Cluster point processes on manifolds

The probability distribution μcl of a general cluster point process in a Riemannian manifold XX (with independent random clusters attached to points of a configuration with distribution μμ) is studied via the projection of an auxiliary measure μˆ in the space of configurations γˆ={(x,ȳ)}⊂X×X, where x∈Xx∈X indicates a cluster “centre” and ȳ∈X:=⨆nXn represents a corresponding cluster relative to xx. We show that the measure μcl is quasi-invariant with respect to the group Diff0(X) of compactly supported diffeomorphisms of XX, and prove an integration-by-parts formula for μcl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. General constructions are illustrated by examples including Euclidean spaces, Lie groups, homogeneous spaces, Riemannian manifolds of non-positive curvature and metric spaces.