JournalsjemsVol. 19, No. 11pp. 3489–3548

Quantitative results on the corrector equation in stochastic homogenization

  • Antoine Gloria

    Université Libre de Bruxelles, Belgium and Team MEPHYSTO, Villeneuve d'Ascq, France
  • Felix Otto

    Max Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
Quantitative results on the corrector equation in stochastic homogenization cover
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Abstract

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions d2d \geq 2. In previous works we studied the model problem of a discrete elliptic equation on Zd\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>2d > 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.

Cite this article

Antoine Gloria, Felix Otto, Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. 19 (2017), no. 11, pp. 3489–3548

DOI 10.4171/JEMS/745