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This work is devoted to the long-standing open problem of homogenization of 2D perfect incompressible fluid flows, such as the 2D Euler equations with impermeable inclusions modeling a porous medium, and such as the lake equations. The main difficulty is the homogenization of the transport equation for the associated fluid vorticity. In particular, a localization phenomenon for the vorticity could in principle occur, which would rule out the separation of scales. Our approach combines classical results from different fields to prevent such phenomena and to prove homogenization towards variants of the Euler and lake equations: we rely in particular on the homogenization theory for elliptic equations with stiff inclusions, on criteria for unique ergodicity of dynamical systems, and on complex analysis in form of extensions of the Rad\'o-Kneser-Choquet theorem., Comment: 32 pages

We establish quantitative homogenization results for the popular log-normal coefficients. Since the coefficients are neither bounded nor uniformly elliptic, standard proofs do not apply directly. Instead, we take inspiration from the approach developed for the nonlinear setting by the first two authors and capitalize on large-scale regularity results by Bella, Fehrmann, and Otto for degenerate coefficients in order to leverage an optimal control (in terms of scaling and stochastic integrability) of oscillations and fluctuations., Comment: 27 pages

The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits). The aim of this article is threefold. First we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension $d=1$, establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to by-pass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimension (provided results are correctly reformulated) to which we give precise entries to the recent literature.

Consider the Schr\"odinger operator $-\triangle+\lambda V$ with non-negative iid random potential $V$ of strength $\lambda>0$. We prove existence and uniqueness of the associated landscape function on the whole space, and show that its correlations decay exponentially. As a main ingredient we establish the (annealed and quenched) exponential decay of the Green function of $-\triangle+\lambda V$ using Agmon's positivity method, rank-one perturbation in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.

This work relates quantitatively homogenization to Anderson localization for acoustic operators in disordered media. By blending dispersive estimates for homogenized operators and quantitative homogenization of the wave equation, we derive large-scale dispersive estimates for waves in disordered media that we apply to the spreading of low-energy eigenstates. This gives a short and direct proof that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic media, and it further provides new lower bounds on {the localization length} of possible eigenstates in case of quasiperiodic or random media., Comment: New title, added general form of large-scale dispersive estimates, 30 pages

Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present work is a new ansatz for the long-time two-scale expansion inspired by spectral analysis. Based on this spectral ansatz, we extend and refine all previous results in the field, proving homogenization up to optimal timescales with optimal error estimates, and covering all the standard assumptions on heterogeneities (both periodic and stationary random settings)., Comment: published version

We consider a suspension of active rigid particles (swimmers) in a steady Stokes flow, where particles are distributed according to a stationary ergodic random process, and we study its homogenization in the macroscopic limit. A key point in the model is that swimmers are allowed to adapt their propulsion to the surrounding fluid deformation: swimming forces are not prescribed a priori, but are rather obtained through the retroaction of the fluid. Qualitative homogenization of this nonlinear model requires an unusual proof that crucially relies on a semi-quantitative two-scale analysis. After introducing new correctors that accurately capture spatial oscillations created by swimming forces, we identify the contribution of the activity to the effective viscosity. In agreement with the physics literature, an analysis in the dilute regime shows that the activity of the particles can either increase or decrease the effective viscosity (depending on the swimming mechanism), which differs from the well-known case of passive suspensions., Comment: 59 pages

This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we turn to our new results on the asymptotic expansion of the effective viscosity in the dilute regime. This includes an optimal proof of Einstein's viscosity formula for the first-order expansion, as well as the continuation of this expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: suitable renormalizations are needed and are captured by means of diagrammatic expansions., Comment: 14 pages

In this note, we provide a short and robust proof of the Clausius-Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and can be applied to dilute systems in many other contexts., Comment: 15 pages

This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions however prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization (using a variational formulation and $\Gamma$-convergence to avoid delicate pressure issues). On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness., Comment: 30 pages. This version corrects a mistake on John's domains, the results are unchanged

Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with $p$-growth. This work is dedicated to a quantitative two-scale expansion result. By treating the range of exponents $2\le p <\infty$ in dimensions $d\le 3$, we are able to consider genuinely nonlinear elliptic equations and systems such as $-\nabla \cdot A(x)(1+|\nabla u|^{p-2})\nabla u=f$ (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 optimal quantitative two-scale expansion result we derive (this is also new in the periodic setting)., Comment: final version, 82 pages

