + g, ρ) + ρ r d+2 Cρ |h -Cρ h| 2 + |g -Cρ g|

2

, (2.4.9)

  function in C ρ , we have by the Theorem 11 for the exponent α

	Exc(∇u -∇w + ξ, r)	r ρ	2α	Exc(∇u -∇w + ξ, ρ).	(2.4.11)
	In addition, we have the following energy estimate
	ˆCρ	ˆCρ		2	ˆCρ		2
	|∇w| 2	h -		h	+	g -	g	.	(2.4.12)
			Cρ			Cρ
	Indeed, by testing (2.4.10) by w itself, we get

  Using the triangle inequality in L 2 (C r ) and the definition of the excess in the form of Lemma 10. For all measurable function f : R d → R + there exists two constants c and C which depends only on the dimension d such that

	2.4. Appendix								141
	Substep 2.2. We prove (2.4.5). The starting point is (2.4.7) in the more general form : for all
	r ≥ r *
	sup ρ∈[r,R]	(	R ρ	) 2α Exc(∇u + g, ρ) Exc(∇u + g, R)
				+ sup ρ∈[r,R]	(	R ρ	) 2α	Cρ	|g -	Cρ	g| 2 + |h -	Cρ	h| 2 .	(2.4.18)
	The estimates (2.4.15) and (2.4.18) combined with the triangle inequality yield
	|ξ r | 2 + Exc(∇u + g, r) |ξ R | 2 + Exc(∇u + g, R)
				+ sup ρ∈[r * ,R] (	R ρ	) 2α	Cρ	|g -	Cρ	g| 2 + |h -	Cρ	h| 2 .	(2.4.19)
				ˆ
				c
				B r * (x) (x)

Cr

|∇u + g| 2 |ξ r | 2 + Exc(∇u + g, r),

and |ξ R | 2 + Exc(∇u + g, R) C R |∇u + g| 2 ,

we may finally pass from (2.4.19) to (2.4.5).

We finally recall the following property of average of r * . The proof can be found in

[START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] Esti. (139)

].

  Chapter 2. Optimal decay of the parabolic semigroup for linear elliptic systems with for correlated coefficient fields Thus, by testing δu into (2.2.11) and v T into (2.4.26), we deduce that

	ˆgr (y)a(y)	ˆT 0	∇δu(t, y)dt dy = ˆ∇v T (0, y) • (a (y) -a (y))e dy ˆˆT
			+	(a(y) -a (y))∇u(a , t, y) • ∇v T (t, y)dt dy
			0 ˆˆT
			-	(a(y) -a (y))∇u(a , t, y) • ∇v T (t, y)dt dy.	(2.4.27)
			0
	The first r.h.s term of (2.4.27) is dominated with (2.4.24) and gives the second r.h.s term of (2.4.23).
	The second and third r.h.s term of (2.4.27) are dominated the same way (we estimate below the
	term with a ) using (2.4.21), (2.4.22), (2.4.24), (2.2.20) (applied with r = and z = x), the Cauchy-
	Schwarz and Jensen's inequality
	ˆˆT			2
	(2.4.24) χ (2.4.21) + ˆT 1 ( + 1) 2d ˆB +1 (x) ˆT 0 ˆ1 0 B +1 (x) |∇u(a , t, y)||∇v T (t, y)|dt dy |∇u(a , t, y)| 2 dy dt ˆ1 0 B +1 (x) 2 t -1 +1 (x) |∇v T (t, y)|dy dt ˆB 2	|∇v T (t, y)| 2 dy dt
	(2.4.22),(2.2.20) χ	( + 1) 2d
			B +1 (x)

.4.26) 0 (a(y) -a (y))∇u(a , t, y) • ∇v T (t, y)dt dy

  Quantitative homogenization theory for nonlinear elliptic equations and systems 3.3 Perturbative regularity theory for the linearized operator . . . . . . . . 162 3.3.1 The Meyers minimal radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.3.2 Quenched perturbative regularity in the large . . . . . . . . . . . . . . . . . 165 Quenched Meyers' estimate in the large . . . . . . . . . . . . . . . . . . . . 165 Quenched weighted Meyers' estimate in the large . . . . . . . . . . . . . . . 168 3.3.3 Control of the Meyers minimal radius: sensitivity estimate and buckling . . 173 3.3.4 Annealed Meyers' estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.4 Control of correctors: Proof of Theorem 15 . . . . . . . . . . . . . . . . 182 3.5 Control of corrector differences: Proof of Theorem 17 . . . . . . . . . . 186 3.5.1 Reduction argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 3.5.2 CLT-scaling: Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . 188 3.5.3 Control of level sets: Proof of Proposition 7 . . . . . . . . . . . . . . . . . . 193 3.6 Quantitative two-scale expansion: Proof of Theorem 12 . . . . . . . . . 197 3.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 3.7.1 Deterministic PDE estimates and consequences . .

	3.2 Strategy of the proof, extensions, and limitations . . . . . . . . . . . . . 152
	3.2.1 General strategy and auxiliary results . . . . . . . . . . . . . . . . . . . . . 152
	3.2.2 Towards large-scale regularity and a nonlinear theory of fluctuations . . . . 157
	3.2.3 Extensions and limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
	Probability laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Contents 3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.1.1 Growth conditions and classical regularity . . . . . . . . . . . . . . . . . . . 147 3.1.2 Quantitative assumptions and quantitative two-scale expansion . . . . . . . 148 3.1.3 Remarks on the strong monotonicity of ā . . . . . . . . . . . . . . . . . . . 150 Form of the monotone map . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Local regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Non-convex energy functionals? . . . . . . . . . . . . . . . . . . . . . . . . . 162 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 146 Chapter 3. . . . . . . . . . . . . . . 200 3.7.2 Nonlinear systems: Caccioppoli, hole-filling, and Schauder . . . . . . . . . . 200 3.7.3 Linear elliptic systems: Caccioppoli and Lemma 17 . . . . . . . . . . . . . . 201 3.7.4 Qualitative differentiability of correctors on bounded domains . . . . . . . . 202 3.7.5 Periodic setting: Proof of Theorem 13 . . . . . . . . . . . . . . . . . . . . . 204 3.7.6 Statistically isotropic random setting: Proof of Theorem 14 . . . . . . . . . 206 3.7.7 Periodization in law and functional inequalities . . . . . . . . . . . . . . . . 211 Periodization in law of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Convergence of the periodization in law of the correctors . . . . . . . . . . . 213 3.7.8 Large-scale averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

  3.1 & 3.3.2 and the quenched Meyers estimates in the large. The starting point is the energy estimate (3.1.8) in form of E |a ξ |Such a weakening of Meyers' estimates still does not hold in our setting because estimate (3.1.8) cannot be turned into an almost sure bound (with a uniform choice of r). A further weakening consists in letting the radius of the ball B r (x) in (3.2.13) be random and depend on the point x ∈ R d . We thus introduce in Definition 3.3.1 the Meyers minimal radius r , a random field on R d which essentially ensures that, with the notation B (x) := B r (x) (x), Such a form of the Meyers' estimates with a random scale r was first used by Armstrong and Dario in[START_REF] Armstrong | Elliptic regularity and quantitative homogenization on percolation clusters[END_REF] to deal with homogenization in percolation. To obtain an estimate in the spirit of (3.2.13) (with B r (x) replaced by B (x)), we rely on the standard proof of Meyers' estimates going through a reverse Hölder inequality and Gehring's lemma. For uniformly elliptic equations, the reverse Hölder inequality is a consequence of Caccioppoli's inequality and of the Sobolev embedding. Caccioppoli's inequality for a ξ -harmonic functions u (say, -∇ • a ξ ∇u = 0 in the ball B 2R centered at 0) typically takes the form ∼ ffl B 2R 1 + |∇φ ξ + ξ| p 1 + |ξ| p provided 2R ≥ r (0) (as a consequence of the definition of r ). We thus need to first appeal to Hölder's inequality with exponents ( p p-2 , p 2 ) to upgrade the Caccioppoli inequality into

	Chapter 3. Quantitative homogenization theory for nonlinear elliptic equations and
	156							systems
								infx sup x	ffl ffl B (x) a ξ B (x) a ξ	> 0 (with a
	deterministic positive lower bound). ˆBR	∇u • a ξ ∇u	1 R 2	ˆB2R	|u| 2 |a ξ |.
	Assuming that ´B2R u = 0, the next step is to appeal to the Poincaré-Sobolev inequality R 2 ffl d+2 B 2R |∇u| 2d d+2	ffl B 2R |u| 2
				B R	∇u • a ξ ∇u	1 R 2	B 2R	|u| p	2 p .
	Then, assuming that p < 2d d-2 , the Poincaré-Sobolev inequality yields the desired reverse Hölder's inequality ffl B R ∇u • a ξ ∇u ffl B 2R |∇u| p *
	p p-2	1 + |ξ| p , which yields two challenges: it is stochastically-averaged and gives a poor
	integrability for large p. We begin with the stochastically-averaged part. The idea is to relax the condition inf a ξ sup a ξ > 0 into the milder requirement inf ffl Br (x) a ξ sup ffl Br (x) a ξ > 0 for some r > 0, which, in turn,
			m	ˆRd		m	ˆRd	m
		|∇u| 2	2 dx				|∇u| 2 dx	2 +	|g| 2	2 dx.	(3.2.13)
		Br(x)				Br(x)		Br(x)

yields the weaker (yet sufficient) Meyers' estimate at scale r > 0 in form of: For u, g related via -∇ • a ξ ∇u = ∇ • g, we have for some Meyers' exponent m > 2 ˆRd d . The random unbounded coefficient |a ξ | ∼ 1 + |∇φ ξ + ξ| p-2 is however in the way (and not of the Muckenhoupt class), and the estimate (3.1.8) only yields ffl B 2R |a ξ | p p-2 2 p * with exponent p * = d+p dp < 2.

  .6. Quantitative two-scale expansion: Proof of Theorem 12 199 which we applied to ζ k

  which is the desired estimate(3.3.16). For d = 2, in which case q * = 1, we use the one-dimensional Sobolev inequality φ L ∞ (Sr)

	r (3.7.9). The case d = 1 is similar.	ffl Sr |∇φ| +	ffl	Sr |φ| instead of (3.7.10), and (3.3.16) follows from

  It remains to show that ∇ ψ = ∇ φξ,e , which directly follows from the weak formulation of (3.7.12).

										systems
	For all w ∈ H 1 0 (D) (resp. H 1 per (Q L ))
			ˆD ∇w • a (1) h ∇δ h φ ξ = -ˆD ∇w • a (2) h e.			(3.7.18)
							3.7.14)
							c 1 .			(3.7.15)
	On the other hand, by the Schauder estimate [76, Theorem 5.19] applied to (3.7.12), and the bounds
	(3.7.14) and (3.7.15), there exists some γ > 0 (depending on α and d) such that
	∇δ h φ ξ C α (D)	a (1) h	γ C α (D)	ˆD |∇δ h φ ξ | 2	1 2 + a	(2) h C α (D)	(3.7.14),(3.7.15) ≤	c 2 ,	(3.7.16)
	for a constant c 2 depending on c 1 , d and D. By (3.7.16) and Arzela-Ascoli's theorem, there exists ψ ∈ C 1,α 0 (D) (resp. C 1,α per (Q L )) such that (up to a subsequence that we do not relabel)
				∇δ h φ ξ	h↓0 → ∇ ψ in C 0 (D).			(3.7.17)

  1 s φ sξs -ψ ξs )| p s 2(2-p) ˆQ(1 + |ξ s + ∇ψ ξs | p ) Next we argue that { 1 s φ sξs } s≥1 is a bounded sequence in C α (Q),in which case (3.7.22) follows from (3.

	|ξs|=1	s 2(2-p) .	(3.7.23)

  and ∇ψ ξ∞ is continuous, there exists κ > 0 such that the symmetric part b sym ξ∞ of b ξ∞ satisfies b sym ξ∞ | Q\Tr(ξ∞) ≥ κId. Hence, by (3.7.24), there exists s < ∞ such that, for all s ≥ s , a sym sξs | Q\Tr(ξ∞) ≥ 1 2 κId. For all s ≥ s we thus have Quantitative homogenization theory for nonlinear elliptic equations and systems which is positive since R d \ T r (ξ ∞ ) is connected in R d . This proves (3.7.21). Step 2. Proof of (3.7.19) in the remaining range: |ξ 1 |, |ξ 2 | 1 and |ξ 2 | 1, |ξ 1 | 1. On the one hand, since c(x, ξ)ξ ≥ 1 C |ξ| 2 , we have for all ξ 1 , ξ 2 ∈ R d (ā(ξ 1 ) -ā(ξ 2 ), ξ 1 -ξ 2 ) ≥ c|ξ 1 -ξ 2 | 2 , from which (3.7.19) follows for |ξ 1 |, |ξ 2 | 1. On the other, since ā ∈ M(p, 1, p, C), for all ξ 1 , ξ 2

	Chapter 3.
	206
	e • 1 s p-2 Dā(sξ s )e ≥	κ 2 ˆQ\Tr(ξ∞)	|e + ∇ φsξs,e | 2 ≥	κ 2	inf φ∈H 1 per (Q) ˆQ\Tr(ξ∞)	|e + ∇ φ| 2 ,

  1 2 |ξ| 2 + 1 p |ξ| p ). As the following examples show, A is not empty.

