Laplace operators in deRham complexes associated with measures on configuration spaces

Abstract

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 $\mu$ on ${\Gamma }_X$. The measure $\mu$ is supposed to satisfy the condition ${\Sigma }_m^{\prime }$ (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 Weitzenböck type formula connecting the Bochner and deRham Laplacians is obtained. As examples, we consider (mixed) Poisson measures, Ruelle type measures on ${\Gamma }_{{\mathbb{R}}^d}$, and Gibbs measures in the low activity--high temperature regime, as well as Gibbs measures with a positive interaction potential on ${\Gamma }_X$.

43 pages