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 = FnXn, 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 quasiinvariant with respect to the group of compactly supported diffeomorphisms ofX and prove an integration-by-parts formula for µcl. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms