De Rham Cohomology of Configuration Spaces with Poisson Measure

Abstract

The space 1 X of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of square-integrable differential forms over 1 X , equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the L 2

De Rham–Hodge decomposition and vanishing of harmonic forms by derivation operators on the Poisson space

“…We refer the reader to [2], [3] for a different construction of differential forms on the configuration space over a Riemannian manifold with a Poisson measure, where nforms were defined by a lifting of the underlying differential structure on the manifold to the configuration space. See also to [4] for a different approach to the construction of the Hodge decomposition on abstract metric spaces.…”

Traces of semigroups associated with interacting particle systems

“…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 Γ X (see [5][6][7]). In fact, the Witten Laplacian H associated with σ is a "part" of the Hodgede Rham operator on Γ X associated with the Gibbs measure μ.…”

Poisson configuration spaces, von Neumann algebras, and harmonic forms

“…A crucial role here is played by a probability measure on Γ X (in particular, a Poisson or Gibbs measure), which is quasiinvariant with respect to the action of a group of diffeomorphisms of X. This philosophy, inspired by the pioneering works [1] and [2], has been initiated and developed in [3], [4] and has lead to many interesting and important results in the field of stochastic analysis on configuration spaces and its applications, see also references in [5], [6].…”

“…Further, we introduce and compute a regularized index of the Dirac operator associated with the de Rham differential on Γ X . For a more detailed exposition, see [5], [11], [7], [8] and references given there.…”