On the Stochastic Homogenization of Fully Nonlinear Uniformly Parabolic Equations in Stationary Ergodic Spatio-Temporal Media

TL;DR

This paper investigates the stochastic homogenization of fully nonlinear uniformly parabolic equations in ergodic media, establishing almost sure homogenization and providing convergence rates under strong mixing conditions.

Contribution

It extends existing homogenization methods to parabolic equations, introducing new techniques to handle their temporal dynamics and deriving quantitative convergence rates.

Findings

Solutions homogenize almost surely in ergodic spatio-temporal media.

Established a rate of convergence in measure under strong mixing assumptions.

Extended previous elliptic homogenization approaches to parabolic equations.

Abstract

We study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media from the qualitative and quantitative perspective. We show that under suitable hypotheses, solutions to fully nonlinear uniformly parabolic equations in spatio-temporal media homogenize almost surely. In addition, we obtain a rate of convergence for this homogenization in measure, assuming that the environment is strongly mixing with a prescribed rate. A general methodology to study the stochastic homogenization of uniformly elliptic equations was introduced by Caffarelli, Souganidis, and Wang, and the rate of convergence for this homogenization was addressed by Caffarelli and Souganidis. We extend their approach to fully nonlinear uniformly parabolic equations, and we develop a number of new arguments to handle the parabolic structure of the problem.