Invariant measures for stochastic conservation laws with Lipschitz flux in the space of almost periodic functions

TL;DR

This paper investigates the long-term behavior of stochastic scalar conservation laws with Lipschitz flux functions, establishing the existence and uniqueness of an invariant measure within the space of Besicovitch almost periodic functions.

Contribution

It introduces the first analysis of invariant measures for stochastic conservation laws in the space of almost periodic functions under Lipschitz flux assumptions.

Findings

Existence of a unique invariant measure.

Invariant measure supported in Besicovitch almost periodic functions.

Results hold in any space dimension.

Abstract

We study the long-time behavior of almost periodic solutions to stochastic scalar conservation laws in any space dimension, under the assumption of Lipschitz continuity of the flux functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in the space of Besicovitch almost periodic functions.