Laplace operators on differential forms over configuration spaces

Abstract

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given. 2000

$L^2$-Betti Numbers of Infinite Configuration Spaces

“…In [3], [4], the authors started studying differential forms and the corresponding Laplacians (of Bochner and de Rham type) over the configuration space Γ X . The main result of [4] is a description of the space K ( * ) of square-integrable (with respect to the Poisson measure) harmonic forms over Γ X :…”

“…In Section 2 we give (following [3], [4]) a description of the de Rham complex over Γ X and the spaces of harmonic forms.…”

Random Witten Laplacians: traces of semigroups, $L^2$-Betti numbers and index

“…such that A(gγ ) = U g A(γ )U g −1 (5) for any g ∈ G and γ ∈ . This approach has been proposed in [3] in the case of the Witten Laplacians associated with Gibbs measures on configuration spaces.…”

“…In Section 4, we introduce the corresponding operator algebra C = C p (cf. formulae (4), (5)) and prove that it is a von Neumann algebra with the faithful normal semifinite trace TR given by the formula…”

“…Measures of such type appear, via the generalized Mecke identity, in the theory of configuration spaces, and in particular in the theory of Laplace operators on differential forms over X (see [5], [6], [4]). In fact, the Witten Laplacian H associated with σ is a "part" of the Hodge-de Rham operator on X associated with the Gibbs measure µ.…”