Gibbs cluster measures on configuration spaces

The probability distribution gcl of a Gibbs cluster point process in X = Rd (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 ˆg in the space of configurations γˆ = {(x, y¯)} ⊂ X × X, where x ∈ X indicates a cluster “center” and ¯y ∈ X := n Xn represents a corresponding cluster relative to x. We show that the measure gcl is quasi-invariant with respect to the group Diff0(X) of compactly supported diffeomorphisms of X, and prove an integration-by-parts formula for gcl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. © 2012 Elsevier Inc. All rights reserved.