This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting., Comment: 57 pages

We investigate the PDE system resulting from even electromechanical coupling in elastomers. Assuming a periodic microstructure and a periodic distribution of micro-charges of a prescribed order, we derive the homogenized system. The results depend crucially on periodicity (or adequate randomness) and on the type of microstructure under consideration. We also show electric enhancement if the charges are carefully tailored to the homogenized electric field and explicit that enhancement, as well as the corresponding electrostrictive enhancement in a dilute regime., Comment: The title of the article has been changed and two parts have been added: the rigorous proof of arbitrarily large dielectric and elastrostrictic enhancement in a dilute regime (Section 5), and the extension of the results to the case of random inclusions (Appendix A)

In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability. In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods., Comment: 148 pages

This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension $d=3$ should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature., Comment: 54 pages

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $\beta=2$; we capture the correct power of logarithms coming from these two sources of criticality. The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration --- the results would fail if correlation decay were encoded in terms of an $\alpha$-mixing condition. Among other ensembles popular in modelling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields. The optimal growth of the corrector $\phi$ is derived from bounding the size of spatial averages $F=\int g\cdot\nabla\phi $ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of $F$, that is, by estimating the functional derivative $\frac{\partial F}{\partial a}$ of $F$ w.~r.~t.~the coefficient field $a$. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on $F$. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator $-\nabla\cdot a\nabla$. The treatment allows for non-symmetric $a$ and for systems like linear elasticity., Comment: Companion article of arXiv:1409.2678

Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution operator displays fluctuations around its expectation. The recently developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension $d\ge1$ previous results so far limited to dimension $d=1$, and to the continuum setting with strong correlations recent results in the discrete iid case., Comment: 32 pages; companion article to arXiv:1807.11781

Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations., Comment: 30 pages

Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space $\mathbb R^d$. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension $d=3$ suggest that the effective velocity is well-defined for hard spheres distributed according to a weakly correlated and dilute point process, and that the variance of the sedimentation speed is infinite. This constitutes the Caflisch-Luke paradox. In this contribution, we consider a scalar version of this problem that displays the same difficulties in terms of interaction between the differential operator and the randomness, but is simpler in terms of PDE analysis. For a class of hardcore point processes we rigorously prove that the effective velocity is well-defined in dimensions $d>2$, and that the variance is finite in dimensions $d>4$, confirming the formal calculations by Caflisch and Luke, and opening a way to the systematic study of such problems., Comment: 38 pages. Update of bibliography + extended introduction

In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on $\mathbb Z^d$ with random coefficients in a perturbative regime using tools from harmonic analysis. This result is surprising for it goes beyond the expected accuracy suggested by recent results in quantitative stochastic homogenization. In this short article we reformulate Bourgain's result in a form that highlights its interest to the state-of-the-art in homogenization (and especially the theory of fluctuations), and we state several related conjectures., Comment: 12 pages

We study the long-time behavior of the Schr{\"o}dinger flow in a heterogeneous potential $\lambda$V with small intensity 0<$\lambda$$\ll$1 (or alternatively at high frequencies). The main new ingredient, which we introduce in the general setting of a stationary ergodic potential, is an approximate stationary Floquet--Bloch theory that is used to put the perturbed Schr{\"o}dinger operator into approximate normal form. We apply this approach to quasiperiodic potentials and establish a Nehoro{\v s}ev-type stability result. In particular, this ensures asymptotic ballistic transport up to a stretched exponential timescale exp($\lambda$--1/s) for some s>0. More precisely, the approximate normal form leads to an accurate long-time description of the Schr{\"o}dinger flow as an effective unitary correction of the free flow. The approach is robust and generically applies to linear waves. For classical waves, for instance, this allows to extend diffractive geometric optics to quasiperiodically perturbed media.

