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We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions . In previous works we studied the model problem of a discrete elliptic equation on . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions 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.
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Antoine Gloria, Felix Otto, Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. 19 (2017), no. 11, pp. 3489–3548DOI 10.4171/JEMS/745