Poincare Inequality on the Path Space of Poisson Point Processes

TL;DR

This paper establishes a Poincare inequality for the path space of Poisson point processes, introducing a jump-type Dirichlet form and highlighting differences from Gaussian settings.

Contribution

It proves quasi-invariance under random shifts and constructs a new Dirichlet form on Poisson path space, contrasting with Wiener space.

Findings

Quasi-invariance under random shift maps

Construction of a jump-type Dirichlet form

Poincare inequality holds, log-Sobolev does not

Abstract

The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincare inequality but not the log-Sobolev one.