Quantitative stochastic homogenization of elliptic equations with unbounded coefficients

TL;DR

This paper develops a quantitative approach to stochastic homogenization for elliptic equations with unbounded, non-uniformly elliptic coefficients, providing convergence rates under certain integrability and ergodicity conditions.

Contribution

It extends subadditive methods to handle unbounded coefficients and establishes convergence rates with minimal assumptions on the coefficient field.

Findings

Derived convergence rate estimates for solutions

Established ergodicity and stationarity conditions

Applicable to unbounded coefficient fields

Abstract

In this paper, we consider stochastic homogenization of elliptic equations with unbounded and non-uniformly elliptic coefficients. Extending subadditive arguments, we get an estimate for the rate of the convergence of the solution of the Dirichlet problem under the condition that coefficients in the unit cube have a certain exponential integrability. For the coefficient field $\mathbf{a}$ in this paper, we only assume a constant decrease at a constant distance of the maximal correlation as an assumption of ergodicity, and stationarity with respect to $\mathbb{Z}^d$-translations.