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

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''.

This paper will appear in a special issue of Journal of Convex Analysis in the occasion of the 70th birthday of Giuseppe Buttazzo. arXiv admin note: substantial text overlap with arXiv:2306.12145; text overlap with arXiv:2303.06415