Coarse-graining and quantitative stochastic homogenization of parabolic equations in high contrast

TL;DR

This paper establishes quantitative homogenization results for high contrast parabolic equations with random space-time coefficients, providing bounds on the homogenization length scale under decorrelation assumptions.

Contribution

It introduces a parabolic coarse-graining framework that extends elliptic homogenization techniques to the parabolic setting with high contrast coefficients.

Findings

Homogenization length scale is bounded by an exponential of a squared logarithm plus a square root term.

The framework generalizes previous elliptic results to parabolic equations.

Quantitative bounds depend on decorrelation assumptions of the random coefficients.

Abstract

We prove quantitative homogenization results for high contrast parabolic equations with random coefficients depending on both space and time. In particular, we prove that under a sufficient decorrelation assumption the homogenization length scale is bounded by $\exp(C\log^2(1+\Lambda/\lambda)) + C\sqrt{\lambda}$. The proof is based on a parabolic coarse-graining framework which generalizes the results of Armstrong and Kuusi in the elliptic setting.