Homogenization of a stochastically forced Hamilton-Jacobi equation

TL;DR

This paper investigates how rapidly oscillating stochastic noise influences the homogenization of Hamilton-Jacobi equations, revealing a deterministic limit and quantifying the noise's enhancement effect.

Contribution

It introduces new probabilistic estimates for large-scale regularity and analyzes the noise's impact on homogenization, including a sensitivity analysis for small amplitude noise.

Findings

Homogenized equation is deterministic.

Noise has a quantifiable enhancement effect.

Identifies scaling where macroscopic effects emerge.

Abstract

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale H\"older regularity of the solutions, which are of independent interest.