Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

TL;DR

This paper establishes the homogenization of a specific class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potentials, using probabilistic representations and large deviation techniques.

Contribution

It provides an explicit characterization of the effective Hamiltonian in terms of the tilted free energy of Brownian motion in a random potential, extending homogenization theory to nonconvex cases.

Findings

Homogenization proven for the class of equations.

Explicit formula for the effective Hamiltonian.

Connection to tilted free energy of Brownian motion.

Abstract

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.