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the $\Gamma$-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the $\Gamma$-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis)., Comment: 50 pages, 1 figure

We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations., Comment: 32 pages

Since the seminal contribution of Geymonat, M\"uller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Guti\'errez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting., Comment: 31 pages, 2 figures, slightly changed the presentation of the main results

Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In a companion article, we introduced the notion of multiscale (first-order) functional inequalities, which extend standard functional inequalities like Poincar\'e, covariance, and logarithmic Sobolev inequalities in the probability space, while still ensuring strong concentration properties. We also developed a constructive approach to these functional inequalities, proving their validity for prototypical examples including Gaussian fields, Poisson random tessellations, and random sequential adsorption (RSA) models, which do not satisfy standard functional inequalities. In the present contribution, we turn to second-order Poincar\'e inequalities \`a la Chatterjee: while first-order inequalities quantify the distance to constants for nonlinear functions $Z(A)$ in terms of their local dependence on the random field $A$, second-order inequalities quantify their distance to normality. For the above-mentioned examples, we prove the validity of suitable multiscale second-order Poincar\'e inequalities. In particular, applied to RSA models, these functional inequalities allow to complete and improve previous results by Schreiber, Penrose, and Yukich on the jamming limit, and to propose and fully analyze a more efficient algorithm to approximate the latter., Comment: 25 pages

In a companion article we have introduced a notion of multiscale functional inequalities for functions $X(A)$ of an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. These inequalities are multiscale weighted versions of standard Poincar\'e, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields $A$ arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions $X(A)$. This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results., Comment: 24 pages

Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In order to establish concentration properties for nonlinear functions $Z(A)$, it is standard to appeal to functional inequalities like Poincar\'e or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Mat\'ern-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest., Comment: 41 pages

Consider an elliptic operator in divergence form with symmetric coefficients.If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization.When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associatedoperator are unclear.At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization.The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an "eigendefect"), which we express with the help of a new family of extended correctors.We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large timesfor the wave equation in terms of the spatial growth of these extended correctors.On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions).On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators., Comment: Annales Scientifiques de l'{\'E}cole Normale Sup{\'e}rieure, Elsevier Masson, In press

Since the seminal contribution of Geymonat, M{\"u}ller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved by homogenization in linear elasticity. This phenomenon is typically related to microscopic buckling of the composite material. The present contribution is concerned with the interplay between isotropy and strong ellipticity in the framework of periodic homogenization in linear elasticity. Mixtures of two isotropic phases may indeed lead to loss of strong ellipticity when arranged in a laminate manner. We show that if a matrix/inclusion type mixture of isotropic phases produces macroscopic isotropy, then strong ellipticity cannot be lost. R{\'e}sum{\'e}. Nous savons depuis l'article fondateur de Geymonat, M{\"u}ller et Triantafyl-lidis qu'en{\'e}lasticit{\'e}en{\'e}lasticit{\'e} lin{\'e}aire l'homog{\'e}n{\'e}isation p{\'e}riodique ne conserve pas n{\'e}cessairement l'ellipticit{\'e} forte. Ce ph{\'e}nom{\`e}ne est li{\'e} au flambage microscopique des composites. Notre contribution consiste examiner le r{\^o}le de l'isotropie dans ce type de pathologie. Le m{\'e}lange de deux phases isotropes peut en effet conduir{\`e} a cette perte si l'arrangement est celui d'un lamin{\'e}. Nous montrons qu'en revanche, si un arrangement de type ma-trice/inclusion produit un tenseur homog{\'e}n{\'e}is{\'e} isotrope, alors la forte ellipticit{\'e} est con-serv{\'e}e.

Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the \emph{homogenization commutator} and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the \emph{pathwise structure} of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works., Comment: Introduction reorganised

We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux $(\nabla\phi,a(\nabla \phi+e))$ of the corrector $\phi$, when spatially averaged over a scale $R\gg 1$ decay like the CLT scaling $R^{-\frac{d}{2}}$. We establish this optimal rate on the level of sub-Gaussian bounds in terms of the stochastic integrability, and also establish a suboptimal rate on the level of optimal Gaussian bounds in terms of the stochastic integrability. The proof unravels and exploits the self-averaging property of the associated semi-group, which provides a natural and convenient disintegration of scales, and culminates in a propagator estimate with strong stochastic integrability. As an application, we characterize the fluctuations of the homogenization commutator, and prove sharp bounds on the spatial growth of the corrector, a quantitative two-scale expansion, and several other estimates of interest in homogenization., Comment: 114 pages. Revised version with some new results: optimal scaling with nearly-optimal stochastic integrability on top of nearly-optimal scaling with optimal stochastic integrability, CLT for the homogenization commutator, and several estimates on growth of the extended corrector, semi-group estimates, and systematic errors

