Poisson process Fock space representation, chaos expansion and covariance inequalities

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-It 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 Poincar, 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.