Stochastic Homogenization of HJ Equations: a Differential Game Approach

TL;DR

This paper introduces a game-theoretic approach to stochastic homogenization of complex Hamilton-Jacobi equations, providing new methods and quantitative convergence rates for solutions in random environments.

Contribution

It develops a differential game framework for homogenization of non-convex, non-coercive Hamilton-Jacobi equations, extending to Lipschitz Hamiltonians without max-min structure.

Findings

Proved stochastic homogenization for a broad class of Hamilton-Jacobi equations.

Established a quantitative convergence rate for solutions.

Extended the approach to Lipschitz Hamiltonians without max-min representation.

Abstract

We prove stochastic homogenization for a class of non-convex and non-coercive first-order Hamilton-Jacobi equations in a finite-range-dependence environment for Hamiltonians that can be expressed by a max-min formula. Exploiting the representation of solutions as value functions of differential games, we develop a game-theoretic approach to homogenization. We furthermore extend this result to a class of Lipschitz Hamiltonians that need not admit a global max-min representation. Our methods allow us to get a quantitative convergence rate for solutions with linear initial data toward the corresponding ones of the effective limit problem.