Stochastic unfolding and homogenization

TL;DR

This paper introduces a stochastic unfolding method for PDEs with random coefficients, extending two-scale convergence concepts to stochastic homogenization, and demonstrates its application to convex and non-convex models.

Contribution

It develops a stochastic unfolding technique that simplifies the analysis of stochastic homogenization in continuum mechanics, linking it to existing convergence notions.

Findings

Characterizes stochastic two-scale convergence via weak convergence.

Proves a new stochastic homogenization result for Allen-Cahn type equations.

Relates stochastic unfolding to previous stochastic two-scale convergence concepts.

Abstract

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere weak convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a new result on stochastic homogenization for a non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method described in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.