Quantitative results on the corrector equation in stochastic homogenization

Abstract

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.

arXiv comment57 pages, 1 figure

References in corpus (4)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• Quantitative stochastic homogenization of convex integral functionals
• On annealed elliptic Green function estimates
• Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

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