	3.7. Appendix	207
	1. The non-degenerate p-Laplacian operator â
	Example 3.7.1. The following nonlinear maps belong to A:

  (ξ) = ζ L (|ξ|). In particular, we necessarily have āL (ξ) = (ζ L ) (|ξ|) ξ |ξ| , so that, by Lemma 28, (3.7.32) will follow provided we show that t → (ζ L ) (t) is differentiable and satisfies for some c > 0 independent of L for all t > 0 (ζ L ) (t) ≥ c(1 + t p )

	3.7. Appendix		209
	W L p-2 p .	(3.7.33)
	Fix a unit vector e ∈ R d . By definition, we have
	(ζ L ) (t) = āL (te) • e = E	a(x, te + ∇φ L te (x)) • edx .
		B L
	By Lemma 27, correctors are differentiable, and differentiating the above yields using (3.2.9) (posed
	on B L with Dirichlet boundary conditions)
	(ζ L ) (t) = E	Da(x, te + ∇φ L te (x))(e + ∇ φL te,e (x)) • edx
	B L
	= E	Da(x, te + ∇φ L te (x))(e + ∇ φL
	B L

.

7.32)

The advantage of (3.7.32) over

(3.7.26)

is that it involves correctors on a bounded domain rather than on the whole space, for which differentiability with respect to ξ can be easily established (cf.

Lemma 27)

. The advantage of W L over W L (which motivates the choice of the ball B L for the domain) is that, as W , ξ → W L is isotropic in the sense that there exists ζ L : R + → R such that te,e (x)) • (e + ∇ φL te,e (x))dx .

Since a(x, ξ) = â(A(x), ξ), ξ → a(x, ξ) is differentiable and satisfies for some c > 0 and for all

ξ, h ∈ R d Da(x, ξ) : h ⊗ h ≥ c(1 + |ξ| p-2 )|h| 2 .

Hence (using that ´BL ∇ φL te,e = 0), (ζ L ) (t) ≥ cE ffl B L |e + ∇ φL te,e | 2 ≥ c, which yields (3.7.33) provided t 1. It remains to treat the case t 1.

  .7.35) systems From (3.7.34) and (3.7.35) we infer that ζ L satisfies the differential relation t(ζ L ) (t)-pζ L (t) = E Let t ≥ 1 to be fixed later. The solution of this ODE is explicitly given for all t ≥ t by ζ L (t) = ζ L (t ) t p t p ++t p ´t t s -1-p h(s)ds. Differentiating twice (which we can since t → ∇φ L te (x) is differentiable for all x ∈ B L ) , this yields

		ρ 2 (A(x), |te+∇φ L te (x)|)|te+∇φ L te (x)| 2 dx-p	ˆ|te+∇φ L te (x)|	sρ 2 (A(x), s)dsdx ,
		B L	B L	0
	which we rewrite as (ζ L ) (t) -p t ζ L (t) = 1 t h(t) with
	h(t) := E	ρ 2 (A(x), |te + ∇φ L te (x)|)|te + ∇φ L te (x)| 2 dx -p	ˆ|te+∇φ L te (x)|	sρ 2 (A(x), s)dsdx .
	B L	B L	0

  .7.37) (The constant c will change from line to line, but remains independent of t and L.) Hence, by the (deterministic) energy estimate fflB L |∇φ L ξ | p ≤ c(1 + |ξ| p ), (3.7.37) yields for all t ≥ 1 Since R L (t) = (p -1)t -2 h(t) + t -1 h (t), it remains to estimate h (t). By differentiating h, using (3.7.37), and rearranging the terms, Cauchy-Schwarz' inequality yields |h

	3.7. Appendix
			|h(t)| ≤ ct β+2 ,	(3.7.38)
	so that	ˆ∞ t	s -1-p h(s)ds ≤ ct β-(p-2) .
	Since β < p -2, we deduce that for t 0 ∼ 1 large enough (and independent of L) we have for all
	t ≥ t ≥ t 0
			γ L (t) ≥	λ 2p	.	(3.7.39)

Substep 2.3. Choice of t 1 and upper bound on R L (t).

  Substep 2.4. Definition of t and proof of (3.7.33). Set t = t 0 ∨ t 1 , which is independent of L. Step 1 yields (3.7.33) in the regime t ≤ t , whereas in the regime t ≥ t , (3.7.33) follows from (3.7.36) in combination with Substeps 2.2 and 2.3.

	t p-2 . We conclude with the argument in favor of (3.7.40). By testing (3.2.9) with φL 4 te,e ∈ H 1 0 (B L ) we
	obtain	ˆBL	ˆBL
		∇ φL te,e • Da(x, te + ∇φ L te )∇ φL te,e =	∇ φL te,e • Da(x, te + ∇φ L te )e,
	which, by our assumptions on a, entails
		ˆBL	ˆBL
		|∇ φL te,e | 2 (1 + |te + ∇φ L te | p-2 )	|∇ φL te,e |(1 + |te + ∇φ L te | p-2 ),
	and therefore (3.7.40) by Cauchy-Schwarz' inequality and the energy estimate on ∇φ L te .

.7.40)

These last three estimates then combine to |h (t)| ≤ ct p-1-(p-2-β) . With (3.7.38) and the formula for R L (t), this entails for all t ≥ 1 the control |R L (t)| ≤ ct β . In particular, since β < p -2, there exists t 1 ≥ 1 such that for all t ≥ t 1 , we have |R L (t)| ≤ λ(p-1)

  .7.53) systems Proof. We first iteratively use Lemma 29 to construct the coupling ensemble E Ω [•] between the ensembles E[•] and E Ln [•], for n ∈ N (and a sequence L n ↑ +∞) in such a way that it generates Gaussian fields G and G Ln , and that the joint law of (G, G Ln ) is the same under E Ω [•] and E Ln [•]. In particular, we first generate G (with a law determined by E[•]

  3. Proof of (3.7.53). The argument is essentially the same as for (3.7.50), and we only argue for ∇ φξ,e . First, by (3.7.50) and Lemma 13, the linearized extended corrector ( φξ,e , σξ,e ) is well-defined. By the Schauder theory in form of [76, Theorem 5.19] applied to the equation (3.2.9) (the Hölder norm of the coefficients is controlled by Lemma 25 and (3.7.54)) and assumption (3.7.52), sup L≥1 E[ e+∇ φL

	q
	ξ,e

  Now let N ∈ N depending only on d and (x i ) i∈ 1,N ⊂ B 11 8 r be such that Since |z| ≤ 11 8 r, there exists i ∈ 1, N such that |z -x i | ≤ 1 16 r. Thus by the triangle inequality |y -x i | ≤ |y -z| + |z -x i | ≤ |y| 1 -

						systems
				N
				B 11 8 r ⊂	B 1 16 r (x i ).	(3.7.60)
				i=1
	We claim that			N
	B r (x 0 )+ 9 8 r ⊂	B r (x 0 )-3 16 r (x i ).	(3.7.61)
				i=1
	Indeed, let y ∈ B r (x 0 )+ 9 8 r and set z =	11 8 r r (x 0 )+ 9 8 r y. 11 8 r r (x 0 ) + 9 8 r	+	1 16	r.
	Since y ∈ B r (x 0 )+ 9 8 r and since the 1 16 -Lipschitz property of r entails r (x 0 ) ≥ r (0) -1 16 |x 0 | ≥ r 3 -r 16 = 13 48 r ≥ 1 4 r, we have
	|y| 1 -	11 8 r r ,ξ,L (x 0 ) + 9 8 r	≤ r (x 0 ) -	1 4	r,
	and consequently
	|y -
						9 8 r ≤ 35 8 r (0) so that for all
	x ∈ B r 8 ,	|B (x 0 )| ∼ |B r (x 0 )+ 9 8 r (x)|.
	Combined with (3.7.59) this yields
		f		m		f	m	dx.
	B (x 0 )			B r 8	B r (x 0 )+ 9 8 r (x)

  Lemma 34. Let x 0 ∈ R d and r ≥ 3r (x 0 ), f : R d → R + a measurable function. We have ˆBr(x0)By reverting from balls to cubes, this implies that there exists C ≥ 1 depending only on d such that for all r ≥ Cr (x 0 ) ˆQr(x0)

	3.7. Appendix							217
					B (x)	f dx	ˆQ2r (x 0 )	f,	(3.7.64)
	and	ˆQr(x0)	f	ˆQ2r (x 0 )	B (x)	f dx.	(3.7.65)
					B (x)	f dx	48 r (x 0 ) ˆB 67	f,	(3.7.62)
	and	ˆB 17 12 r (x 0 )	f	ˆB2r (x 0 )	B (x)	f dx.	(3.7.63)

16 r ≤ r (x).

  .[START_REF] Douglas | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] cf.[80, (140)] (which relies on the construction of a Calderón-Zygmund partition of R d based on r ). We start with the proof of (3.7.62). Since r is1 16 -Lipschitz and 3r (0) ≤ r, for all x ∈ B r , r (x) ≤ r (0) + r 16 ≤ 19 48 r so that B (x) ⊂ B 67 48 r . This yields (3.7.62) in form of ˆBr This bilinear form c determines the centred Gaussian measure • , see[START_REF] Martin | An introduction to stochastic pdes[END_REF] Proposition 3.9]. We will assume that there exists an Hilbert space H densely embedded in X and that c is continuous as a bilinear form in H * . More precisely, since by the inclusion H ⊂ X we also mean that the topology of H is finer than that of X, an element ∈ X * can be identified with a bounded linear form on H and thus, by Riesz's representation theorem, with an element of H itself. Hence we haveX ∼ = H * is denoted by | • | H and the inner product by (•, •) H .It is well-known that such a Gaussian measure only exists provided the embedding H ⊂ X is compact in a quantitative way. A simple sufficient criterion, see [33, Example 3.9.7.] serves as a convenient assumption for our application:

	with also the first embedding being dense. What we mean by the domination of c is
					c( , ) ≤ | | 2 H	for all ∈ X * ,		(4.1.2)
	where the norm on H h X ≤ |Ah| H ,			(4.1.3)
	B (x)	f dx =	ˆBr	B (x)	f 1 B 67 48 r dx ≤	ˆRd	B (x)	f 1 B 67 48 r dx	(3.7.66) ∼	ˆRd	f 1 B 67 48 r .
	We now turn to (3.7.63). By (3.7.66),							from
	ˆRd H and thus from (4.1.1) and (4.1.2) that h 2 n (h n , h) 2 X ≤ n µ 2 (4.1.3) that h 2 X ≤ n µ 2 n , which ˆB 17 12 r f ∼ B (x) f 1 B 17 is finite by the characterization of the Hilbert-Schmidt property. Note that by Fernique's theorem 12 r dx. [33, Theorem 2.8.5] we have
	Since r is 1 16 -Lipschitz and 3r (0) ≤ r, if |x| ≥ 2r, r (x) ≤ r (0) + 2 16 r ≤ 11 24 r ≤ 11 B (x) ⊂ R d \ B 85 12 r . Hence, exploiting the indicator function 1 B 17 12 r , the above turns into 48 r ⊂ R d \ B 17 48 |x|, so that g p X ≤ C
		ˆB 17 12 r	f ∼	ˆB2r	B (x)	f 1 B 17 12 r dx ≤	ˆB2r	B (x)	f dx,
	that is, (3.7.63).

* ⊂ H * ∼ = H ⊂ X for a symmetric Hilbert-Schmidt operator A on H, 2 where • X denotes the norm of X. In view of [33, Proposition A.2.6.(iii)], the assumption of symmetry is just for convenience. A quick way to see the connection between (4.1.3) and the existence of a Gaussian measure • goes as follows: By compactness and symmetric property of the Hilbert-Schmidt operator A, there exists an orthonormal basis (h n ) n∈N of H formed by eigenvectors of A with eigenvalues (µ n ) n∈N . Hence we obtain p < ∞ for all p < ∞.

(4.

1.4)

  3 , see[START_REF] Igorevich | Gaussian measures[END_REF] Proposition A.2.9.]. In view of[START_REF] Igorevich | Gaussian measures[END_REF] Proposition A.2.10. (ii)], it suffices to argue that B is of trace class, which we will do now. By boundedness of b on X, B is bounded from X to H. Since as a consequence of (4.1.3), H ⊂ X is compact, B is compact (with respect to H). Hence there exists an orthonormal basis (h n ) n∈N of H formed by eigenvectors of B. In order to show that B is trace class, we need to show summability of the modulus of eigenvalues , Bh n ) H | < ∞, see [33, Proposition A.2.9.(iii)]. The summability of |h n .Bh n | = |b(h n , h n )| in turn follows via the boundedness of b (with respect to X) and the assumption (4.1.3) in form of|b(h n , h n )| ≤ b h n X h n X (4.1.3) ≤ b |Ah n | H |Ah n | H ,(where b denotes the norm of the bilinear form b with respect to X) combined with the characterizing property |A| 2 HS := n |Ah n | 2 < ∞ of a Hilbert-Schmidt operator4 A. orthonormal basis (h n ) n∈N of H. By the symmetry of Ċ and B, which implies that the traces of ĊB and B Ċ are equal, we also have

	+∞ b. ċ = for an arbitrary b. ċ = b(h +∞ +∞ n=0 ċ(h n , Bh n ) +∞ n=0 n=0 |Ah n ||A Ċh n | ≤ |A| 2 HS | ċ|, where | ċ| denotes the norm of the bilinear form ċ with respect to H. Inserting (4.1.7) into (4.1.6) (4.1.5) (4.1.8) we obtain from (4.1.8) the boundedness of the pairing: |(h n It follows from the definition of the trace of ĊB that the pairing can be recovered by |b. ċ| ≤ |A| 2 HS b | ċ|. (4.1.9)
	n=0

n , Ċh n ). (4.

