Homogenization of pathwise Hamilton-Jacobi equations

TL;DR

This paper studies the homogenization behavior of pathwise Hamilton-Jacobi equations driven by rough signals, establishing conditions for homogenization or blow-up, and providing new well-posedness results with explicit estimates.

Contribution

It offers new qualitative and quantitative homogenization results for equations with rough signals and convex Hamiltonians, and introduces a novel well-posedness framework with explicit solution estimates.

Findings

Homogenization occurs with a single rough signal and convex Hamiltonian.

Blow-up or homogenization can occur in multi-signal settings.

New well-posedness results include explicit continuity and equicontinuity estimates.

Abstract

We present qualitative and quantitative homogenization results for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. When there is only one such signal and the Hamiltonian is convex, we show that the equation, as well as equations with smooth approximating paths, homogenize. In the multi-signal setting, we demonstrate that blow-up or homogenization may take place. The paper also includes a new well-posedness result, which gives explicit estimates for the continuity of the solution map and the equicontinuity of solutions in the spatial variable.