# Quantitative results on the corrector equation in stochastic homogenization

2 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01093381
Contributor : Antoine Gloria <>
Submitted on : Wednesday, December 10, 2014 - 3:42:42 PM
Last modification on : Monday, August 20, 2018 - 9:44:02 AM

### Identifiers

• HAL Id : hal-01093381, version 1
• ARXIV : 1409.0801

### Citation

Antoine Gloria, Felix Otto. Quantitative results on the corrector equation in stochastic homogenization. Journal of the European Mathematical Society, European Mathematical Society, 2015, pp.57. ⟨hal-01093381⟩

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