Mesoscopic higher regularity and subadditivity in elliptic homogenization

TL;DR

This paper introduces a novel regularity-based method for stochastic homogenization of elliptic equations, avoiding concentration inequalities and achieving improved convergence and sublinearity estimates through a multiscale approach.

Contribution

It develops a higher regularity theory on mesoscopic scales that enhances convergence rates and controls fluctuations without relying on traditional probabilistic inequalities.

Findings

Achieves faster convergence of the expected energy in homogenization.

Provides quantitative estimates on the sublinearity of the corrector.

Introduces a multiscale Poincaré inequality for better control of spatial averages.

Abstract

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincar\'e or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher ($C^{k}$, $k \geq 1$) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities) which yields, by a new "multiscale" Poincar\'e inequality, quantitative estimates on the sublinearity of the corrector.