Stochastic homogenization of viscous superquadratic Hamilton-Jacobi equations in dynamic random environment

TL;DR

This paper proves the homogenization of viscous superquadratic Hamilton-Jacobi equations in complex random environments, extending previous results to more general diffusion matrices without uniform ellipticity.

Contribution

It introduces a homogenization framework for second order viscous Hamilton-Jacobi equations with superquadratic Hamiltonians in space-time ergodic random environments, relaxing uniform ellipticity assumptions.

Findings

Established homogenization in non-uniformly elliptic settings

Characterized the effective Hamiltonian in general random environments

Extended previous homogenization results to broader classes of diffusions

Abstract

We study the qualitative homogenization of second order viscous Hamilton-Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient) we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion.