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The distribution $\mu_{cl}$ of a Poisson cluster process in$X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poissonmeasure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, withintensity measure defined as a convolution of the background intensity ofcluster centres and the probability distribution of a generic cluster. We showthat the measure $\mu_{cl}$ is quasi-invariant with respect to the group ofcompactly supported diffeomorphisms of $X$ and prove an integration-by-partsformula for $\mu_{cl}$. The corresponding equilibrium stochastic dynamics isthen constructed using the method of Dirichlet forms.

Poisson Cluster Measures: Quasi-invariance, Integration by Parts and Equilibrium Stochastic Dynamics