A path-wise theory of fluctuations in stochastic homogenization

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Résumé : In this talk I will consider the model problem of a discrete elliptic equation with independent and identically distributed conductances. I shall 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 field. 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 corrector and the solution satisfy a functional central limit theorem, and converge to (a variant of) a Gaussian free field. Compared to previous contributions, the approach I shall present yields the first path-wise results, quantifies the CLT in Wasserstein distance, and only relies on arguments that extend to the continuum setting.
This is based on a joint work with M. Duerinckx and F. Otto.