Robustness of the pathwise structure of fluctuations in stochastic homogenization

TL;DR

This paper demonstrates that the pathwise structure of fluctuations in stochastic homogenization remains robust even for continuum elliptic systems with strongly correlated coefficients, extending previous discrete models.

Contribution

It extends the theory of fluctuations based on the homogenization commutator to continuum elliptic systems with correlated coefficients, showing robustness of the pathwise fluctuation structure.

Findings

Two-scale expansion of the homogenization commutator remains accurate at large scales.

Large-scale fluctuations of the field and flux are driven by the commutator fluctuations.

Robustness of the fluctuation structure in more general continuum settings.

Abstract

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 operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator 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 allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting 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.