Precompact probability spaces in applied stochastic homogenization

Abstract

We provide precompactness and metrizability of the probability space Ω for random measures and random coefficients such as they widely appear in stochastic homogenization and are typically given from data. We show that these properties are enough to implement the convenient two-scale formalism by Zhikov and Piatnitsky (2006). To further demonstrate the benefits of our approach we provide some useful trace and extension operators for Sobolev functions on Ω, which seem not known in literature. On the way we close some minor gaps in the Sobolev theory on Ω which seemingly have not been proven up to date.