Laplace operators and diffusions in tangent bundles over Poisson spaces

TL;DR

This paper constructs differential form spaces over Poisson configuration spaces, analyzes Laplacians of Bochner and de Rham types, and provides a probabilistic interpretation of their semigroups, advancing the mathematical understanding of Poisson spaces.

Contribution

It introduces new differential form spaces over Poisson configuration spaces and studies associated Laplacians and semigroups with probabilistic insights.

Findings

Construction of differential form spaces over Poisson spaces

Analysis of Laplacians of Bochner and de Rham types

Probabilistic interpretation of semigroups

Abstract

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation is given.