---
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.'
address:
- |
    Ceremade (UMR CNRS 7534)\
    Université Paris-Dauphine\
    Place du Maréchal De Lattre De Tassigny\
    75775 Paris CEDEX 16, France
- |
    Ceremade (UMR CNRS 7534)\
    Université Paris-Dauphine\
    Place du Maréchal De Lattre De Tassigny\
    75775 Paris CEDEX 16, France
author:
- Scott Armstrong
- Pierre Cardaliaguet
bibliography:
- 'meancurv.bib'
title: 'Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions'
---

Introduction
============

Motivation and informal summary of results
------------------------------------------

In this paper, we study time-dependent, quasilinear, viscous Hamilton-Jacobi equations taking the form $$\label{e.pde}
\partial_t u^{\varepsilon}- {\varepsilon}\operatorname{tr}\left( A\left(\frac{Du^{\varepsilon}}{\left|Du^{\varepsilon}\right|},\frac x{\varepsilon}\right)D^2u^{\varepsilon}\right) + H\left(Du^{\varepsilon},\frac x{\varepsilon}\right) = 0 \quad \mbox{in} \ {\mathbb{R}^d}\times (0,\infty).$$ We give a brief summary of the main assumptions, which are given precisely in Section \[s.statements\], below. The diffusion matrix $A(e,x)$ is assumed to be 