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 uniqueness of "invariant measures", i.e., solutions to the equation L ¯ = 0 where L = \Delta +B \Delta r on R n with B satisfying some mild integrability conditions and ¯ is a probability measure on R n . This solves an open problem posed by S.R.S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L 1 (¯) generates a strongly continuous semigroup having ¯ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L 1 (¯) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the "symmetric case") is in particular studied and conditions are identified ensuring that L ¯ = 0 implies that L is symmetric on L 2 (¯) resp. L ¯ = 0 has a unique solution. We also prove infinite dimensional analogues of the latter two results and a ne...

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...

We prove existence and uniqueness for invariant measures of strongly continuous semigroups on L 2 (X ; ¯), where X is a (possibly infinite--dimensional) space. Our approach is purely analytic based on the theory of sectorial forms. The generators covered are e.g. small perturbations (in the sense of sectorial forms) of operators generating hypercontractive semigroups. An essential ingredient of the proofs is a new result on compact embeddings of weighted Sobolev spaces H 1;2 (ae \Delta dx) on R d (resp. a Riemannian manifold) into L 2 (ae dx). Probabilistic consequences are also briefly discussed. AMS Subject Classification Primary: 31 C 25 Secondary: 47 D 07, 60 H 10, 47 D 06, 60 J 60 Key words and phrases: invariant measures, sectorial forms, compact embeddings, Poincare inequality, Log--Sobolev inequality Running head: Invariant measures for semigroups 1) Department of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia 2) Fakultat fur Mathematik, U...

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models.

This paper consists of lectures given at the C.I.M.E. summer school on Kolmogorov equations held at Cetraro in 1998. The purpose of these lectures was to present an approach to Kolmogorov equations in infinite dimensions which is based on an L p ()--analysis of the corresponding diffusion operators w.r.t. suitably chosen measures. The main ideas and aims are explained, and an as complete as possible presentation is given of what has been achieved in this respect over the last few years.

We give a (to some extent pedagogical) survey on recent results about Dirichlet forms on infinite-dimensional "manifold-like" state spaces including path and loop spaces as well as spaces of measures. The latter are associated with interacting Fleming-Viot processes resp. infinite particle systems. Also some new results, further developing the Dirichlet form approach to infinite particle systems, are enclosed. Finally, a brief summary of other research activities in the theory of Dirichlet forms is given and some prospects for the future are indicated. AMS Subject Classification (1991) Primary: 31C25 Secondary: 58B99, 58G32, 60G57, 60J45, 60J60, 60K35, 81S20, 81T08, 82B26, 82B31 Key words and phrases: Dirichlet forms, infinite-dimensional manifolds, path and loop spaces, Fleming-Viot processes, infinite particle systems, Euclidean quantum fields, lattice Gibbs states, ergodicity Running head: Dirichlet forms on infinite-dimensional manifolds 1) Fakultat fur Mathematik, Un...