  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.
