The additive structure of elliptic homogenization

TL;DR

This paper introduces a new approach to stochastic homogenization for elliptic equations by demonstrating the additivity of energy-related quantities, enabling optimal quantitative estimates and central limit theorems for correctors.

Contribution

It establishes the additivity of energy densities in elliptic homogenization, leading to optimal convergence estimates and Gaussian free field limits for correctors.

Findings

Proves additivity of energy densities in stochastic homogenization.

Derives optimal quantitative estimates on convergence of correctors.

Shows convergence of correctors to a Gaussian free field in the large-scale limit.

Abstract

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in \cite{AKM}: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.