Robustness of the pathwise structure of fluctuations in stochastic homogenization
Résumé
We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution. In a previous contribution, for discrete elliptic equations with independent and identically distributed conductances, we developed a theory of fluctuations based on the notion of homogenization commutator, defined as the flux minus the homogenized coefficients times the field of the solution: we proved that the two-scale expansion of this special quantity is accurate at leading order when averaged on large scales (as opposed to the two-scale expansion of the field and flux taken separately) and that the large-scale fluctuations of both the field and the flux can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and with strongly correlated coefficient fields (Gaussian-like with a covariance function that displays an arbitrarily slow algebraic decay at infinity). Our main result shows that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.
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