Using a natural “Riemannian
geometry-like” structure on the configuration space
Γover R
d, we prove that for a large class
of potentials
φthe 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 on
Γone can define a corresponding divergence div
Γφsuch that the canonical Gibbs
measures are exactly those
measuresμfor which
Γ and div
Γφare dual operators on
L2(
Γ,
μ). One consequence is that for such
μthe corresponding Dirichlet forms E
Γμare defined. In addition, each
of them is shown to be associated with a conservative diffusion process on
Γwith invariant measure
μ. The corresponding generators are extensions
of the operator
ΔΓφ:=div
ΓφΓ. The diffusions can be characterized in terms
of a martingale problem and they can be considered as a Brownian motion on
Γperturbed by a singular drift. Another main result
of this paper is the following: If
μis a canonical Gibbs measure, then it is extreme (or a “pure phase”) if and only if the corresponding weak Sobolev space
W1, 2(
Γ,
μ) on
Γis 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 R
dis even replaced by a Riemannian manifold
X.