Quantitative results on the corrector equation in stochastic homogenization

• DOI: 10.4171/jems/745
• Publication Date: 2017-10-03T21:40:10Z
• Authors: Antoine Gloria; Felix Otto
• Keywords: Stochastic Homogenization; Variance Estimate; [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]; Stochastic homogenization; 01 natural sciences; [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]; corrector equation; Mathématiques; Mathematics - Analysis of PDEs; FOS: Mathematics; 35B27, 39A70, 60H25, 60F99; [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]; 0101 mathematics; Corrector Equation; variance estimate; Analysis of PDEs (math.AP)

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions

d \geq 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.