1.6)

We now recall the boundedness property of this pairing. For later use, we note that by (4.1.3), the summands in (4.1.6) are estimated as

|b(h n , Ċh n )| ≤ b |Ah n | H |A Ċh n | H . (

4

.1.7) Using the fact that, since Ċ is a continuous operator in H, A Ċ is an Hilbert-Schmidt operator in H with |A Ċ| HS ≤ |A| HS | Ċ| (where | Ċ| denotes the operator norm on H), see [33, Proposition A.2.10.], we learn by Cauchy-Schwarz' inequality that the dominating series in (4.1.7) is summable

  .2.7)We now argue as for (4.1.9). Let C L be the Riesz representation of c L in H and observe that for all n, m ∈ N c L (h n , h m ) = (h m , C L h n ) H . Thus, we deduce from the definition (4.2.4) of P N and the fact that P N is symmetric that, for all n > N (writing A * the adjoint operator of A)

	m>N

  2 1 2 , vanishes as M ↑ ∞ uniformly in N and L. By the identities (4.2.12) and (4.2.13) together with (4.2.14) this establishes (4.2.11).

  Lemma 36. We keep the notations of Subsection 4.3.1 and Lemma 35. We consider the spaces X and H as in (4.3.6) and (4.3.7). Assume that B is an Hilbert-Schmidt operator in X. Then, b admits a (unique) kernel k ∈ L 2 dµ⊗dµ (R d × R d ),that is for all g, g ∈ X is defined in (4.3.8) and the sequence (λ n ) n∈N ⊂ R d is now squared summable. In addition, using the L 2 (R d )-based structure of X, we have for all g ∈ X

		b(g, g ) = ˆˆk(x, y)g(x)g(y)dµ(x)dµ(y).	(4.3.9)
	Moreover, for all x ∈ R d , δ x : g ∈ H → g(x) is a continuous linear form on H and Formula (4.3.2)
	specifies to	b. ċ = ˆˆk(x, y) ċ(δ x , δ y )dµ(x)dµ(y).	(4.3.10)
	Proof. We use the notations of the proof of Lemma 35. Recall that from (4.3.4), we have the
	decomposition
		+∞
		b =	λ n n ⊗ n ,	(4.3.11)
		n=0
	where n n

  .15) where δ x : g ∈ H → g(x). This integral makes sense since, by definition9 of | • | H , |δ x | (1 + |x| 2 ) k so that since h n ∈ X, ´(1 + |x| 2 ) k h n (x)dµ(x) < +∞.Consequently, using Formula (4.3.2) from Lemma 35 combined with (4.3.9), we obtain Finally, by continuity of ċ inH (see (4.1.2)) one has | ċ(δ x , δ y )| ≤ |δ x | H |δ y | H (1 + |x| 2 ) k (1 + |y| 2 ) k so that (x, y) → ċ(δ x , δ y ) belongs to L 2 dµ⊗dµ (R d × R d ).To conclude, (4.3.16) simplifies to (4.3.10) by exchanging the sum and the integrals as well as the definition of the kernel k in (4.3.14).We end this chapter by the following corollary, which is a direct consequence of Price's formula of Theorem 26 and the paring (4.3.10). It states the version of Price's formula that we use in Chapter 5.

	b.	ċ (4.3.2) =	+∞	λ n ċ( n , n )
			n=0
		(4.3.15) =	+∞	λ n ˆˆh n (x)h n (y) ċ(δ x , δ y )dµ(x)dµ(y).	(4.3.16)
				n=0

  .2.1)Instead of directly estimating ξ * • āξ L -ξ * • a hom ξ, we will estimate its derivative with respect to L, that is d dL ξ * • āξ L . The reason is that by general Gaussian calculus (in form of Price's formula of Chapter 4 applied to the ensemble • L of (periodic) fields g that depends on a parameter L, we have for any F = F (g)

	d dL	F L =	1 2 ˆRd	dx	ˆRd	dy	∂ 2 F ∂g(-x)∂g(-y) L	∂c L ∂L	(x -y),	(5.2.2)

  and (5.4.36) below. Next, we show that the annealed estimates of Lemma 38 imply ˆ[0,L) d |∇v| 2pArgument for (5.4.30) and (5.4.31) Notice first that (5.4.30) is a direct consequence of Lemma 38 applied to v, which satisfies(5.4.34), with weight 1. Therefore, it remains to establish(5.4.31). In this perspective, we introduce the (gradient) field h j such that the r. h. s. of(5.4.36) reads P e j •a * ∇v = ∇ • h j . As a consequence of annealed unweighted estimates on ∇(-∇ • a * ∇) -1 ∇• (namely, Lemma 38 with weight µ = 1), we getˆ[0,L) d |∇w j | 2pAs a consequence of annealed weighted estimates on ∇(-∇ • a * ∇) -1 ∇• (namely, Lemma 38 with weight µ = | • | 2 L ) applied to (5.4.34), we have ˆ[0,L) d |x| 2 L |∇v| 2p Combining it with the Hardy inequality (5.4.38) for h j and with (5.4.39) yields Inserting this and (5.4.40) into (5.4.37), and employing once more (5.4.40) in the triangle inequality gives (5.4.31). for (5.4.38) W. l. o. g. we may assume that L = 1, and we consider random periodic functions of vanishing average u. The annealed Hardy inequality (5.4.38) relies on three ingredients. Inserting this into (5.4.44) yields the desired (5.4.38) (noting that by periodicity, we can replace [-1 2 , 1 2 ) by [0, 1) d ). We now establish successively (5.4.41), (5.4.42), and (5.4.43). First, (5.4.41) comes by applying the following Hardy inequality for compactly supported functions v: |∇ |v| 2p 1 2p | = | |v| 2p 1 2p -1 |v| 2p-1 ∇|v| | ≤ |∇v| 2p 1 2p . Similarly, we get (5.4.42) from the usual Poincaré inequality applied to the function |u| 2p 1 2p

									ensemble instead of its realizations
	Last, we make use of an annealed Poincaré-Wirtinger estimate
		ˆ[-1 2 , 1 2 ) d	|u| 2p 1 2p		ˆ[-1 2 , 1 2 ) d	|∇u| 2p 1 2p .	(5.4.43)
	1 p L Using (5.4.41) for v := ηu where η is a cut-off function of [-1 1 2 ˆ[0,L) d |h j | 2p + |v| 2p 2 , 1 2 ) d into [-3 1 p 4 , 3 4 ) d , we have by 1 2 . (5.4.37) periodicity of u L We now claim the following annealed Hardy inequality: ˆ[0,L) d |v| 2p 1 p L 1 2 |x| 2 L |∇v| 2p 1 p L 1 2 . (5.4.38) ˆ[-1 2 , 1 2 ) d |u| 2p 1 p ˆ[-1 2 , 1 |x| 2 p + ˆΩ |u| 2p 1 p , 1 |∇u| 2p 1 2 ) d Inserting (5.4.42) and then (5.4.43) into the above estimate yields ˆ[0,L) d ˆ[-1 2 , 1 2 ) d |u| 2p 1 p ˆ[-1 2 , 1 2 ) d |x| 2 1 |∇u| 2p 1 p + ˆ[-1 2 , 1 2 ) d 2 |∇u| 2p 1 2p . (5.4.44)
	1 p L Since d > 2, we may employ the Cauchy-Schwarz inequality to get 1 2 ˆ[0,L) d |x| 2 L |f | 2 1 2 , and therefore, by (5.4.38), there holds ˆ[0,L) d |v| 2p 1 p L 1 2 ˆ[0,L) d |x| 2 L |f | 2 1 2 . ˆ[-1 2 , 1 2 ) d |∇u| 2p 1 2p 2 ≤ ˆ[-1 2 , 1 2 ) d |x| 2 1 |∇u| 2p 1 p ˆ[-1 2 , 1 2 ) d ˆ[-1 2 , 1 2 ) d |x| 2 1 |∇u| 2p 1 p .	|x| -2 1	(5.4.39) (5.4.40)
	Moreover, by the annealed weighted estimates on ∇ 2 (-) -1 [113, Theorem 7.1], we obtain
		ˆ[0,L) d	|x| 2 L |∇h j | 2p ˆRd	1 p L	1 2	ˆRd	ˆ[0,L) d	|x| 2 L |∇v| 2p	1 p L	1 2 .
				|v| 2			|x| 2 |∇v| 2 ,
	to v	ˆ[0,L) d 2p , and noticing that by the Hölder inequality with exponents ( 2p |h j | 2p 1 p L 1 2 ˆ[0,L) d |x| 2 L |f | 2 1 2 |u| 2p 1 2p-1 , 2p)
	Argument First, if |v| 2p 1 p is compactly supported, we have
			ˆ	|v| 2p 1 p	ˆ|x| 2 |∇v| 2p 1 p .	(5.4.41)
	1 p L 4 ) d , the following annealed Poincaré estimate holds: 1 2 |f | 2 1 2 , 4 , 1 2 ) d \[-1 2 , 1 Next, for Ω := [-1 ˆ[0,L) d ˆ[0,L) d |∇w j + ve j | 2p 1 p L 1 2 ˆ[0,L) d |x| 2 L |f | 2 1 2 , ˆΩ |u| 2p 1 p 1 2 ˆΩ |∇u| 2p 1 p 1 2 + ˆΩ |u| 2p 1 2p .	(5.4.30) (5.4.31) (5.4.42)

  • ε ) -a)∇u ε , (6.2.1) is a key quantity to understand the fluctuations, since it encodes the convergence of ∇u ε and a( • ε )∇u ε in the sense that Ξ ε 0 if and only if ∇u ε ∇u and a( • ε )∇u ε a∇u.

Now, by defining the standard homogenization commutator Ξ = (Ξ i ) i∈ 1,d by Ξ i := a(∇φ e i + e i ) -a(∇φ e i + e i ),

Einstein's original result contained an error, which was corrected with help of his student and collaborator Ludwig Hopf in[START_REF] Einstein | Berichtigung zu meiner arbeit: Eine neue bestimmung der moleküldimensionen[END_REF].

The general notation • is used to either mean the average on the periodic cell ffl Q or the expectation E[•] depending on the context.

The divergence form of the right-hand side allows us to consider also the case Ω = R d . Note that if Ω is bounded, we can always rewrite the right-hand (provided it has zero mean) in divergence form. We precise that boundary layers are not discussed in this thesis and we fully focus on the homogenization error. In coordinates the system readsd j=1 ∂j((a( • ε , ∇uε))ij) = d j=1 ∂jfij, for all i ∈ {1, ..., m}.

Note that we do not assume a dependence of q in uε and in particular our results do not apply to scalar conductivity models of the form q = A( x ε )u q ε (s + |∇uε| p-2 )∇uε for some s ≥ 0, q > 0 and p ≥ 2.

We omit the domain of integration when we integrate on the whole space R d .

We use the notation L 1 (dP) := {F measurable | E[|F |] < +∞}.

More generally, by density ifF ∈ L q loc (R d , L 1 (dP)), F (r•, a) r↑+∞ E[F ] in L q loc (R d ), P-almost surely.

Without this choice, we only get the stationarity of the probability distribution with respect to Z d translations.

Similar model can be consider as well using vector valued white noise, see[START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF] Definition 5.5].

X ∼ Y means that the random variables X and Y have the same distribution.

For β = d, a logarithmic correction |m(x)| ≤ c(1 + |x|) -1 2 (d+β) log -1 2 (1 + |x|) is needed.

Indeed, (1.1.5) implies(1.1.15).

Because of measurability issues, the definition of the two-scale expansion (only in the nonlinear setting) has to be slightly modified using a piecewise constant approximation. We refer to Theorem 12 for more details.

We can not directly apply the assumption (1.1.9). However, a standard piecewise constant approximation of f combined with the assumption (1.1.9) show the convergence. The details are left to the reader.

The notation A ∼ B means that there exist a constant γ such that 1 γ B ≤ A ≤ γB.

Note that since u ≥ 0, we only need to construct the correctors in the directions ξ ∈ R + in order to express the two-scale expansion(1.1.31), that we do for simplicity.

For d = 1, one can make explicit computations (as in Subsection 1.1.4) and show that the convergence rate of the homogenization error is of order √ ε.

The choice f = 1 π arctan π• is taken for convenience but any function such that |f | < 1, f (0) = 0 and f (0) = 1 would work as well.

We use the notation dist(U, V) = inf (x,y)∈U ×V |x -y| for the distance between two sets.

We may rigorously define ∂ osc x,1 F as the measurable envelope of (1.1.60).

This model corresponds to a stationary linear diffusion, corresponding to the linearization of (1.1.4). We also make the choice to work in the whole space to avoid boundary layer problems.

For notational simplicity, we consider scalar Gaussian field, however, the Gaussian field g may take values in any finite dimensional linear space.

This formal argument can be made rigorous if the ensemble • itself is discrete and periodic with a very large period with respect to L.

It can be generated via the model g = m W, for m ∈ L

(R d ) and W a white noise. See Section 1.1.2.2 We use the symbol • to address both the ensemble and to denote its expectation operator.

1 √ T

We recall that Ẇ 1,p (R d ) := {u ∈ L p (R d )|∇u ∈ L p (R d )}.

