Stochastic two-scale convergence and Young measures

TL;DR

This paper compares different notions of stochastic two-scale convergence, introduces stochastic two-scale Young measures to relate mean and quenched limits, and discusses examples highlighting the applicability of stochastic unfolding.

Contribution

It introduces stochastic two-scale Young measures and compares mean, quenched, and unfolding approaches to stochastic two-scale convergence.

Findings

Stochastic two-scale Young measures effectively compare mean and quenched limits.

Examples demonstrate stochastic unfolding's advantages over quenched convergence.

The paper clarifies relationships among various stochastic two-scale convergence concepts.

Abstract

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikeli\'c and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.