Analysis and Geometry on Configuration Spaces: The Gibbsian Case

Using a natural “Riemannian geometry-like” structure on the configuration spaceΓover Rd, we prove that for a large class of potentialsφthe corresponding canonical Gibbs measures on 1 can be completely characterized by an integration by parts formula. That is, if ∇Γ is the gradient of the Riemannian structure onΓone can define a corresponding divergence divΓφsuch that the canonical Gibbs measures are exactly those measuresμfor which ∇Γ and divΓφare dual operators onL2(Γ, μ). One consequence is that for suchμthe corresponding Dirichlet forms EΓμare defined. In addition, each of them is shown to be associated with a conservative diffusion process onΓwith invariant measureμ. The corresponding generators are extensions of the operatorΔΓφ :=divΓφ ∇Γ. The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brownian motion onΓperturbed by a singular drift. Another main result of this paper is the following: Ifμis a canonical Gibbs measure, then it is extreme (or a “pure phase”) if and only if the corresponding weak Sobolev spaceW1, 2(Γ, μ) onΓis 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.,φ≡0) where the underlying space Rdis even replaced by a Riemannian manifoldX.