Rademacher’s Theorem on Configuration Spaces and Applications

We consider an L-Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρ-Lipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular fulfilled by the classical Poisson measures and 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 ρ 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 ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form. 0. Introduction. Let ΓX be the configuration space over a Riemannian manifold X. In this paper, we consider a class of probability measures on ΓX , which in particular contains certain Ruelle type Gibbs measures and mixed Poisson measures. Using a natural ‘non-flat’ geometric structure of ΓX , recently analyzed in Albeverio, Kondratiev and Röckner (1996a), (1996b), (1997), and (1998), one can define weak derivatives and introduce the related Sobolev spaces. Here we are interested in a more detailed description of this concept of differentiability. Similar to the case of H-differentiability on Wiener space, it turns out that not all values which a function u takes in a small neighborhood of some γ ∈ ΓX are relevant for its weak gradient ∇u(γ), but only those which are located in certain ‘directions’. Here, of course, the word ‘direction’ needs to be defined because of the absence of any linear structure on ΓX even if X = R . Our way of making the above precise is to prove an infinite dimensional version of the celebrated theorem of Rademacher (1919) stating that Lipschitz functions on R are differentiable almost everywhere and and in the weak sense. On abstract Wiener space and its generalizations, similar results were obtained by Kusuoka (1982a), (1982b), Enchev and Stroock (1993), and Bogachev and Mayer-Wolf (1996). On configuration space, the correct Lipschitz condition is defined through an L-Wasserstein type distance function ρ, which, for non-compact X, divides ΓX into uncountably many disjoint ‘fibers’, each of the form {ω | ρ(γ, ω) <∞}. A consequence of our Rademacher type theorem then is that only the behavior of u in small ρ-balls around γ matters for the value of ∇u(γ). Research carried out in part at MSRI and supported in part by Deutsche Forschungsgemeinschaft and by NSF grant DMS-9701755.