Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics

The distribution μcl of a Poisson cluster process in X=Rd (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X=⊔nXn, 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 μcl is quasi-invariant with respect to the group of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for μcl. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.