Homogenization of Stochastic Conservation Laws with Multiplicative Noise

TL;DR

This paper develops a homogenization framework for stochastic conservation laws with oscillatory coefficients and multiplicative noise, introducing stochastic two-scale Young measures to analyze the behavior of solutions.

Contribution

It introduces a novel homogenization approach for stochastic conservation laws using stochastic two-scale Young measures, especially addressing equations with special stochastic solutions.

Findings

Existence of stochastic two-scale Young measures established.

Homogenization results for two types of stochastic conservation laws.

Identification of stochastic solutions as key elements in analysis.

Abstract

We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.