The new feature in[START_REF] Bella | On the regularity of minimizers for scalar integral functionals with (p, q)-growth[END_REF], which establishes C k,α -regularity for local solutions of ∇ • a(∇u) = 0, is the largest range of exponents 2 ≤ p < 2(d-1) d-3 compared to previous contributions -more general results with right-hand sides and nonlinear Calderón-Zygmund theory, which are expected to hold true as well, have not been established yet.

This condition, which comes from an argument of[START_REF] Bella | Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations[END_REF] and is used in the main part of this paper dedicated to large-scale Meyers estimates, is also crucially used in[START_REF] Bella | On the regularity of minimizers for scalar integral functionals with (p, q)-growth[END_REF].

Indeed, for g = |BR| -1 1B R , the right-gand side of (3.2.7) scales like R -d 2 .

We use the symbol • to address both the ensemble and to denote its expectation operator.

We recall that it means that A is compact and for any orthonormal basis (hn) n∈N , n |Ahn| 2 H < +∞. Equivalently, if A is symmetric, n µ 2 n < +∞ where (µn) n∈N denotes the eigenvalues of A.

[START_REF] Allaire | Homogenization and two-scale convergence[END_REF] Recall that tr( ĊB) := +∞ n=0 (hn, ĊBhn)H , for an arbitrary orthonormal basis (hn) n∈N of H. We say that ĊB is of trace class if tr( ĊB) < +∞ (and then, it is independent of the choice of the orthonormal basis).

This definition of the Hilbert-Schmidt norm is independent of the choice of orthonormal basis of H, see[START_REF] Igorevich | Gaussian measures[END_REF] Proposition A.2.6.].

We identify here the Riesz representation of a linear form and the linear form itself via H * ∼ = H.

Meaning that b is continuous with respect to the weak topology in X.

Since X has an Hilbert structure, it is equivalent to say that B is a compact operator on X.

The uniqueness is deduced by density of the family {(x, y) → N n=1 gn(x)gn(y)|(gn)n ⊂ L 2 (R d ), N ≥ 1} in L 2 (R d ).

We can see it via the inequality (which follows by the chain rule)|(1 + | • | 2 ) k (1 -∆) k g| L 2 |(1 + | • | 2 ) k (g, ∇g, ..., ∇ 2k g)| L2 and the Sobolev embedding |g(x)| |(g, ∇g, ..., ∇ 2k g)| L 2 (B 1 (x)) .

For notational simplicity, we consider scalar Gaussian field, however, the Gaussian field g may take values in any finite dimensional linear space.

By uniqueness (up to additive constants), we have φ = i ξiφi.

Stationarity is understood here in the sense of shift covariance ∇φ(a(• + h), x) = ∇φ(a, x + h), for any h ∈ R d .

We use here formula (4.3.17) formally, where the kernel is formally given by the formal derivatives ∂ 2 F (x, y) = ∂ 2 F ∂g(-x)∂g(-y) .

While we work with the assumption A * = A and thus have a * = a, keeping primal and dual medium apart reveals more of the structure.

Since we are only interested in the mixed gradient of the Green function, the dimension d = 2 poses no problems here.

The language of quenched and annealed arises from metallurgy estimate and made its to model with disorder in statistical mechanics

We recall that X = g ´|g| 2 dµ < +∞ , where µ :x ∈ R d → (1 + |x| 2 ) -4k .

Observe that if we anchor the second correctors as in(5.3.36), we have to do the same procedure in Lemma 39 to obtain a similar estimate as (5.3.39) (where note that the location of the support of g is not important).

Remerciements

Using then (3.3.3) in form of ρ ≥ r (0) ≥ r ,ξ,L (0, c 2 ), the definition (3.3.2), and Jensen's inequality, this yields the reformulation of (3.3.5)

(c 2 + 1)(1 + |ξ| p ).

Quenched perturbative regularity in the large

Quenched Meyers' estimate in the large

Recall that a ξ := Da(•, ξ + ∇φ ξ ). The elliptic operator -∇ • a ξ ∇ has unbounded coefficients, whose growth depends on the nonlinear corrector ∇φ ξ : There exists (c, C) ∈ R + × R + , depending on λ and p, such that for all

where

In addition, by (3.3.5) in Lemma 15 we have for all r ≥ r µ ξ

The main result of this section is the following quenched Meyers estimate in the large.

Theorem 18 (Quenched Meyers' estimate in the large). Under Hypothesis 3.1.2, for all ξ ∈ R d , there exists m > 2 depending on d, p and |ξ|, such that for all exponents 2 ≤ m ≤ m, and all Q L -periodic functions g and u related via The same result holds with a ξ replaced by a * ξ (the pointwise transpose field). We follow the standard strategy based on a reverse Hölder inequality and Gehring's lemma to prove this Meyers estimate. We start with the reverse Hölder inequality: Lemma 16 (Reverse Hölder inequality). Let Hypothesis 3.1.2 hold. Set q = p p-2 . For all x ∈ R d , r ≥ r (x), and all g and u related via (The choice of 67 48 and 17 12 is convenient for the sequel, but obviously not essential.) The same result holds with a ξ replaced by a * ξ (the pointwise transpose field). Not surprisingly, this estimate follows from the Caccioppoli and the Poincaré-Sobolev inequalities. As opposed to the case of uniformly bounded coefficients, the weight µ ξ is in the way (and cannot be treated as a Muckenhoupt weight, which it is not). In order to get the entire range of exponents 2 ≤ p < ∞ in dimension d = 3, we have to be careful in the Caccioppoli inequality. Inspired by [25, Lemma 1], we optimize with respect to the cut-off in Caccioppoli's inequality, which allows us to appeal to Poincaré-Sobolev in dimension d -1 rather than d (and therefore improve the integrability).

Lemma 17. Let q ∈ [1, +∞), assume that q > d- 1 2 if d ≥ 3, and let q * be given by (3.3.14). For 0 < ρ < σ < +∞, v ∈ W 1,q * (B σ ) and µ ∈ L q loc (R d ), the quantity

The proof of Lemma 17, which closely follows the proof of [25, Lemma 1], is postponed to Appendix 3.7.1. We now prove Lemma 16.

Proof of Lemma 16. Without loss of generality, we may assume x = 0 and ´B 17 (3.3.17)

We then apply Lemma 17 with exponent q = p p-2 for d ≥ 3 and q = 1 for d = 2, to the effect that J ( 67 48 r, 17 12 r, µ ξ , u) r -2d d-1 µ ξ L q (B 17

The main result of this section is the following control of r , which implies Proposition 7 in combination with Lemma 22. Proposition 8. There exists an exponent γ > 0 depending on |ξ| such that for all q ≥ 1, E L [r q ] 1 q q γ . Proof. We split the proof into three steps. In the first step, we control the level set {r = R} for all dyadic R using averages of the corrector gradient. In Step 2, we use Proposition 6 to reformulate the right-hand side using moments of r itself, and then buckle in Step 3 by exploiting the gain of integrability provided by the hole-filling exponent β > 0.

Step 1. Control of level sets of r . We claim that there exists a constant c > 0 (depending on ξ, d, p) such that for all dyadic R ∈ [1, L], and all 0 < κ, ε < 1 and q ≥ 1

∇ φ 2 dx q , (3.5.27) where C ,R := R -ε µ ξ 2 L ∞ (B 4R ) . By the defining property (3.5.19) of r (with a constant C to be chosen below), we have

By the Caccioppoli inequality of Lemma 26, (3.5.29) yields

so that, provided C is chosen large enough in (3.5.19), we have x) φ(y)dy dx. By the triangle inequality, Poincaré's inequality in L 2 (B R ), and the definition of C ,R , the above turns into

∇ φ 2 dx (3.5.28)

∇ φ 2 dx .

Chapter 4

Price's formula for infinite dimensional Gaussian measure We extend the so-called Price's formula (well known in finite dimension) to the case infinite dimensional Gaussian field. It gives a general formula for the derivative of a functional with respect to the covariance function, which is one of the main tool of Chapter 5. This Chapter is based on joint work in preparation with Felix Otto.

The Chapter is organized as follows: In Section 4.1 we recall the framework of infinite dimensional Gaussian field. In Section 4.2 we state and prove the extension of the Price's formula to the infinite dimensional Gaussian setting of Section 4.1. In Section 4.3, we discuss about the paring between two bilinear forms and we prove that under additional assumptions, the paring can be rewriting in a natural way that will be used in the next chapter.

Reminder on infinite dimensional Gaussian measures

We recall in this section the framework of infinite dimensional Gaussian field that we will use in this chapter and in Chapter 5. For more details, we refer to the general book of Gaussian measures of Bogachev [START_REF] Igorevich | Gaussian measures[END_REF].

We consider a centred Gaussian measure 1 • on a Banach space X, which is defined by imposing that the push-forward under any bounded linear form ∈ X * is a centred one-dimensional Gaussian. By the Cauchy-Schwarz inequality in L

which gives (4.1.11). We will show in Section 4.3 that Formula (4.1.11) also holds in the infinite dimensional setting if we assume that b is compact and admits a decomposition of the form (4.1.10) (that we will specify when X has an Hilbert structure for simplicity).

Statement and proof of the extension of Price's formula

for some exponent p < ∞. Then L → F L is continuously differentiable with

Proof. First of all, since (4.2.1) implies

we learn from (4.1.4) that F L is well-defined. We then split the proof into two steps. We first prove by an approximation argument that it is enough to establish (4.2.2) when d 2 F has a finite rank, that we prove in the second step.

Step 1. Finite rank approximation. We will establish (4.2.2) by making use of an finite rank approximation, which reduces to prove the classical Price formula in finite dimension. The finite rank approximation is established with help of the inclusion X * ⊂ H and projection. Indeed, since X * is dense in H, there exists an orthonormal basis (h n ) n∈N ⊂ X * of H. We use it to define for every N ∈ N the linear (projection) map, for all g ∈ X P

About the paring of bilinear forms and application

We discuss in this section about the paring (4.1.6). We first show that under the additional assumption that b in compact 6 , we can rewrite (4.1.6) in a more canonical way as we did for the finite dimensional case in (4.1.11). Then, in view of an application to the next chapter, we will specify the spaces X in H as L 2 -based spaces and show that if b is in addition Hilbert-Schmidt in X, the paring can be rewritten with help of a kernel for b.

Simplification of the paring (4.1.6)

For simplicity, we assume that X has an Hilbert structure and we strengthen the assumption (4.1.3) by assuming that (

where (•, •) X denotes the inner product in X. We consider a bounded symmetric bilinear form b on X (and we denote by B its Riesz representation in X) and a bounded symmetric bilinear form ċ on H ⊃ X * (where we denote by Ċ its Riesz representation in H). We have the following lemma.

Lemma 35. Assume that b is compact 7 on X. There exist a bounded sequence (λ n ) n∈N ⊂ R and an orthonormal basis

with for all n ∈ N,

Proof. Since B is compact and symmetric on X (because b is), there exist a bounded sequence In Chapter 5, the family of covariance function L → c L will be constructed by a periodization procedure from c. The purpose of our application of Formula (4.2.2) is to vary the period L. We identify a random variable g on the torus [0, L) d with an L-periodic random variable g on the whole space R d . Thus, we build X and H as weighted L 2 (R d )-based spaces. We consider an exponent k > d 4 and a weight µ : x ∈ R d → (1 + |x| 2 ) -4k as well as the corresponding measure dµ = µ(x)dx. We define the spaces

and

that we endow with the two inner products, for all g, g ∈ X and g, g ∈ H (g, g ) X = ˆg(x)g (x)dµ(x), as well as

Obviously, (4.3.1) holds with

We claim that A is an Hilbert-Schmidt operator in H. In view of the definition (4.3.8), it is equivalent to prove that the conjugate operator à of A given by: for all g ∈ H

which can be seen as a convolution operator with a kernel k :

. Therefore, Ã is an integral operator where its kernel is given by

In order to show that

which is finite by the smoothness of K and the decay of its Fourier transform.

The assumptions of Subsection 4.3.1 are thus satisfied. We now rewrite the Formula (4.3.2) for the specific X and H defined above.

Assumption and statement of rigorous result

We restate in this section the assumption of the chapter as well as our main result. For an introduction to this result, we refer to Section 1.2.6. We start by recalling the important quantities. Let L ≥ 1 and a : R d → R d×d be λ-uniformly elliptic (for some λ > 0):

Suppose that the coefficient field a is L-periodic, meaning that a(x+Lk) = a(x) for all x and k ∈ Z d . Given a Cartesian coordinate direction i = 1, • • • , d and denoting by e i the unit vector in the i-th direction, we consider the corrector (up to additive constants) φ

(1) i as the L-periodic solution of

i + e i ) = 0, (5.1.2)

to which we associate the corresponding homogenized matrix:

i + e i ).

(5.1.3)

We then recall the general assumptions.

5.2. Theorem 29: refinement and main ideas 233 Assumption 28. Let • be a stationary and centered Gaussian ensemble of scalar 1 fields g on R d , as determined by the covariance function c(x) := g(x)g(0) . We assume that there exists an α > 0 such that

We identify • with its push forward under the map

where A : R → R d×d is such that the coefficient field a is λ-uniformly elliptic, see (5.1.1). We assume that

We then recall the definition of the periodize ensemble • L .

Definition 5.1.1. For given L < ∞, let • L be the stationary and centered Gaussian ensemble of scalar fields g defined by the covariance function

Clearly, the covariance function c L and thus the realizations g are L-periodic. As in Assumption 28 we identify • L with its push forward under (5.1.6).

The purpose of the chapter is to prove the following.

