Stochastic Homogenization of Monotone Systems of Viscous Hamilton-Jacobi Equations with Convex Nonlinearities

TL;DR

This paper studies the homogenization process of monotone viscous Hamilton-Jacobi systems with convex nonlinearities in a stochastic setting, showing how microscopic systems average to deterministic equations.

Contribution

It introduces methods for analyzing both collapsing and non-collapsing systems of Hamilton-Jacobi equations in a stochastic homogenization framework.

Findings

Systems collapse to scalar equations under homogenization.

Methods applicable to non-collapsing systems as well.

Results establish deterministic limits for stochastic systems.

Abstract

We consider the homogenization of monotone systems of viscous Hamilton-Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic scale tends to zero, average to a deterministic scalar Hamilton-Jacobi equation. However, our methods also apply to systems which do not collapse and, as the microscopic scale tends to zero, average to a deterministic system of Hamilton-Jacobi equations.