PUB - Publikationen an der Universität Bielefeld

Using a natural "Riemannian geometry-like" structure on the configuration space Gamma over R-d we prove that for a large class of potentials phi the corresponding canonical Gibbs measures on Gamma can be completely characterized by an integration by parts formula. That is, if Vr is the gradient of the Riemannian structure on Gamma one can define a corresponding divergence div(phi)(Gamma) such that the canonical Gibbs measures are exactly those measures mu for which del(Gamma) and div(phi)(Gamma) are dual operators on L-2(Gamma, mu). One consequence is that for such mu the corresponding Dirichlet forms E-mu(Gamma) are defined. In addition, each of them is shown to be associated with a conservative diffusion process on Gamma with invariant measure mu. The corresponding generators are extensions of the operator Delta(phi)(Gamma):= div(phi)(Gamma)del(Gamma). The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brownian motion on Gamma perturbed by a singular drift. Another main result of this paper is the following: If mu is a canonical Gibbs measure, then it is extreme (or a "pure phase") if and only if the corresponding weak Sobolev space W-1,W-2(Gamma,mu) on Gamma is irreducible. As a consequence we prove that for extreme canonical Gibbs measures the above mentioned diffusions are time-ergodic. In particular, this holds for tempered grand canonical Gibbs measures ("Ruelle measures") provided that the activity constant is small enough. We also include a complete discussion of the free case (i.e., phi = 0) where the underlying space R-d is even replaced by a Riemannian manifold X: (C) 1998 Academic Press.