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

Bogachev, L. & Daletskii, A.

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 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.

cluster point process; Poisson measure; configuration space; quasi-invariance; integration by parts; Dirichlet form; stochastic dynamics