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

Abstract

Abstract. 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.Résumé. 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.

Special quasirandom structures: A selection approach for stochastic homogenization

“…Both estimates have been established for the discrete variant of the problem. A similar decay of the statistical error has also been established for the continuous case we consider in the present article (see [13,Theorem 1] and [20, Theorem 1.3 and Proposition 1.4]).…”

Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations

“…As a consequence of Lemma 4, there exist N ∼ 1 linear functionals {F n } n=1,··· ,N whose rescaled versions F n,r satisfy the boundedness property (14) such that for any r ≥ 2R we have the implication…”

The additive structure of elliptic homogenization

“…1 Several months after this paper was submitted and posted to arXiv and before it was accepted, Gloria and Otto completed a substantial revision [22] of [21] in which they prove Theorem 1 as well as Theorem 2. Their analysis is based on a quantity they call the "homogenization commutator" which is closely related to the quantity J considered here.…”