# An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity

## TL;DR

This paper presents a specific example demonstrating that stochastic homogenization can fail for viscous Hamilton-Jacobi equations when the Hamiltonian is non-convex, highlighting limitations of existing homogenization theories.

## Contribution

It provides the first known explicit example of failure in stochastic homogenization for viscous Hamilton-Jacobi equations without convexity.

## Key findings

- Homogenization fails in the constructed example.
- Non-convexity of the Hamiltonian is critical for failure.
- Highlights limitations of current homogenization approaches.

## Abstract

We give an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with non-convex Hamiltonian.