Cluster point processes on manifolds

The probability distribution μ_cl of a general cluster point process in a Riemannian manifold X (with independent random clusters attached to points of a configuration with distribution μ) is studied via the projection of an auxiliary measure ˆμ in the space of configurations ˆ Γ = {(x,y)} ⊂ X × X, where x ∈ X indicates a cluster "centre" and y ∈ X represents a corresponding cluster relative to x. We show that the measure μ_cl is quasi-invariant with respect to the group Diff0(X) of compactly supported diffeomorphisms of X, 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.