Theorem 29. Let d > 2 and A be symmetric. Under the Assumption 28 on • , for all L, and with the Definition 5.1.1 of • L we have for the expectation ā L of ā defined in (5.1.3)

More precisely, we prove the refinement version of Theorem 30.

Theorem 29: refinement and main ideas

The two ingredients for Theorem 29 are a suitable representation formula for ā L , see Subsection 5.2.1, and its asymptotics through stochastic homogenization, here on the level of the mixed derivatives of the Green's function, see Subsection 5.2.2. We need the second-order version of stochastic homogenization because of an inversion symmetry. We refine Theorem 29 in Subsection 5.2.3 by identifying the leading-order error term, see Theorem 30. In Subsection 5.2.4 we will argue that the leading-order error typically does not vanish, by exploring the regime of small ellipticity contrast. T -∇ • a∇, see Proposition 11. In order to access the cancellations, we will perform a re-summation. Assuming for simplicity for this exposition that • has unit range of dependence, so that c is supported in the unit ball, we have that c L (z = 0) does not depend on L ≥ 2. Hence the second r. h. s. term in (5.2.7) does not contribute. By L-periodicity of the correctors, (5.2.7) can be re-summed to

where from now on we use Einstein's convention of summation over repeated spatial indices, here n ∈ {1, • • • , d}. Formula (5.2.9) is our final representation. Clearly, the sum over k is still not absolutely convergent. However, as we shall see in the next subsection, it converges after homogenization.

Approximation by second-order homogenization

In this subsection, we turn to the asymptotics of the representation (5.2.9) for L ↑ ∞. In particular, we shall argue why first-order homogenization is not sufficient and give an efficient introduction into second-order correctors.

As there is no contribution from k = 0, and since by our finite range assumption (for the sake of this discussion), z is constrained to the unit ball, the argument z + Lk of the Green's function satisfies |z + Lk| L. Hence we may appeal to homogenization to replace G(x, y) by G(x -y), where Ḡ denotes the fundamental solution of -∇ • ā∇. This appears like periodic homogenization as long as L is fixed, but in fact amounts to stochastic homogenization since we are interested in L ↑ ∞. Since we are interested in its gradient, we need to replace G by the two-scale expansion of G. (See below for more details on the two-scale expansion.) Since we are interested in the mixed gradient, the two-scale expansion acts on both variables. Hence in a first Ansatz, we approximate

where φ * j (1) denotes the solution of (5.2.4) with ξ * replaced by e j . To leading order, this yields by the periodicity of correctors

(5.2.11)

Applying k∈Z d k l to the r. h. s., we see that it vanishes by parity w. r. t. inversion k -k. This is an indication that the first order two-scale expansion (5.2.10) is not sufficient and that we have to go to a second-order expansion, which we shall describe now.

We need to replace the first-order version of the two-scale expansion of G by its second-order version. We recall the two-scale expansion in its first-order version: Given an ā-harmonic function ū, one considers u = (1 + φ [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] i ∂ i )ū as a good approximation to an a-harmonic function. Indeed, it follows from (5.1.2) that when ū is a first-order polynomial, u is exactly a-harmonic. In fact, this is a characterization of the first-order correctors φ

(1) i . Second-order correctors φ

(2) ij can be characterized in a similar way: For every ā-harmonic second-order polynomial ū, we postulate that u = (1 + φ (1)

ij depends on the choice of the additive constant in φ [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] i , which we now fix through

(5.2.12)

Since for our second-order polynomial ū we have

i e j + ∇φ

i e j +∇φ

(2) ij ) = 0, and using that ∇ • ā∇ū = 0, we obtain the following standard PDE characterization of φ

i + e i ) -āe i ).

(5.2.14)

Note that (5.2.14) is uniquely solvable (up to additive constants) for a periodic φ

ij because the r. h. s. of (5.2.14) has vanishing average in view of (5.1.3). The definition of φ * ij (2) for the dual medium a * is analogous.

In view of (5.2.13), we thus replace (5.2.10) by

(5.2.15)

It is here that the assumption of symmetry of A is convenient: Otherwise, the instance of G in the first r. h. s. term of (5.2.15) would have to be replaced by G + G (2) where G (2) is the (1 -d)-

is the second-order homogenized coefficient, see (5.3.20). Since G (2) , as a dipole, is odd w. r. t. point inversion, its contribution does not vanish as for G, c. f. (5.2.11). For the analogue of (5.2.11) we now turn to the first-order Taylor expansion (recall k = 0)

i e m + ∇φ

jm )(z). 7 This does not characterize all components φ

(2) ij separately but only the trace-free and symmetric part of this tensor, where the trace is defined w. r. t. ā. Since we apply the two scale expansion only to ā-harmonic functions like G, this is not an issue.

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations By the inversion symmetry of G and its homogeneity, this implies

i )(0) ⊗ (e j + ∇φ * j (1) )(z)

i e m + ∇φ

+(e i + ∇φ

jm )(z) .

(5.2.16)

In view of ā ≈ a hom we finally replace G, which is still random, by the deterministic G hom that may be pulled out of • L when inserting (5.2.16) into (5.2.9). Hence we obtain the approximation

where the five-tensor field Q L is defined through a combination of three covariances of quadratic expressions in correctors, see Definition 5.2.1, and where the four-tensor Γ hom is formally given by the (borderline) divergent lattice sum k∈Z d k n ∂ ijm G T,hom (k), which in line with the remark at the end of Subsection 5.2.1 we replace by

with G T,hom denoting the fundamental solution of 1 T -∇ • a hom ∇.

Refinement of rigorous result

We start with the full definition of the tensor field Q L appearing in (5.2.17). with a replaced by a * . For given vectors ξ and ξ * we continue to write φ (1) = ξ i φ i and φ * (1) = ξ i φ * (1) i . Consider the random tensor fields

i e m + ∇φ

jm ) • a (ξ + ∇φ (1) ) (z).

(5.2.20)

For any L we consider the ensemble • L from Definition 5.1.1 and define

Here comes the more precise version of Theorem 29, which consists in making (5.2.17) rigorous:

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations

Re-summation

Following Subsection 5.2.2, we now appeal to second-order homogenization, which allows for a resummation. As a by-product of the re-summation, we may pass to the limit T ↑ ∞ in (5.3.5). The difficulty with passing to the limit T ↑ ∞ lies in the {|z| ≥ L}-part of the integral in (5.3.5). We thus fix a smooth cut-off function

and to split the z-integral into the benign near-field part ´Rd dzη L (z) and the delicate far-field part ´Rd dz(1 -η L )(z). On the far-field part, we appeal to the two-scale expansion (5.2.15). Hence we have to monitor the homogenization error

i e m + ∇φ

where as before G denotes the fundamental solution for the constant-coefficient operator -∇ • ā∇.

The translation invariance of G together with the periodicity of φ (1) and φ (2) allows for a resummation. As in Subsection 5.2.2, we feed in a zeroth-and first-order Taylor expansion of G. This gives rise to the analogue of (5.2.18), namely

where G T denotes the fundamental solution of 1 T -∇ • ā∇. The existence of this limit follows by the same arguments given for (5.2.27). The Taylor expansion generates the additional error terms

(2)

(5.3.9)

Thanks to this re-summation, the subtlety of the T ↑ ∞ is limited to the not absolutely convergent sum in (5.3.7). The sums in (5.3.8) and (5.3.9) are absolutely convergent since both summands decay as |k| -(d+1) for |k| |z| L , see (5.3.13) and (5.3.14) for a more quantitative discussion. Equipped with these definitions, we are now able to express the limit T ↑ ∞ of (5.3.5): Proposition 12. Let Γ be as in (5.3.7), (1) and (2) as in (5.3.8) & (5.3.9), and E as in (5.3.6).

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations

We start with the second term and estimate (1) , see (5.3.8): In the range |k| ≥ |z| L , we obtain from Taylor applied to (1

Hence the contribution to the sum from this range is dominated by min{|z| 2 L -(d+2) , |z|L -(d+1) }. In the other range |k| ≤ |z| L , the contribution from the middle term vanishes by parity, the contribution from the last term is estimated by |z|L -(d+1) (by the same argument that shows that the limit (5.3.7) exists), and the first term in the summand is estimated by |k|L -d so that its contribution to the sum is also dominated by |z|L -(d+1) . Since this second range is only present for |z| ≥ L, we obtain in conclusion

(5.3.13)

For the estimate of (2) , see (5.3.9), we proceed in a similar way and obtain the stronger estimate

We now see that Assumption 28 is just what we need: By (5.1.4) with n = 1 (and w. l. o. g. α ≤ 1 2 ) we obtain for the second term in (5.3.10) ˆRd dz

(2)

Here and in the sequel means ≤ up to a multiplicative constant that only depends on d, λ, and the constants implicit in (5.1.4) and (5.1.6) of Assumption 28.

We now turn to the third term on the r. h. s. of (5.3.10). It follows from Proposition 13 i) and Proposition 14, together with (5.1.6) in Assumption 28, that

Inserting (5.2.8) we obtain the following estimate

Since by integrability of |∇c|, which is ensured by (5.1.4), the z-integral is estimated by (L|k|) -(d+ 3 2 ) , the sum converges and is estimated by L -(d+ 3 2 ) , which as desired is o(L -(d+1) ). We finally turn to the first term in (5.3.10). On the one hand, from the qualitative result (which can be seen as a consequence of Corollary 43) Since on the other hand, Γ is uniformly bounded (recall that ā is confined to the set (5.1.1)), and by (5.3.15), it is enough to show that the limits

ijm (z) L not only exist but can be characterized in terms of • . This is the content of Corollary 31 iii) in the following subsection.

We finally turn to the fourth term on the r. h. s of (5.3.10). We split the term into the near-zero contribution ´[0,1) d dz and the far-zero contribution ´Rd \[0,1) d dzη L . On the near-zero contribution, we decompose as

(5.3.16) On the one hand, the first r. h. s term of (5.3.16), thanks to our assumption (5.1.4), is of order L -d-1-2α . On the other hand, by Hölder continuity of a, φ (1) and ∂c L ∂L as well as the decay of ∇∇G(0, z), the integrand of the second r. h. s term of (5.3.16) is dominated by |z| -d+γ L -d-1-2α (for some γ > 0 depending on α) and thus this term is of order L -d-1-2α . On the far-zero contribution, from the decay of ∇∇G(0, z) and (5.1.4), the integrand is dominated by |z| -d L -d-1-2α and thus this term is also of order L -d-1-2α .

Stochastic corrector estimates up to second order

As just discussed, the proof of Theorem 30 will rely on estimates of not only the first-order corrector φ [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] i , but also its second-order version φ

(2) ij , see part i) of Proposition 13. Since the period L of the ensemble • L tends to infinity, these have to be of stochastic nature. This is the reason for the restriction to d > 2 (which is just a more telling way of saying d ≥ 3 since it is rather d = 2 that is borderline): For d = 2, the first-order corrector in the whole-space ensemble • is not stationary, so that one looses (pointwise) control even of a centered second-order corrector. Only for d > 2 one has the middle item in (5.3.23) (see for instance Theorem 15 of Chapter 3). For the (limiting) wholespace ensemble • , such higher-order corrector estimates have first been establishes in [START_REF] Gu | Scaling limit of fluctuations in stochastic homogenization[END_REF] (however sub-optimal in odd dimensions) and [27, Theorem 3.1] (see [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]Proposition 2.2] for a treatment of any order). These works, like ours, rely on Malliavin calculus and a suitable spectral gap estimate, as is available under Assumption 28 (see Proposition 9 of Chapter 3). Unfortunately, we cannot simply quote [START_REF] Bella | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors[END_REF] since we need the estimate for the periodized ensembles • L (uniform for L ↑ ∞, of course).

For Proposition 14, we need to also estimate the flux correctors, both first and second-order, which we shall recall now. (We also refer to [56, Section 2] for a compact introduction into all Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations higher-order correctors). It follows from (5.1.2) and (5.1.3) that a(∇φ

i + e i ) -āe i is divergencefree, periodic, and of zero average. Hence it allows for, in the language of d = 2, a periodic stream function, or in the language of d = 3, a periodic vector potential. For general d, it can be represented in terms of a periodic tensor field σ i with a(∇φ

inm ,

(5.3.17)

where for a (skew symmetric) tensor field σ, we write (∇•σ) m := ∂ n σ mn , as an instance of an exterior derivative. Observe that (5.3.17) does not determine σ

i . Indeed, σ

i , which can be interpreted as an alternating (d-2)-form, is only determined up to a (d-3)-form. For estimates like in Proposition 13, we choose a suitable (and simple) gauge, that is

(5.3.18)

Note also that (5.2.14) can be reformulated in divergence form

i e j .

(5.

3.19)

This shows that there is a second-order analogue of (5.3.17): For every coordinate direction i, let the matrix ā(2) be defined through

for any j = 1, • • • , d, and the periodic tensor field σ

ij through a(∇φ

i e j + ∇ • σ

(2) ij and σ

(2)

ijnm .

(5.3.21)

The merits of the flux correctors σ

(1) i and σ

(2) ij will become clear in Subsection 5.3.5. In fact, in that context it will be convenient to have yet one more object, namely the periodic solution ω i of

i .

(

ii) The random tensor fields σ

(1) i and σ

(2) ij can be constructed such that

iii) We have for any periodic deterministic vector field g

where µ

(2)

for 2 ≤ r and d = 3 ln

We have for all z

d (|z|).

(5.3.27)

Here means ≤ up to a multiplicative constant that only depends on d, λ, and the constants implicit in (5.1.4) and (5.1.6) of Assumption 28. The subscript p indicates an additional dependance.

