A regularity theory for random elliptic operators

TL;DR

This paper extends large-scale regularity results for elliptic operators with random coefficients, establishing qualitative and quantitative regularity estimates based on a minimal radius that depends on the randomness and mixing properties of the coefficients.

Contribution

It introduces an intrinsic large-scale regularity theory for elliptic systems with random coefficients, generalizing previous results to a broader class of stochastic environments.

Findings

Almost sure finiteness of the minimal radius under ergodicity

High stochastic integrability of the minimal radius with mixing assumptions

Optimal moment bounds for Gaussian-type coefficient fields

Abstract

Since the seminal results by Avellaneda \& Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong \& Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the \emph{intrinsic large-scale} regularity of Avellaneda \& Lin (namely, intrinsic large-scale Schauder and Calder\'eron-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field $r_*$ which we call the minimal radius. This regularity theory is \textit{qualitative} in the sense that $r_*$ is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is \textit{quantifiable} in the sense that $r_*$ has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing \emph{optimal} moment bounds on $r_*$ for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.