Gibbs cluster measures on configuration spaces

الأصل · EN

The 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 g in the space of configurations gamma={(x,y)}⊂ X, where x∈ X indicates a cluster "center" and y:=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. These results are quite general; in particular, the uniqueness of the background Gibbs measure g is not required. The paper is an extension of the earlier results for Poisson cluster measures %obtained by the authors [J. Funct. Analysis 256 (2009) 432-478], where a different projection construction was utilized specific to this "exactly soluble" case.