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov. The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by the first author and Shen for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincar\'e-type inequality., Comment: 30 pages, minor revision. To appear in Arch. Ration. Mech. Anal

We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has $p$-growth from below (with $p>d$, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space-variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals., Comment: 64 pages, 2 figures

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius-Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions., Comment: 47 pages

We study a random conductance problem on a $d$-dimensional discrete torus of size $L > 0$. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance $A_L$ of the network is a random variable, depending on $L$, and the main result is a quantitative central limit theorem for this quantity as $L \to \infty$. In terms of scalings we prove that this nonlinear nonlocal function $A_L$ essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of $A_L$., Comment: 37 pages

Since the seminal results by Avellaneda \& Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong \& Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the \emph{intrinsic large-scale} regularity of Avellaneda \& Lin (namely, intrinsic large-scale Schauder and Calder\'eron-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field $r_*$ which we call the minimal radius. This regularity theory is \textit{qualitative} in the sense that $r_*$ is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is \textit{quantifiable} in the sense that $r_*$ has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing \emph{optimal} moment bounds on $r_*$ for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations., Comment: We split the original paper into two parts: regularity theory and quantitative estimates. This part gives a digested version of the regularity theory

This paper is the companion article of [Gloria, M3AS, 21 (2011), No. 3, pp 1601-1630]. One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio $\frac{\varepsilon}{\rho}$, where $\rho$ is a typical macroscopic lengthscale and $\varepsilon$ is the typical size of the heterogeneities. In the present work, we make a systematic use of regularization and extrapolation to reduce this resonance error at the level of the approximation of homogenized coefficients and correctors for general non-necessarily symmetric stationary ergodic coefficients. We quantify this reduction for the class of periodic coefficients, for the Kozlov subclass of almost periodic coefficients, and for the subclass of random coefficients that satisfy a spectral gap estimate (e.g. Poisson random inclusions). We also report on a systematic numerical study in dimension 2, which demonstrates the efficiency of the method and the sharpness of the analysis. Last, we combine this approach to numerical homogenization methods, prove the asymptotic consistency in the case of locally stationary ergodic coefficients and give quantitative estimates in the case of periodic coefficients.

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the $L^2$-norm in probability of the \mbox{$H^1$-norm} in space of this error scales like $\epsilon$, where $\epsilon$ is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green's function by Marahrens and the third author.

In the present contribution we establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for $d>2$. The main difference with respect to the first part of [Gloria-Otto, arXiv:1409.0801] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.

In a recent work, Blanc, Le Bris, and Lions defined a notion of increment-stationarity for random point sets, which allowed them to prove the existence of a thermodynamic limit for two-body potential energies on such point sets (under the additional assumption of ergodicity), and to introduce a variant of stochastic homogenization for increment-stationary coefficients. Whereas stationary random point sets are increment-stationary, it is not clear a priori under which conditions increment-stationary random point sets are stationary. In the present contribution, we give a characterization of the equivalence of both notions of stationarity based on elementary PDE theory in the probability space. This allows us to give conditions on the decay of a covariance function associated with the random point set, which ensure that increment-stationary random point sets are stationary random point sets up to a random translation with bounded second moment in dimensions $d>2$. In dimensions $d=1$ and $d=2$, we show that such sufficient conditions cannot exist.

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions., Comment: 57 pages, 1 figure

We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on $\mathbb{R}^d$ for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality and for periodic coefficients, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. As a main application we obtain optimal estimates on the fluctuations of solutions of linear elliptic PDEs with "noisy" diffusion coefficients, an uncertainty quantification result. As a direct corollary of the decay estimates we also prove that for these classes of coefficients the H\"older exponent of the celebrated De Giorgi-Nash-Moser theory can be taken arbitrarily close to 1 in the large (that is, away from the singularity)., Comment: 43 pages