Scaling limits and homogenization of mixing Hamilton-Jacobi equations

TL;DR

This paper investigates the homogenization of nonlinear Hamilton-Jacobi equations with oscillatory mixing dependence, revealing that the effective equations are stochastic with white noise in time, and establishes regularity and stability results.

Contribution

It introduces new homogenization results for mixing Hamilton-Jacobi equations and proves regularity and path stability properties of stochastic Hamilton-Jacobi equations.

Findings

Homogenized equations are stochastic Hamilton-Jacobi equations driven by white noise.

Established regularity and path stability results for stochastic Hamilton-Jacobi equations.

Demonstrated homogenization in various settings with oscillatory spatio-temporal dependence.

Abstract

We study the homogenization of nonlinear, first-order equations with highly oscillatory mixing spatio-temporal dependence. It is shown in a variety of settings that the homogenized equations are stochastic Hamilton-Jacobi equations with deterministic, spatially homogenous Hamiltonians driven by white noise in time. The paper also contains proofs of some general regularity and path stability results for stochastic Hamilton-Jacobi equations, which are needed to prove some of the homogenization results and are of independent interest.