# On the existence of correctors for the stochastic homogenization of viscous hamilton-jacobi equations

## TL;DR

This paper proves the existence of correctors in stochastic homogenization for viscous Hamilton-Jacobi equations, extending results to degenerate, non-convex, and geometric cases in stationary ergodic media.

## Contribution

It establishes the existence of correctors under broad conditions, including cases where homogenization is not initially known, for a wide class of Hamilton-Jacobi equations.

## Key findings

- Correctors exist for all extreme points of the convex hull of the effective Hamiltonian's sublevel sets.
- Existence of correctors implies homogenization in new settings, including geometric equations and radially symmetric environments.
- Homogenization holds for many directions in non-convex Hamiltonians and general stationary ergodic media.

## Abstract

We prove, under some assumptions, the existence of correctors for the stochastic homoge-nization of of " viscous " possibly degenerate Hamilton-Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type equations including motion by mean curvature, in radially symmetric environments and for all directions. Correctors also exist and, hence, homogenization holds for many directions for non convex Hamiltoni-ans and general stationary ergodic media.