The distribution $\mu_{ cl}$ 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 $\mu_{ cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $\mu_{ cl}$. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.
Categories: math.FA math.PR