An optimal error estimate in stochastic homogenization of discrete elliptic equations

Abstract

This paper is the companion article to [Ann. Probab. 39 (2011) 779-856]. We consider a discrete elliptic equation on the d-dimensional lattice Z d with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix A hom = a hom Id is characterized by ξ · A hom ξ = (ξ + ∇φ) · A(ξ + ∇φ) for any direction ξ ∈ R d , where the random field φ (the "corrector") is the unique solution of −∇ * · A(ξ + ∇φ) = 0 in Z d such that φ(0) = 0, ∇φ is stationary and ∇φ = 0, · denoting the ensemble average (or expectation).In order to approximate the homogenized coefficients A hom , the corrector problem is usually solved in a box Q L = [−L, L) d of size 2L with periodic boundary conditions, and the space averaged energy on Q L defines an approximation A L of A hom . Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation A L converges almost surely to A hom as L ↑ ∞. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size 2L, but replace the elliptic operator by T −1 − ∇ * · A∇ with (typically) T ∼ L 2 , as standard in the homogenization literature. We then replace the ensemble average by a space average on Q L , and estimate the overall error on the homogenized coefficients in terms of L and T .

The additive structure of elliptic homogenization

“…In these works, the space is the discrete lattice Z d , d ≥ 3, and a key assumption is that the probability measure has an underlying product structure which makes available tools such as concentration inequalities, the Chatterjee-Stein [9,10] method of normal approximation and the Helffer-Sjöstrand representation of correlations [26,39,32]. Each of these papers makes essential use of, and refines, the optimal quantitative estimates first proved in [19,20,16].…”

“…The first has its origins in an unpublished paper of Naddaf and Spencer [33], and is based on probabilistic machinery more commonly used in statistical physics [32], namely concentration inequalities, such as spectral gap or logarithmic Sobolev inequalities, which provide a way to quantitatively measure the dependence of the solutions on the coefficients. This approach has been developed extensively by Gloria, Otto and their collaborators [19,20,23,16,15,28], who proved optimal quantitative bounds on the scaling of the first-order correctors (including their sublinear growth and spatial averages of their energy density). In particular, they were the first to obtain estimates for the correctors at the critical scalings, albeit with suboptimal stochastic integrability (typically finite moment bounds) and with somewhat restrictive ergodic assumptions.…”

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

“…Nous démontrons des résultats quantitatifs sur l'approximation par périodisation du correcteur en homogénéisation stochastique deséquations aux dérivées partielles elliptiques linéaires sous forme divergence, lorsque les coefficients de diffusion satisfont une hypothèse de trou spectral en probabilité et en dimension d > 2. Ce travail s'inspire de [5], qui est une version complète dans le cas continu de [6,7] (qui traiteégalement le cas d = 2 de manière optimale). La différence majeure avec [5] est qu'on utilise directement la théorie de De Giorgi-Nash-Moserà la place des fonctions de Green.…”

“…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. This work is based on [5], which is a complete continuum version of [6,7] (with in addition optimal results for d = 2). The main difference with respect to the first part of [5] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.…”

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

“…The obstacle is that the various qualitative proofs of the quenched invariance principle rely on an appeal to the ergodic theorem which is difficult to quantify. On the other hand, in the uniformly elliptic setting (when all bonds are open and a ∈ [λ, 1]), there is now a very precise quantitative theory due to Gloria and Otto [20,21] (see also [18]) with implications to random walks explained in [15]. However, it is not obvious to see how to extend the methods of these papers to the case of percolation clusters since they rely heavily on uniform ellipticity and seem to require the geometry of the random environment to posses some homogeneity down to the smallest scales.…”