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

Abstract

We prove homogenization for a class of viscous Hamilton-Jacobi equations in the stationary & ergodic setting in one space dimension. Our assumptions include most notably the following: the Hamiltonian is of the form G(p) + βV (x, ω), the function G is coercive and strictly quasiconvex, min G = 0, β > 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 (θ 1 (β), θ 2 (β)), 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 β on (θ 1 (β), θ 2 (β)), and strictly monotone elsewhere.