The structure of fluctuations in stochastic homogenization

Mitia Duerinckx, Antoine Gloria, Felix Otto

Published 2016-02-04Version 1

We establish a path-wise theory of fluctuations in stochastic homogenization. More precisely we consider the model problem of a discrete elliptic equation with independent and identically distributed conductances. We identify a single quantity, which we call the corrected energy density of the corrector, that drives the fluctuations in stochastic homogenization in the following sense. On the one hand, when properly rescaled, this quantity satisfies a functional central limit theorem, and converges to a Gaussian white noise. On the other hand, the fluctuations of the corrector and the fluctuations of the solution of the stochastic PDE (that is, the solution of the discrete elliptic equation with random coefficients) are characterized at leading order by the fluctuations of this corrected energy density. As a consequence, when properly rescaled, the solution satisfies a functional central limit theorem for $d\ge 2$, and the corrector converges to (a variant of) the (whole-space massless) Gaussian free field for $d>2$. Compared to previous contributions, our approach, based on the corrected energy density, unravels the complete structure of fluctuations. It holds for $d\ge 2$, yields the first path-wise results, quantifies the CLT in Wasserstein distance, and only relies on arguments that extend to the continuum setting.