Equilibrium stochastic dynamics of Poisson cluster ensembles

The distribution μ of a Poisson cluster process in Χ=Rd (with n-point clusters) is studied via the projection of an auxiliary Poisson measure in the space of configurations in Χn, with the intensity measure being the convolution of the background intensity (of cluster centres) with the probability distribution of a generic cluster. We show that μ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of Χ, and prove an integration by parts formula for μ. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms