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...

Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating sub--Markovian C0 --semigroups on L 1 (U; ¯) as well as associated diffusion processes. We give sufficient conditions on the coefficients so that there exists only one extension of L generating a C0 --semigroup and apply the results to prove uniqueness of the invariant measure ¯. Our results imply in particular that if ' 2 H 1;2 loc (R d ; dx), ' 6= 0 dx--a.e., the generalized Schrödinger operator (\Delta + 2' \Gamma1 r' \Delta r;C 1 0 (R d )) has exactly one extension generating a C0 --semigroup if and only if the Friedrich's extension is conservative. We also give existence and uniqueness results for ...

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 review results on the existence and the long term behaviour of non--autonomous linear evolution equations. Emphasis is put on recent results on the asymptotic behaviour using a semigroup approach.

Introduction A. Let ¯ be a Radon measure on an infinite dimensional smooth manifold E. Associated to ¯ there are various additional structures on E. This is seen from the example of Gaussian spaces where E is a separable Banach space inducing an abstract Wiener space structure on E, from the example of path and loop spaces on finite dimensional Riemannian manifolds with measures induced by Brownian motions and Brownian bridges which are usefully analyzed using special "tangent spaces" [18], from the notions of "differentiability " of measures leading to classes of "admissible" vector fields describing the directions in which ¯ can be differentiated [4], and from very general considerations [11]. Here we describe a class of vector fields determined by ¯ and the differential structure of E which also have a claim to be called "admissib

this paper we systematically develop a stochastic calculus for generalized Dirichlet forms (cf. [St1]). In particular, we show Fukushima's decomposition of additive functionals and an Ito-type formula in this framework. The class of generalized Dirichlet forms is much larger than the well-studied class of symmetric and coercive Dirichlet forms as in [FOT 94] resp. [MR 92] and time dependent Dirichlet forms as in [O 92]. It contains examples of an entirely new kind (cf. Section 6, [St1]). Therefore, the results obtained in this paper lead to extensions of the corresponding results in the "classical" theories. In particular the proofs are "locally" completely different (cf. e.g. Theorem 2.3 and Theorem 2.5; though for the reader's convenience we tried to follow the line of argument in [FOT 94] as closely as possible). This difference has several reasons: First of all we do not assume any sector condition; in certain cases we have to handle E-quasi-lower-semicontinuous

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...

Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly non-locally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock-Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier-Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

In this paper, we study lower order perturbations of a symmetric second order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the in general not sub-Markovian semigroup associated with the perturbed operator and prove that the perturbed semigroup is also hypercontractive under some exponential integrability conditions on the coefficients. 1 Introduction Let X be a locally convex Hausdorff topological vector space which is Souslinean and let (H; !; ?H ) be a Hilbert space continuously and densely embedded into X . Let ¯ be a probability measure on (X; B(X)), where B(X) stands for the Borel oe-algebra. The starting point of this paper is a second order symmetric differential operator Lu(x) = 1 2 div(A(x)ru(x)) (1) on L 2 (X; ¯) determined by a quadratic form (E ; D(E)), which satisfies a Log-Sobolev inequality. This is equivalent to the associated semigroup being hypercontractive (see e.g. [8]). Consider now a lower order pertu...