Stochastic homogenization of interfaces moving with changing sign velocity

TL;DR

This paper studies the averaged behavior of interfaces in random environments with oscillatory velocities that change sign, using homogenization of Hamilton-Jacobi equations to understand their large-scale dynamics.

Contribution

It extends previous periodic analyses to random media, showing convergence of solutions to deterministic effective Hamiltonians in a stochastic setting.

Findings

Solutions converge in $L^$-weak $*$ to a combination of initial data and effective Hamiltonian solutions.

The effective Hamiltonian(s) are determined by the properties of the random media.

The approach generalizes homogenization results to non-coercive Hamiltonians with sign-changing velocities.

Abstract

We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous non-coercive Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions and Souganidis (2009). Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton-Jacobi equation converge in $L^\infty$-weak $*$ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.