Stochastic homogenization of $L^\infty$ variational problems

TL;DR

This paper establishes a homogenization result for $L^ Infty$ variational problems in stationary ergodic random environments, using a generalized distance function linked to absolute minimizers.

Contribution

It introduces a new generalized distance function and connects it to absolute minimizers to prove homogenization in complex random settings.

Findings

Homogenization of $L^ Infty$ variational problems in ergodic environments

Connection between generalized distance functions and absolute minimizers

Homogenization results derived from properties of the associated eikonal equation

Abstract

We present a homogenization result for $L^\infty$ variational problems in general stationary ergodic random environments. By introducing a generalized notion of distance function (a special solution of an associated eikonal equation) and demonstrating a connection to absolute minimizers of the variational problem, we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments.