Gibbs cluster measures on configuration spaces

The probability distribution g_cl of a Gibbs cluster point process in X = R^d (with i.i.d. random clusters attached to points of a Gibbs configuration with distribution g) is studied via the projection of an auxiliary Gibbs measure ĝ in the space of configurations ^γ={(x,\bar{y})}, where x∈X indicates a cluster "center" and y∈\mathfrak{X}=\sqcup_{n} X^n represents a corresponding cluster relative to x. We show that the measure g_cl is quasi-invariant with respect to the group Diff_0(X) of compactly supported diffeomorphisms of X, and prove an integration-by-parts formula for g_cl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.