While part i) of Proposition 13 is explicitly used in Section 5.3.3, the usage of the other parts more indirect: Part ii) is used in Corollary 31, part iii) is used to estimate the second-order homogenization error in Lemma 39; and part iv) is used to apply this to the Green's function, see Proposition 14.

The proof of Proposition 13 essentially follows the strategy of [101, Section 4] and extends it from first-order to second-order correctors; the passage from • to • L is only a minor change. In this paper, we will only establish the most important ingredient for Proposition 13, namely the characterization of stochastic cancellations of the gradient of the correctors in Lemma 37. While (5. For any deterministic periodic vector field f and any p < ∞ we have

where we denote by | • | L the periodic distance on T L . We note that (5.3.29) also holds with

q with q = 2d d+2 . However when passing from (5. In establishing (5.3.29), we use the same approach as [101, Proposition 4.1] for (5.3.28), namely we identify and estimate the Malliavin derivative of the l. h. s. and then appeal to the spectral gap estimate. However, while for the first-order result (5.3.28), a buckling is required, it is not necessary for its second-order counterpart (5.3.29). One can avoid it by appealing to the quenched Calderon-Zygmund estimate, see [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF] For all p ≥ 1 and γ ∈ (-d, d(p -1)), we have for µ :

where we recall that |x| L := inf k∈Z d |x + kL|.

An inspection of the proof of [101, Proposition 7.1 ii)] shows that the argument extends to the case with a weight in the corresponding Muckenhoupt class. Indeed, the only essential new ingredient is that this weighted annealed estimate holds for the constant coefficient operator, i. e. the analogue of [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Lemma 7.4]. This in turn follows from [START_REF] Lorist | On pointwise r -sparse domination in a space of homogeneous type[END_REF]Theorem 7.1]. Alternatively, one can derive the weighted estimate from the unweighted one and the dualized Lipschitz estimate Lemma 41, following the strategy of [80, Corollary 5]. Precisions will be added in the future version [START_REF] Clozeau | Bias in the representative volume element method: periodize the ensemble instead of its realizations[END_REF].

The limit L ↑ ∞ for the first r. h. s. term in (5.3.10) relies on the following purely qualitative consequence of Proposition 13.

i with (φ

i + e i ) = 0 a. s.. ii) For i, j = 1, • • • , d there exists a unique stationary random field ∇φ

i + e i ) -āe i ) a. s..

(5.3.31)

iii) We have |φ

.32)

where also the r. h. s. integrands are defined by the formulas (5.3.22) and (5.2.20).

The important element of part i) of Corollary 31 is the stationarity of φ

Estimate of homogenization error to second order, application to the Green function

A second main role of the corrector estimates of Proposition 13, in particular the estimate of the flux correctors, is to provide an estimate of the homogenization error. On our second-order level, this connection relies on identity (5.3.35) involving the two-scale expansion (5.3.34), which we recall now. Suppose that u and ū are related via

i ∇∂ i ū) = 0.

(5.3.33)

Consider the error in the second-order two-scale expansion

Then σ

(2) ij allows to write the residuum in divergence form:

Now the advantage of A and thus a being symmetric becomes apparent: It implies that the symmetric part of the three-tensor with entries ā(2) imn vanishes (see e. g. [56, Lemma 2.4]). Since (5.3.33) may be rewritten as -∇ • ā∇ū (2) = a

(2) imn ∂ imn ū, we may assume ū(2) = 0 under our symmetry assumption. Hence (5.3.34) simplifies to

and (5.3.35) may be rewritten as

We are allowed to pass to the centered versions of the second order (flux) corrector, by which we mean that (φ

ij ) is replaced by (φ

ij (0)), which we do with (5.3.25) in mind, since a change by an additive constant does not affect anything stated so far, and in particular not formula (5. Here p has the same meaning as in Proposition 13.

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations This pointwise estimate (5.3.39) relies on a decomposition of the r. h. s. of (5.3.37) into pieces supported on dyadic annuli. For each piece, we first apply the energy estimate, into which we feed (5.3.25) and the energy estimate for ∇ 2 ū, see (5.3.38), and then apply Lemma 41. For simplicity, we have taken the worst case in (5.3.26), namely d = 3; the finer estimates would come with an additional logarithm.

The main goal of this subsection is to estimate the homogenization error on the level of the Green's function, see Proposition 14. This type of homogenization result with singular r. h. s. has been worked out on the level of the first-order approximation in [START_REF] Bella | Quantitative stochastic homogenization: local control of homogenization error through corrector[END_REF]Corollary 3] and extended to second-order in [29, Theorem 1], where these estimates are derived from estimates on (φ

i ) and (φ

ij ) of the type of Proposition 13, however in a pathwise way, see [START_REF] Bella | Effective multipoles in random media[END_REF]Proposition 1]. While equipped with Proposition 13, we could post-process [29, Theorem 1] to obtain Proposition 14, we take a different, and shorter, route in this paper. Note that [29, Theorem 1] is not formulated in terms of the Green's function G, but in terms of decaying a-harmonic functions in exterior domains. Recovering a statement on the Green's function would require [28, Lemma 4], which we restate as Lemma 40 below for the convenience of the reader.

In this paper, we use Lemma 40, or rather Corollary 32, in a more substantial way than it is used in [28, Corollary 3]. Here comes an outline of the argument for Proposition 14: We apply Lemma 39 with the origin replaced by a general point x 0 . Writing u(x) = ´Rd dy g(y) • ∇ y G(x, y) and ū(x) = ´Rd dy g(y)

i (x 0 ))

i (x 0 )e j + ∇φ

3 (R) sup |∇ 2 g| with 2R the diameter of suppg; here we used the centering of φ

(2) ij in x 0 . We now fix a point y 0 with |y 0 -x 0 | ≥ 4 and replace both instances of G(x 0 -y) by what we obtain from applying the two-scale expansion operator in the y-variable

Provided g is supported in B R (y 0 ) with R := 1 2 |y 0 -x 0 | ≥ 2, this preserves the estimate: While for three out of the four extra terms, this follows directly from parts i) through iii) of Proposition 13, we need part iv) and an integration by parts in y for the contribution coming from φ * ( 1)

i (x 0 )). Keeping only first and second-order terms and recalling the definition (5.3.6), this yields ˆRd dy g(y) • E m (x 0 , y)

for any g supported in B R (y 0 ). By construction, up to third-order terms, R d -{x 0 } y → E m (x 0 , y) is a linear combination of a gradient of an a * -harmonic function, namely ∂G ∂xm (x 0 , y), and gradients of two-scale expansions of ā * -harmonic functions, namely of ū(y

ij (x 0 )). Hence we may appeal once more to (5.3.37), this time in the y-variable and thus for the dual medium, and with the origin replaced by y 0 . We decompose the r. h. s. of (5. Here comes the crucial Lemma that converts weak into strong control.

Lemma 40. Let • be an ensemble of λ-uniformly elliptic coefficient fields 8 . Let the random function u be a-harmonic in the ball B R of radius R. Then we have for all x and H -n

x (the case W -2,1

x of (5.3.42) is obtained for n > d 2 + 2). However, it strengthens [28, Lemma 4] by restricting the r. h. s. functional to smooth functions g with compact support, i.e., functions that vanish to appropriate order at the boundary. Nevertheless, it requires only a minor modification of the proof. It is obtained as a combination of two ingredients. First, by the Caccioppoli estimate and by an L 2

x interpolation estimate, we may estimate the l. h. s. of (5.3.42) by the L 2

x norm of w for ∆ 2n w = u. Second, appealing to the fact that the Dirichlet operator ∆ 2n has finite trace for 2n > d, we may obtain (5.3.42). This second step differs from [28, Lemma 4], where the Fourier decomposition was explicitly used to solve ∆ 2n w = u (thus, losing the property of compact support). This argument also shows that the second derivative on g, that we need here for our second-order homogenization, could be replaced by any order (properly non-dimensionalized).

We use Lemma 40 only in combination with a second inner regularity estimate, Lemma 41, which amounts to a Lipschitz estimate. Lipschitz estimates are central in the large-scale regularity theory in homogenization as initiated by Avellaneda and Lin in the periodic context, and as introduced by Armstrong and Smart [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF] to the random context. Here p,p has the same meaning as in Proposition 13. 8 We will apply it to • L for some fixed γ > 0. The derivative (5.3.5) is reinterpreted by d dL F (0) L (where the dependence on a is made implicit). Let (η n ) n∈N be a mollifying sequence and define the approximation of the coefficient field a by, for all n ∈ N a n : g ∈ X → η n A(η n g).

(5.4.2)

Now define the approximation

(5.4.3)

We now show that we can apply the version of Price's formula of Corollary 27 to F n , for all n ∈ N.

To do so, we check that F n is twice Fréchet differentiable in the space 10 X defined in (4.3.6), that g ∈ X → d 2 F n (g) is bounded (as a function taking value into the set of continuous bilinear form on X) and that d 2 F (g) is Hilbert-Schmidt for all g ∈ X. We argue in the two following substeps.

Twice Fréchet differentiability and boundedness of g → d 2 F (g). We prove that F n is twice Fréchet differentiable in X and that g ∈ X → d 2 F n (g) is bounded. From (5.4.1) and (5.4.3), the second variation of F n is given by (we use the notations a n := η n A (η n g) and a n = η n A (η n g) and skip the dependence in n for the massive corrector φ

T depending on a n ): for all g, δg, δg ∈ X

where 1 T δφ

(1)

T + ξ)(η n δg), (5.4.5) and

T (δg ))(η n δg)

T + ξ)(η n δg)(η n δg ).

(5.4.6)

We now verify that for all g ∈ X, d 2 F n (g) is a continuous bilinear form on X. Fix g ∈ X. First of all, thanks to the outer convolution in (5.4.2), one has

In addition, using local Schauder's theory to the equation (5.3.3) and the exponentially localized energy estimates (see Proposition 15) one has

T ] 0,α α,[a] 0,α ,T 1.

(5.4.7)

Then, using the exponentially localized energy estimate (see Proposition 15) followed by the local Schauder's theory to the equation (5.4.5), we obtain

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations By the same ingredients, it follows from (5.4.6) that

T,α,γ,n δg X δg X .

(5.4.9)

From the estimates 11 (5.4.7), (5.4.8), (5.4.9) and (5.4.5), we deduce that F n is twice Fréchet differentiable in X with for all g, δg, δg ∈ X

Hilbert-Schmidt property. We now prove that for all g ∈ X, d 2 F n (g, z) in Hilbert-Schmidt. Fix g ∈ X. We prove that

Note that (5.4.10) is formally obtained by evaluating (5.4.4) for δg = δ x and δg = δ y (where we recall that δ

For notational convenience, we evaluate δφ

T and δ 2 φ

(

T on Dirac distributions, which makes sense in (5.4.5) and (5.4.6) thanks to the convolution with η n . The fact that

) can be easily check by computing, for all δg, δg ∈ L 2 (R d ), ´´dx dy∂ 2 F n (z, x, y)δg(x)δg(y) and noticing that ´dx∇δφ T (δ x , δ y , z)δg(x)δg(y) are the solution of (5.4.5) and (5.4.6) respectively. We now check that

We only treat the last right-hand side term of (5.4.10), the others are treated the same way. From the same ingredients as for (5.4.8) and (5.4.9), one has |∇δ 2 φ (1)

(1)

since η n is compactly supported.

We are now in position to apply the version of Price's formula of Corollary 27:

where by definition c L (δ x , δ y ) = c L (x, y) (see Assumption 28). We now slightly rewrite (5.4.12) by using stationarity. Clearly, F has the shift invariance property F (a, z + h) = F (a(• + h), z) (from the uniqueness of the solution of (5.3.3)). Combined with the trivial shift invariance of a n , it translates to the shift invariance of F n which in turn translates to d 2 F n in form of 11 And a purely qualitative bit more which amounts to estimating the third derivatives. 12 Note that if

) in the sense of (4.3.9).

Proofs 255

for all h ∈ R d and g, δg, δg ∈ X. At the level of the kernel, it reads (where we make explicit the dependance in g ∈ X)

Consequently, using the stationarity of • L , one deduces that

(5.4.13)

With the identity (5.4.13) together with the fact that by stationarity of • one has c L (x, y) = c L (x -y), we arrive at (using the change x -y z and the fact that the argument for (5.4.11) also show that

(5.4.14)

We now compute

Fn (see (5.4.10). We proceed as in Subsection (5.2.1). We note that

T (δ x , δ 0 , z)) .

(5.4.15)

Ideed, from (5.4.5) and (5.4.6) we have for u :

Thus, by testing the above equation with φ * (1) T

and by testing δ 2 φ(δ x , δ 0 , •) in (5.3.3) (with a n replaced by a * n ) 13 , we obtain 1 T ˆφ * (1)

Taking the difference of the two identities above yields (5.4.15). Finally, using the definition of ∂ 2 Fn and (5.4.10), (5.4.15) turns into T ) as well as the fact that c L is even (which implies that the two contributions in the first right-hand side of (5.4.16) are identical), we arrive at

T (δ 0 , •))

T + ξ) L . 13 Which are both admissible since they decay exponentially by Schauder theory and the exponentially localized energy estimates, see Proposition 15

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations

We conclude by rewriting the first term of the above equation by duality, using the solution of

and equation (5.4.5), such a way that finally

T + ξ) L .

We then obtain (5.3.5) by letting n ↑ +∞, which easily follow from Schauder's theory (which gives compactness) and continuity of all the functions.

