In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "non-flat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and Laplace-Beltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...

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The convex set M a of quasi-invariant measures on a locally convex space E with given "shift"-Radon-Nikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be non-empty. A specification (of lattice type) is constructed so that M a coincides with the set of the corresponding Gibbs states. As a consequence, via a well-known method due to Dynkin-Follmer a unique representation of an arbitrary element in M a in terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (E ; D(E )) and their associated semigroups (T t ) t?0 on L 2 (E; ) are discussed. Under a mild positivity condition it is shown that 2 M a is extreme if and only if (E ; D(E )) is irreducible or equivalently, (T t ) t?0 is ergodic. This implies time-ergodicity of associated diffusions. Applications to Gibbs states of classical and quantum lattice models as well as those occuring in Euclidean...

Using a natural "Riemannian-geometry-like" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is the gradient of the Riemannian structure on \Gamma one can define a corresponding divergence div \Gamma OE such that the canonical Gibbs measures are exactly those measures ¯ for which r \Gamma and div \Gamma OE are dual operators on L 2 (\Gamma; ¯). One consequence is that for such ¯ the corresponding Dirichlet forms E \Gamma ¯ are defined. In addition, each of them is shown to be associated with a conservative diffusion process on \Gamma with invariant measure ¯. The corresponding generators are extensions of the operator \Delta \Gamma OE := div \Gamma OE r \Gamma . The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brown...

We prove that there exists a diffusion process whose invariant measure is the two-dimensional polymer measure g . The diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. We prove the closability of the appropriate pre-Dirichlet form which is of gradient type, using a general closability result by two of the authors. This result does not require an integration by parts formula (which does not hold for the two-dimensional polymer measure g ) but requires the quasi-invariance of g along a basis of vectors in the classical Cameron-Martin space such that the RadonNikodym derivatives (have versions which) form a continuous process. We also show the Dirichlet form to be irreducible or equivalently that the diffusion process is ergodic under time translations. AMS Subject Classification Primary: 60 J 65 Secondary: 60 H 30 Key words: two-dimensional polymer measure, closability, Dirichlet forms, diffusion processes, ergodicity, quasi-inv...

In this paper we provide new sufficient conditions for the existence of weak solutions to a class of semilinear stochastic equations, which at the same time guarantee that the solution is a symmetric ergodic right process. The invariant measure (which is also the symmetrizing measure) is given explicitly. The emphasis is on the fact that due to the symmetry in our situation, the assumptions on the nonlinear term can be relaxed considerably.

We extend classical results of Holley-Stroock on the characterization of extreme Gibbs states for the Ising model in terms of the irreducibility (resp. ergodicity) of the corresponding Glauber dynamics to the case of lattice systems with unbounded (linear) spin spaces. We first develop a general framework to discuss questions of this type using classical Dirichlet forms on infinite dimensional state spaces and their associated diffusions. We then describe concrete applications to lattice models with polynomial interactions (i.e., the discrete P(φ)d-models of Euclidean quantum field theory). In addition, we prove the equivalence of extremality and shift-ergodicity for tempered Gibbs states of these models and also discuss this question in the general framework.