Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions

TL;DR

This paper proves the first qualitative stochastic homogenization results for a broad class of second-order, degenerate, quasilinear Hamilton-Jacobi equations, including geometric motions and viscous non-convex cases, with quantitative error estimates.

Contribution

It provides the first proof of qualitative stochastic homogenization for these equations and offers quantitative error estimates with algebraic rates.

Findings

First qualitative stochastic homogenization proof for these equations.

Quantitative error estimates with algebraic convergence rates.

Applicability to geometric motions and viscous non-convex Hamilton-Jacobi equations.

Abstract

We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing geometric motions of level sets as well as a large class of viscous, non-convex Hamilton-Jacobi equations. The main results include the first proof of qualitative stochastic homogenization for such equations. We also present quantitative error estimates which give an algebraic rate of homogenization.