Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations

TL;DR

This paper demonstrates that in certain random Hamilton-Jacobi equations, the global minimizer is hyperbolic and the viscosity solutions are locally smooth near this minimizer, extending previous results to higher dimensions.

Contribution

It generalizes earlier findings on hyperbolicity and regularity of solutions to higher-dimensional random Hamilton-Jacobi equations.

Findings

Global minimizer is hyperbolic almost surely.

Viscosity solutions are smooth near the global minimizer.

Extends previous results to dimensions d ≥ 2.

Abstract

We show that for a family of randomly kicked Hamilton-Jacobi equations, the unique global minimizer is hyperbolic, almost surely. Furthermore, we prove the unique forward and backward viscosity solutions, though in general only Lipshitz, are smooth in a neighbourhood of the global minimizer. Our result generalizes the result of E, Khanin, Mazel and Sinai (\cite{EKMS00}) to dimension $d\ge 2$, and extends the result of Iturriaga and Khanin in \cite{IK03}.