Proof of Proposition 12: Limit T ↑ ∞

The strategy of proof is simple. First, we reorder the terms of the derivative of d dL ξ * • a T ξ L in order to make appear the "massive" analogue (that is, involving the massive operator 1 T -∇ • a∇) of the r. h. s. of (5.3.10). For this first step, we essentially make precise the computations done in Section 5.2.2. Second, we systematically make use of the dominated convergence theorem to obtain the convergence of each term to its massless counterpart.

Reordering the derivative of ξ * • a T ξ L . We shall first establish that, under Assumptions 28, formula (5.3.10) holds on the level of the massive equation, that is

T ijm (z)ξ +

(1) 5.4.17) where the objects E T ,

T Lijn ,

T Lijlm , Q

T ij and Q

T ijm are defined by the "massive" analogue of (5.3.6), (5.3.8), (5.3.9), (5.2.19) and (5.2.20) respectively (where G, G shall be replaced by G T , G T ). Moreover, Γ T ijmn is defined by (5.3.7).

The proof of (5.4.17) follows from straightforward computations on (5.3.5), which are all legitimate on account of the exponential decay of the massive Green's function, see (5.3.1). Recall

Invoking (5.3.23) and recalling (5.4.30) and (5.4.31) finally yields the desired estimate (5.3.29).

Argument for (5.4.29) We give ourselves infinitesimal (periodic) perturbation δg of g. In view of (5.1.2) and (5.2.12), it generates a perturbation δφ

(1) i characterized by

i + e i ) = 0 and

In view of (5.2.14), this in turn generates the perturbation δφ

i e j ) + δga (∇φ

i e j ) = P e j • a∇δφ [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] i + δga (∇φ [START_REF] Abdulle | Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields[END_REF] i + e i ) ,

i e j + δga (∇φ

i + e i ) .

( 5.4.35) This in turn prompts the introduction of a second auxiliary periodic function w j of zero mean -∇ • a * (∇w j + ve j ) = P e j • a * ∇v, (5.4.36) so that by testing (5.4.32) with w j and (5.4.36) with δφ

i , we may eliminate δφ

(1) i in (5.4.35):

i + e i ) .

By definition of the Fréchet derivative, this amounts to (5.4.29).

Proofs

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations Therefore, inserting (5.4.50) into (5.4.49) and using the injection 2 ⊂ ∞ , we get

Next, assuming that L ≥ 2 √ N , from (5.1.7) and (5.1.4), we have for all x ∈ [0,

(5.4.52)

which we may insert into (5.4.51), to the effect of

Next, we enrich the ensemble • ΩL into • Ω L in such a way that the latter generates Gaussian fields g and g L of respective covariance c and c L that are coherent with G N and G N L , respectively, in the sense of the first equality of (5.4.47) (and in such a way that g conditioned by

. By a triangle inequality, this coupling

Hence, we may now use (5.4.48) to cope with the first two r. h. s. term, whereas we appeal to (5.4.53) to estimate the third r. h. s. term, obtaining

(where we assumed that √ N ≥ R and L ≥ 2 √ N ). We now optimize the latter inequality by setting

Interpolating this result with (5.4.46) concludes the proof.

Capitalizing on Lemma 42, we show that, under the coupling • Ω L , the correctors φ (1) and φ (2) are as well close to their counterparts φ

L and φ

(2)

L . More precisely:

Lemma 43. For any R ≥ 1 and p < +∞ we have the following convergence:

(a, ∇φ (1) , φ (1) , ∇σ (1) , σ (1) , ∇φ (2) -a L , ∇φ

L , φ

L , ∇σ

L , σ

L , ∇φ

L )

p

where the functions with subscript L and with no subscript refer to the functions previously associated with • L and • , respectively.

Proofs 265

The proof of (5.4.54) relies on a more general and abstract result (that we then apply succesively to all quantities in (5.4.54)). We now prove the following general result: Lemma 44. Under Assumptions 28, suppose that, for all the ensembles • L with L ∈ [1, +∞), we are given two square-integrable stationary random fields on T L , namely f and ∇u, that are related through the equation

and that satisfy the following bounds

for a fixed constant C. In addition, suppose that we have two stationary fields f and ∇u on R d associated with • that are related through the equation (5.4.55) and satisfy

along the sublinear growth property

Let • Ω L denote the coupling ensembles between • and • L that are built in Lemma 42. (We denote with a subscript L and without a subscript the coupled random variables previously associated with • L and • , respectively.) If, given a sequence L n ↑ ∞, we have the convergence

then there holds ∇u Ln -∇u 2

Proof of Lemma 44. The proof relies on classical mechanisms of weak convergence and elliptic theory. It falls in three steps. First, we build a global coupling ensemble • Ω between all the ensembles • and • Ln , and we denote by Ω an associated probability space. Second, we extract a weak limit u ∞ of u Ln in L 2 Ω, H 1 loc R d , and show it solves (5.4.55) along with (5.4.58); thus, by a Liouville theorem u ∞ -u is constant. Last, we establish the strong local L 2 convergence of ∇u Ln to ∇u, namely (5.4.60).

Step 1: We may iteratively use the construction of Lemma 42 to build the coupling ensemble

• Ω between the ensembles • and • Ln for n ∈ N in such a way that it generates Gaussian fields g and g Ln , and where the joint law of (g, g Ln ) is the same under • Ω as under • Ω Ln (defined in Lemma 42). That is, we first generate g of law determined by • and then we successively build the Gaussian fields g Ln in such a way that the conditional probabilities P Ω (g Ln |g) are all independent of each others.

Note that, since A is Lipschitz regular (see (5.1.6)), then (5.4.45) can be turned into

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations we deduce from the fact that ∇φ (1) -∇φ

L satisfies (5.4.54) and from (5.4.68) that

and get the desired convergence

(5.4.70)

Step 3.

We justify that we may apply Lemma 44 (with the slight modification that a is replaced by the identity matrix in (5.4.55)) for f := ∂ j (e n • a(e i + ∇φ

imn . By Step 2 above, we know that f satisfies (5.4.59). Therefore, we can apply Lemma 44 and, by a similar reasoning than in Step 2, we obtain successively the two convergences ∇σ (1) -∇σ

(5.4.71)

Step 4. Finally, we set f := aφ

i e j and u := φ

ij (0). We check that (5.4.55), (5.4.56), (5.4.57) and are satisfied by appealing to Proposition 13. Also, we deduce (5.4.59) from (5.4.70) and (5.4.71). Therefore, as in Step 2, we may apply Lemma 44, to the effect of ∇φ (2) -∇φ

(

We conclude by gathering the estimates (5.4.67), (

and, thanks to Step 1, finally establish (

Equipped with Lemma 43, we finally turn to the proof of:

Proof of Corollary 31. Since the proof is the same for Q

ij and Q

ijm , we only show that for all z and ξ

In this perspective, we introduce the coupling ensembles • Ω L of Lemma 42 as well as the notations

and we compute

Thanks to Lemma 43, we have the following convergences: for all z and p < +∞

Moreover, by (5.4.65), all the components of Q

ij have finite p-moments; namely for all z

Therefore, the Hölder inequality entails

which implies in turn (5.4.73). This concludes the proof of Corollary 31.

Proofs

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations

In order to proceed, we need k 1 λ k < ∞, which means that the inverse of the Dirichlet-2n has finite trace, which in turn follows from the finiteness of the corresponding Green's function along the diagonal, which means that Dirac distributions are in H -2n (B 2 ), which amounts to the Sobolev embedding H 2n 0 (B 2 ) ⊂ C 0 0 (B 2 ), and thus holds provided 2n > d, which we henceforth assume. Hence by the density of C ∞ 0 (B 2 ) in H 2m 0 (B 2 ) w k we obtain the annealed inner regularity estimate

It remains to post-process (5.4.78). Provided 2n > d 2 + 2 we may appeal to Sobolev's embedding applied to ∇ 2 w in order to upgrade (5.4.78) to

(5.4.79)

Since we may w. l. o. g. assume ´B2 u = 0, we may restrict to w with ´w = 0. A standard argument in the theory of distributions yields the existence of a vector field

Hence (5.4.79) may be upgraded to the desired

(5.4.80)

Proof of Lemma 39

We decompose the r. h. s. of (5.3.37) according to a family of dyadic annuli. Namely, we set

so that we can also decompose ∇w = ∞ k=-∞ ∇w k , where ∇w k is the square-integrable solution to

Noting that for any k there holds -∇ • a∇w k = 0 in B 2 k-1 , we mainly applied stochastic corrector estimates (see Proposition 13) together with annealed estimates (see Lemma 41, and annealed

(5.3.25) µ

(2)

(5.4.81) where q ≥ 2 and p < p < p < p < ∞. We mention that the second line is due to [101, Proposition 7.1], while the fourth line follows from a (random) coordinate transformation 15 coupled with [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Lemma 7.4]. We distinguish the later proof into large-scale cases R ≥ 2 and smallscale cases R < 2. Firstly, we handle the large scale case, and set k 0 to be an integer such that

) sup |∇ 2 g| µ

3 (R) sup |∇ 2 g|,

(5.4.82) provided q 1 > 2d and 2 ≤ q 2 < 2d. We can similarly address the small-scale case,

) sup |∇ 2 g| µ (2) 3 (R) sup |∇ 2 g|, whenever q 3 > d and 2 ≤ q 4 < d. This completes the proof.

Proof of Proposition 14: Homogenization error

We fix any andx, y ∈ R d with x = y. Let the error V x 0 ,y 0 (x, y) be (fully) the two-scale expansion of the Green function, defined by

jn (y 0 ))∂ jn G(x -y),

(5.4.83) 15 The ideas is to transform the coefficient ā to ∆ and then appeal to annealed Calderón-Zygmund estimates on -∆. Consequently, the loss of stochastic integrability caused by the random transformations of the coordinate. Concretely, it relies on the following computations. Let y = Qx such that Q -1 ā(Q * ) -1 = Id, and then there holds ∇y = (Q -1 ) * ∇x. Let ũ(x) = u(Qx) with g(x) = Q -1 g(Qx). Thus, the equation -∇ • ā∇ū = 0 can be formulated by

Then, one may apply [START_REF] Josien | The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization[END_REF]Lemma 7.4] to the above equation to have the desired estimate.

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations where we follow the similar rule of the notations in (5.2.15), and this leads to ∇∇V x 0 ,y 0 (x, y) = ∇∇G(x, y)

im (x) + φ

(1)

im (x) -φ jn (y 0 ))e l ∂ jnl G(x -y).

(5.4.84) Thus, recalling the definition of E(x, y) in (5.3.6), we have

i e m + ∇φ e n + ∇φ * ( 2)

jn (y 0 ))e l (y).

We additionally mention that since the symmetry assumption on a, it is not very necessary to use the superscript " * " to indicate the adjoint objects. However, it is convenient to distinguish the variables in later computations. Then, taking x = x 0 and y = y 0 , it follows from the above equality and Proposition 13 that |∇∇V x 0 ,y 0 (x 0 , y 0 ) -E(x 0 , y 0 )| p (5.4.85)

Obviously, to obtain the desire estimate (5.3.41) it suffices to show

(5.4.86)

Consequently, the stated estimate (5.3.41) simply follows from replacing x 0 , y 0 with x, y, respectively.

Arguments for (5.4.86). We first consider the gradient of V x 0 ,y 0 (x, y) in (5.4.83) with respect to the variable x, and then fix x = x 0 . Hence, ∇V x 0 ,y 0 (x 0 , •) is expanded as follows:

jn (y 0 ))∂ jn ūx 0 (x 0 , •), (5.4.87) where ūx 0 (•, •) is defined by

i (x)e m + ∇φ

(5.4.88) Moreover, we observe that

By the two-scale expansion arguments introduced in Subsection 5.3.5, we arrive at jn (y 0 )) ∇∂ jn ūx 0 (x 0 , •).

(5.4.90)

Let k 0 be such that 2 k 0 -1 < R ≤ 2 k 0 , and it suffices to study the equation (5.4.89) in a ball with the radius 2 k 0 centered at y 0 (since we are then interesting in evaluating at y = y 0 ). To estimate it, we decompose its right-hand side according to a family of dyadic annuli, which are

(5.4.91)

Then we set W = ∇V x 0 ,y 0 (x 0 , •) -k 0 k=-∞ Z k , and it satisfies a * -harmonic condition on B 2 k 0 (y 0 ), i.e., -∇ • a * ∇W = 0 in B 2 k 0 (y 0 ).

(5.4.92)

We will handle the equations (5.4.91) and (5.4.92), separately. As a preparation, recalling the definition of (5.4.88), for all positive integer j and p < ∞, we derive from Proposition 13 and decay estimates of G that sup y∈B R (y 0 ) ∇ j ūx 0 (x 0 , y) (5.4.94)

For the last inequality above, in order to use Minkowski's inequality and Jensen's inequality, we will obtain the second line of the following computations (5.4.95), and then on account of Proposition 13, a similar calculation as given for (5.4.82) leads to the last line of (5.4.95).

jn (y 0 )

(5.3.25),(5.3.26), (5.4.93)

(5.4.95) in which we also ask for the condition 2 ≤ p < ∞ in first inequality, and recall the relationship 2 k 0 -1 < R ≤ 2 k 0 for the last one. Beyond the pointwise estimate (5.4.94), we still require some weak norm estimate later on, and now establish it as follows. In fact, the argument is quite similar to that given (5.4.95), and the main difference is to employ annealed Calderón-Zygmund estimates Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations [101, Proposition 7.1] for the second line below.

provided q ≥ 2d, and the third line is due to the same ideas given for (5.4.82). In the sequel, the above estimate implies the weak norm estimate, i.e., for any g ∈ C ∞ 0 (B R (y 0 )),

|∇ 2 g(y)|,

(5.4.96) where we also employ Poincaré's inequality in the last step.

