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

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) + 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this study, homogenization of a class of viscous Hamilton-Jacobi equations in the stationary and ergodic setting in one space dimension is proven. The approach involves showing the existence of a unique sublinear corrector with certain properties for directions outside of a bounded interval. The effective Hamiltonian is determined to be coercive, equal to beta within a specific interval, and strictly monotone elsewhere.