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arXiv · 0803.4496

Poisson Cluster Measures: Quasi-invariance, Integration by Parts and Equilibrium Stochastic Dynamics

Leonid Bogachev·Alexei Daletskii
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Abstract

The distribution $μ_{cl}$ of a Poisson cluster process in $X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure $μ_{cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $μ_{cl}$. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.

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Leonid Bogachev, Alexei Daletskii. 2008-10-07. Poisson Cluster Measures: Quasi-invariance, Integration by Parts and Equilibrium Stochastic Dynamics. https://arxiv.org/abs/0803.4496

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