Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments

TL;DR

This paper establishes the homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded, stationary ergodic environments, revealing a new formula for the effective Hamiltonian and connecting it to an eikonal equation.

Contribution

It introduces a novel strategy for almost sure homogenization, completing a program that previously only achieved homogenization in probability, applicable to a broad class of problems.

Findings

Proves deterministic homogenization in unbounded environments.

Derives a new formula for the effective Hamiltonian.

Connects the effective Hamiltonian to an eikonal-type equation.

Abstract

We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.