Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields

TL;DR

This paper investigates the large-scale behavior of elliptic systems with correlated random coefficients, using a parabolic approach to optimize decay estimates of the corrector's gradient and flux, with implications for numerical methods.

Contribution

It introduces an optimal decay estimate for the corrector's gradient and flux in stochastic homogenization with correlated coefficients, advancing theoretical understanding and numerical analysis.

Findings

Proved optimal decay estimates for the corrector's gradient and flux.

Provided tools for analyzing numerical methods in stochastic homogenization.

Enhanced understanding of large-scale behavior in systems with slowly decaying correlations.

Abstract

We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R larger than 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.