On Bourgain's approach to stochastic homogenization

TL;DR

This paper advances Bourgain's harmonic-analytic approach to stochastic homogenization, extending it to continuum settings, Gaussian coefficients, and fluctuations, leading to improved precision and broader applicability.

Contribution

It extends Bourgain's approach to continuum and Gaussian settings, and develops fluctuation analysis, enhancing the theoretical framework of stochastic homogenization.

Findings

Robustness of Bourgain's approach in continuum with exponentially mixing coefficients

New Malliavin calculus proof for Gaussian coefficients

Construction of weak correctors up to order 2d

Abstract

In 2018, Bourgain pioneered a novel perturbative harmonic-analytic approach to the stochastic homogenization theory of discrete elliptic equations with weakly random i.i.d. coefficients. The approach was subsequently refined to show that homogenized approximations of ensemble averages can be derived to a precision four times better than almost sure homogenized approximations, which was unexpected by the state-of-the-art homogenization theory. In this paper, we grow this budding theory in various directions: First, we prove that the approach is robust by extending it to the continuum setting with exponentially mixing random coefficients. Second, we give a new proof via Malliavin calculus in the case of Gaussian coefficients, which avoids the main technicality of Bourgain's original approach. This new proof also applies to strong Gaussian correlations with power-law decay. Third, we extend Bourgain's approach to the study of fluctuations by constructing weak correctors up to order $2d$, which also clarifies the link between Bourgain's approach and the standard corrector approach to homogenization. Finally, we draw several consequences from those different results, both for quantitative homogenization of ensemble averages and for asymptotic expansions of the annealed Green's function.