Poisson process Fock space representation, chaos expansion and covariance inequalities
Abstract
We consider a Poisson process $\eta$ on an arbitrary measurable space with an arbitrary sigmafinite 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 WienerIto chaos expansion. We apply these results to extend wellknown 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 HarrisFKGinequalities for monotone functions of $\eta$.
 Publication:

arXiv eprints
 Pub Date:
 September 2009
 DOI:
 10.48550/arXiv.0909.3205
 arXiv:
 arXiv:0909.3205
 Bibcode:
 2009arXiv0909.3205L
 Keywords:

 Mathematics  Probability;
 Mathematics  Complex Variables;
 60G55;
 60H07
 EPrint:
 25 pages