We now turn to address the estimate on ∇W in the equation (5.4.92). In view of Lemmas 40 and 41, there holds

(5.4.98), (5.4.96) µ

3 (R)R -d-2 , (5.4.97) where the last step relies on the following estimate ˆdy∇∇V x 0 ,y 0 (x 0 , y) • g(y) Arguments for (5.4.98). Let u with ū and g ∈ C ∞ 0 (B R (y 0 )) be associated by the equation (5.3.38). Recalling the expansion (5.3.36), we may define w x 0 as follows.

and then by differentiating in both sides w. r. t. the x-variable combined with an integration by parts, we arrive at ∇w x 0 (x) (5.4.88) = -ˆdy ∇∇G(x, y) -∇ x ūx 0 (x, y) g(y).

(5.4.99) jn (y 0 ))e l ∂ jnl ūx 0 (x, y).

(5.4.100) Thus, we split the integral in the l. h. s. of (5.4.98) as follows (where in the last line we use an integration by parts): ˆdy∇∇V x 0 ,y 0 (x 0 , y) • g(y)

(5.4.100) = ˆdy ∇∇G(x 0 , y) -∇ū x 0 (x 0 , y) • g(y)

jn (y 0 ))∂ jn ūx 0 (x 0 , y)g(y)

(5.4.99)

jn (y 0 ))∂ jn ūx 0 (x 0 , y)(∇ • g)(y).

(5.4.101) It follows from Lemma 39 that 16 ,

We rewrite the second term of the right-hand side of (5.4.101) by means of ω * j defined by (5.3.22), so that an integration by parts yields ˆdy φ * (1) j

Therefore, there holds

L |∇ 2 g| Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations 5.5 Appendix

Regularity of the coefficients

We give in this section the argument for (5.3.2). Without loss of generality, we may assume that x = 0.

By the stationary of the ensemble • and the inequality (5.1.4), we have

where η is a smooth cut-off with support inside B 2 . By a Besov embedding [18, Prop. 2.71 p. 99], we can replace the Besov space B s p,p (R d ) in (5.5.2) by the Besov space B β ∞,∞ for all β ∈ (0, 1) which coincides with the Hölder space C 0,β (R d ) (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]p. 99,Examples]). Hence, g satisfies (5.3.2). Last, the relation a = A(g) along with (5.1.6) implies that a also satisfies (5.3.2).

Convergence of the massive quantities

We give in this section the argument for (5.3.12).

We only prove the convergence of ∇G T , the other convergences will follow by the same type of arguments. Let β ∈ (0, 1), x = y ∈ R d and r := |x -y|/2. In the following proof, the symbol implicitly depends on γ, β, r and a C 0,β (T L ) but not on T . Thanks to the Arzelà-Ascoli theorem and the dominated convergence theorem, it is enough to prove the two following statements:

and lim sup

(5.5.4)

We first give arguments for (5.5.3). Let f ∈ C ∞ c (B r (y)), and the square-integrable functions u T and ∇u satisfying

(5.5.5)

Using the energy estimates, there holds

(5.5.6)

Appendix 277

The difference u T -u solves

(5.5.8)

Moreover, Since ∇u T is uniformly bounded in L 2 (R d ), we may extract a weak limit ∇u ∞ ∈ L 2 (R d ) when T ↑ ∞. By (5.5.5) and using the above estimates, we deduce that -∇ • a∇u ∞ = ∇ • f . By the Liouville principle associated with the operator -∇ • a∇ (recall that a is T L -periodic), we finally obtain that ∇u ∞ = ∇u. Therefore, we deduce that

Combining this with (5.5.8), we get that

From the pointwise convergence in x and the formula

we obtain the convergence (5.5.3). We now give the arguments for (5.5.4). We have from the definition of the Green function G T (since a is symmetric)

Therefore, using interior Schauder estimate combined with the Caccioppoli inequality we obtain

(5.5.9)

Moreover, we already know from [START_REF] Grüter | The green function for uniformly elliptic equations[END_REF] (with the very slight difference that there is here an extra massive term, which is of no harm for the result) that, for all x = y and for all T ≥ 1, there holds |G T (x , y )| |x -y | 2-d . Therefore, (5.5.4) follows from inserting the above estimate into (5.5.9).

Localized energy estimates

We recall the exponential localization for the massive operator 1 T -∇ • a∇. For a proof, we refer to [68, Lemma 36].

Chapter 5. Bias in the Representative Volume Element method: periodize the ensemble instead of its realizations in the distributional sense in R d . Suppose that there exists a k > 0 such that

Then, there exists a γ > 0 depending on and λ such that ˆdx

Chapter 6

Perspectives and remaining open questions In Chapter 2, we prove optimal time decay estimates for the first order semigroup defined in (1.2.2) related to the first order correctors. A natural extension would be to go one order further and to consider the second order semigroup u

ij ) ij defined by:

The second order semigroup is related to the second order correctors (φ

), which are the distributional and sub-quadratic 1 (up to an additive constant) solutions in R d of

The second order correctors provide an improvement of the convergence of the homogenization error whence the two-scale expansion is truncated at second order, see [START_REF] Bella | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors[END_REF]. The second order approximation finds also its interest in the study of numerical methods, as in [START_REF] Lu | Optimal artificial boundary condition for random elliptic media[END_REF] where the authors mentioned

Therefore, the contribution of the right-hand side of (6.1.2) in the regime ≤ √ T is given by ˆ√T Therefore, the contribution of the right-hand side of (6.1.2) in the regime > √ T is given by: ˆ+∞ T -β 4 . (6.1.4)

The combination of (6.1.3) and (6.1.4) finally gives E[(u (2) ) q ]

Similarly, we can obtain

We see that for β < d, ∇u (2) is integrable only if β > 2. The extension in the non-perturbative regime as well as its application to artificial boundary conditions is left for future investigation.

Those results are so far restricted to the linear setting. The investigation of the fluctuations in the nonlinear setting is left for future investigation (for ideas and details, we refer to Section 3.2.2 of Chapter 3).

• The convergence of the two-scale expansion that we prove in Chapter 3 (see Theorem 12) is stated in the whole space and therefore we truly focused on the homogenization error and we avoid the question of boundary layers. One could state and prove a similar result on a bounded domain with Dirichlet boundary conditions in which case the bound would be of the order of the square root of what we have in Theorem 12. A way to refine this result is to construct a correction up to the boundary and introduce a new "boundary" corrector u ε,bl :

-∇ • a( • ε , ∇u ε,bl ) = 0 in Ω, u ε,bl (x) = -φ ∇u ( x ε ) on ∂Ω. (6.2.2)

One expect that the "modified" two-scale expansion:

), reconstruct correctly the oscillations on the boundary of Ω, where the advantage is now that φ ∇u ( • ε ) + u ε,bl satisfies a Dirichlet boundary condition. In addition, since the boundary layer u ε,bl is not explicit with respect to ε, one has to capture the asymptotic behaviour of u ε,bl by homogenizing the system (6.2.2). The homogenization of (6.2.2) started in the linear and periodic case with the works of Bakhvalov, Panasenko, Bensoussan, Lions, Papanicolaou, Oleinik, Shamaev and Yosifian in [START_REF] Sergeevich | Homogenisation: averaging processes in periodic media[END_REF][START_REF] Bensoussan | Boundary layers and homogenization of transport processes[END_REF][START_REF] Lions | Some methods in the mathematical analysis of systems and their control[END_REF][START_REF] Arsenievna Oleïnik | Mathematical problems in elasticity and homogenization[END_REF] for very specific domains, and was then extended for instance by Allaire, Gérard-Varet, Masmoudi, Shen, Zhuge, Armstrong, Kuusi, Mourrat and Prange [START_REF] Gérard | Homogenization in polygonal domains[END_REF][START_REF] Gérard | Homogenization and boundary layers[END_REF][START_REF] Shen | Boundary layers in periodic homogenization of neumann problems[END_REF][START_REF] Armstrong | Quantitative analysis of boundary layers in periodic homogenization[END_REF] for more general domains. In the random setting, the homogenization error up to the boundary has been addressed in [START_REF] Armstrong | Quantitative stochastic homogenization and largescale regularity[END_REF]Section 6.4] however similar results in the nonlinear setting or the homogenization of (6.2.2) has been solved. An interesting work could be to investigate this question and it would provide a good complement to the quantitative theory developed in Chapter 3.

Non-convex variational models: beyond nonlinear systems with monotone coefficients

In the variational context, that is when a = ∇ ξ W and m = d in (1.1.4), as it appears in the context of nonlinear elasticity, we essentially dealt with uniformly convex W in Chapter 3 and the system (1.1.4) can be seen as the Euler-Lagrange equation of the minimization of the functional (we consider Dirichlet boundary conditions for the discussion) W (x, ξ) = +∞. (6.2.4)

• The noninterpenetration of the matter:

W (x, ξ) = ∞ if det ξ < 0. (6.2.5)

The standard example of such integrand potential is given by W (x, ξ) = V (x, ξ) + A(x)h(detξ), (6.2.6) with ξ → V (•, ξ) a convex stationary ergodic integrand, A ∈ L ∞ (Ω) an ergodic and stationary random field and h : R → [0, +∞] is a continuous convex function with h(t) = +∞ for t ≤ 0 and h(t) < +∞ for t > 0.

In this model, we are interested by minimizing the value of the energy and we understand the homogenization process by finding a suitable limit to (inf F ε ) ε . The natural notion of convergence in this variational context is the De Giorgi Γ-convergence (see for instance [START_REF] Maso | An introduction to Γ-convergence[END_REF]). More precisely, the aim is to find a functional F hom such that (F ε ) ε Γ-converge to F hom (for the L p (Ω)-topology), meaning that:

• For all (u ε ) ε ⊂ W 1,p 0 (Ω) and u ∈ W 1,p 0 (Ω) such that u ε → ε↓0 u in L p (Ω) and

• For all (u ε ) ε ⊂ W 1,p 0 (Ω) there exists u ∈ W 1,p 0 (Ω) such that u ε → ε↓0 u in L p (Ω) and

The qualitative homogenization of (F ε ) ε has been intensively studied since the 1980 , starting from the works of Braides and Müller in [START_REF] Braides | Homogenization of some almost periodic coercive functional[END_REF][START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF] for integrand W with p-growth (that is for all x ∈ Ω, |W (x, ξ)| ∼ 1 + |ξ| p ), and then extended in a lot of works by, for instance, Braides, Defranceshi, Garroni, Messaoudi, Michaille, Jikov, Kozlov, Oleinik, Anza Hafsa, Mandallena, Gloria and Duerinckx in [START_REF] Braides | Homogenization of multiple integrals[END_REF][START_REF] Braides | Homogenization of periodic nonlinear media with stiff and soft inclusions[END_REF][START_REF] Messaoudi | Stochastic homogenization of nonconvex integral functionals[END_REF][START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF][START_REF] Anza | Homogenization of nonconvex integrals with convex growth[END_REF][START_REF] Duerinckx | Stochastic homogenization of nonconvex unbounded integral functionals with convex growth[END_REF]. However, those results do not take into account the two conditions (6.2.4) and (6.2.5), and it constitutes a very challenging and open question to prove an homogenization result which is compatible. Some works in this direction has been proved in [START_REF] Hafsa | Homogenization of nonconvex unbounded singular integrals[END_REF][START_REF] Hafsa | Omar and leghmizi, mohamed lamine and mandallena, jean-philippe. on a homogenization technique for singular integrals[END_REF] in the periodic case, for model of type (6.2.6) for V with p-growth, the periodic functions x → V (x, •) and x → A(x) are continuous almost-everywhere and there exists T > 0 such that h : R → [0, +∞] is continuous and convex on (-T, +∞), h(t) = +∞ for t ∈ [-T, 0] and h(t) = T for t < -T . It would be interesting to extend this result to the stochastic setting. However, this result will not be physically relevant since the assumptions on h are only compatible with (6.2.4) and not with (6.2.5) (but only in the weaker form of W (x, ξ) = +∞, if det ξ ∈ [-T, 0] for some T possibly very large).

Concerning the quantitative homogenization theory of (F ε ) ε , it is currently unknown in the all situations above and constitute a difficult open question. The main reason is that the non-convexity of W does not imply the existence of a single corrector, which constitutes a strong obstruction to proving quantitative homogenization based on two-scale expansion (as we did in Chapter 3). Even some relaxation of convexity (as, for instance, polyconvexity) does not allow to prove the existence of a corrector (see for instance the counterexample of Müller in [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF], see also [START_REF] Barchiesi | New counterexamples to the cell formula in nonconvex homogenization[END_REF]). However, some works in this direction has been done in [START_REF] Neukamm | Quantitative homogenization in nonlinear elasticity for small loads[END_REF][START_REF] Neukamm | Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains[END_REF] for small deformations, in the periodic setting. The authors have observed that for small deformations, one can construct a single corrector (bound on the extensive use of null Lagrangians) and reconstruct the oscillations of a minimizer of F ε via a two-scale expansion. An interesting extension would be to understand this work in the stochastic setting of Chapter 3 and get quantitative estimates in the case of small deformations.