PUB - Publikationen an der Universität Bielefeld

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 = R-d. It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient del(Gamma), divergence div(Gamma), and Laplace-Beltrami operator H-Gamma = -div(Gamma)del(Gamma) are constructed. The associated volume elements, i.e., all measures mu on Gamma(X) w.r.t. which del(Gamma) and div(Gamma) become dual operators on L-2(Gamma(X); mu), 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 forms E-mu(Gamma) on L-2(Gamma(X); mu) are, therefore, defined. Each is shown to be associated with a diffusion process which is thus the Brownian motion on Gamma(X) and which is subsequently identified as the usual independent infinite particle process on X. The associated heat semigroup (T-mu(Gamma)(t))(t>0) is calculated explicitly. It is also proved that the diffusion process, when started with mu, is time-ergodic (or equivalently delta(mu)(Gamma) is irreducible or equivalently (T-mu(Gamma)(t))(t>0) is ergodic) if and only if mu is Poisson measure pi(zdx) with intensity z dx for some z greater than or equal to 0. Furthermore, it is shown that the Laplace-Beltrami operator H-Gamma = -div(Gamma)del(Gamma) on L-2(Gamma(X); pi(zdx)) is unitary equivalent to the second quantization of the Laplacian -Delta(X) on X on the corresponding Fock space circle plus(n greater than or equal to 0)L(2)(X; z dx)(circle times n). As another direct consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields on X on Poisson space. Finally, generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on X are described, in particular, the case where the mixed Poisson measures mu are replaced by Gibbs measures of Ruelle-type on Gamma(X). (C) 1998 Academic Press.