Abstract
We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Itô 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 Poincaré 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 η.
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Mathew D. Penrose partially supported by the Alexander von Humboldt Foundation through a Friedrich Wilhelm Bessel Research Award.
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Last, G., Penrose, M.D. Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150, 663–690 (2011). https://doi.org/10.1007/s00440-010-0288-5
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DOI: https://doi.org/10.1007/s00440-010-0288-5
Keywords
- Poisson process
- Chaos expansion
- Derivative operator
- Kabanov–Skorohod integral
- Malliavin calculus
- Poincaré inequality
- Variance inequalities
- Infinitely divisible random measure