Stochastic homogenization of a class of quasiconvex viscous Hamilton-Jacobi equations in one space dimension

TL;DR

This paper proves homogenization for a class of viscous Hamilton-Jacobi equations with quasiconvex Hamiltonians in one dimension, providing explicit formulas for the effective Hamiltonian and analyzing its properties.

Contribution

It introduces a novel approach to homogenization for quasiconvex viscous Hamilton-Jacobi equations, including explicit characterization of the effective Hamiltonian.

Findings

Effective Hamiltonian is coercive and equals β on a specific interval.

Unique sublinear correctors exist outside a bounded interval of directions.

The effective Hamiltonian is strictly monotone outside the interval.

Abstract

We prove homogenization for a class of viscous Hamilton-Jacobi equations in the stationary and ergodic setting in one space dimension. Our assumptions include most notably the following: the Hamiltonian is of the form $G(p) + \beta V(x,\omega)$, the function $G$ is coercive and strictly quasiconvex, $\min G = 0$, $\beta>0$, the random potential $V$ takes values in $[0,1]$ with full support and it satisfies a hill condition that involves the diffusion coefficient. Our approach is based on showing that, for every direction outside of a bounded interval $(\theta_1(\beta),\theta_2(\beta))$, there is a unique sublinear corrector with certain properties. We obtain a formula for the effective Hamiltonian and deduce that it is coercive, identically equal to $\beta$ on $(\theta_1(\beta),\theta_2(\beta))$, and strictly monotone elsewhere.