Poisson process Fock space representation, chaos expansion and covariance inequalities

TL;DR

This paper develops a Fock space representation and chaos expansion for Poisson processes, providing explicit formulas and extending variance inequalities to general Poisson processes with broad applications.

Contribution

It introduces an explicit Fock space representation and chaos expansion for Poisson processes, extending variance inequalities and covariance identities to general settings.

Findings

Explicit Fock space representation for Poisson processes

Extension of variance inequalities to general Poisson processes

Covariance identities and Harris-FKG inequalities for monotone functions

Abstract

We consider a Poisson process $\eta$ on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of $\eta$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris-FKG-inequalities for monotone functions of $\eta$.