Rademacher's theorem on configuration spaces and applications

Abstract

We consider an $L^2$-Wasserstein type distance $\rho$ on the configuration space ${\Gamma }_X$ over a Riemannian manifold $X$, and we prove that $\rho$-Lipschitz functions are contained in a Dirichlet space associated with a measure on ${\Gamma }_X$ satisfying some general assumptions. These assumptions are in particular fulfilled by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of $\rho$ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that $\rho$ is optimal in the sense that it is the intrinsic metric of our Dirichlet form.