PUB - Publikationen an der Universität Bielefeld

Let Gamma(X) denote the space of all locally finite configurations in a complete, stochastically complete, connected, oriented Riemannian manifold X, whose volume measure m is infinite. In this paper, we construct and study spaces L(mu)(2)Omega(n) of differential n-forms over Gamma(X) that are square integrable with respect to a probability measure It on Gamma(X). The measure it is supposed to satisfy the condition Sigma(m)(') (generalized Mecke identity) well known in the theory of point processes. On L(mu)(2)Omega(n), we introduce bilinear forms of Bochner and deRham type. We prove their closabilty and call the generators of the corresponding closures the Bochner and deRham Laplacian, respectively. We prove that both operators contain in their domain the set of all smooth local forms. We show that, under a rather general assumption on the measure mu, the space of all Bochner-harmonic mu-square-integrable forms on Gamma(X) consists only of the zero form. Finally, a Weitzenbock type formula connecting the Bochner and deRham Laplacians is obtained. As examples, we consider (mixed) Poisson measures, Ruelle type measures on Gamma(Rd), and Gibbs measures in the low activity-high temperature regime, as well as Gibbs measures with a positive interaction potential on Gamma(X). (C) 2002 Elsevier Science B.V. All rights reserved.