Invariant measures for stochastic conservation laws on the line

TL;DR

This paper studies a stochastic conservation law with solution-dependent diffusivity and random forcing, proving the existence and uniqueness of a spatially-homogeneous invariant measure for each mean, extending prior results on the stochastic Burgers equation.

Contribution

It establishes the existence and uniqueness of invariant measures for a broad class of stochastic conservation laws with complex nonlinearities and forcing.

Findings

Unique ergodic invariant measure for each mean in a non-explicit set

Generalization of previous results on stochastic Burgers equation

Markov process admits a spatially-homogeneous invariant measure

Abstract

We consider a stochastic conservation law on the line with solution-dependent diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth, spatially-homogeneous kick-type random forcing. We show that this Markov process admits a unique ergodic spatially-homogeneous invariant measure for each mean in a non-explicit unbounded set. This generalizes previous work on the stochastic Burgers equation.