Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations

TL;DR

This paper proves that in stochastic homogenization of linear elliptic PDEs, correctors grow sublinearly with optimal stochastic estimates, especially for weakly decorrelated, Gaussian-like coefficient fields, enhancing understanding of their behavior.

Contribution

It provides the first optimal stochastic estimates for the sublinear growth of correctors under weak decorrelation and Gaussian-like conditions, using advanced probabilistic and PDE techniques.

Findings

Established sublinear growth of correctors with optimal stochastic moments.

Derived quantitative sublinearity estimates based on decorrelation scales.

Applied Malliavin calculus and concentration inequalities to obtain results.

Abstract

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and "essentially Gaussian" coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for $a$-harmonic functions.