Stochastic homogenization of a class of quasiconvex and possibly degenerate viscous HJ equations in 1d

TL;DR

This paper establishes homogenization results for a class of degenerate viscous Hamilton-Jacobi equations with quasiconvex Hamiltonians in one dimension, considering random media with specific conditions on the coefficients.

Contribution

It proves homogenization for degenerate viscous HJ equations with quasiconvex Hamiltonians in 1D under explicit random media conditions, extending previous results to degenerate and non-rigid environments.

Findings

Homogenization holds for equations with vanishing diffusion coefficients.

The results apply to a broad class of random media satisfying a scaled hill condition.

Degeneracy and quasiconvexity do not prevent homogenization in this setting.

Abstract

We prove homogenization for possibly degenerate viscous Hamilton-Jacobi equations with a Hamiltonian of the form $G(p)+V(x,\omega)$, where $G$ is a quasiconvex, locally Lipschitz function with superlinear growth, the potential $V(x,\omega)$ is bounded and Lipschitz continuous, and the diffusion coefficient $a(x,\omega)$ is allowed to vanish on some regions or even on the whole $\mathbb{R}$. The class of random media we consider is defined by an explicit scaled hill condition on the pair $(a,V)$ which is fulfilled as long as the environment is not ``rigid''.