Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample

TL;DR

This paper presents a counterexample demonstrating that stochastic homogenization can fail for nonconvex Hamilton-Jacobi equations, challenging the assumption that homogenization always occurs under standard conditions.

Contribution

It provides the first known example where stochastic homogenization does not hold for a Hamilton-Jacobi equation with a nonconvex Hamiltonian, despite meeting standard assumptions.

Findings

Homogenization fails in the constructed nonconvex example.

Standard assumptions are insufficient for homogenization in nonconvex cases.

Convexity is crucial for stochastic homogenization to occur.

Abstract

We provide an example of a Hamilton-Jacobi equation in which stochastic homogenization does not occur. The Hamiltonian involved in this example satisfies the standard assumptions of the literature, except that it is not convex.