promoting access to White Rose research papers Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/ This is an author produced version of a paper published in Journal of Geometry and Physics White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74924/ Published paper: Bogachev, L and Daletskii, A (2013) Cluster point processes on manifolds. Journal of Geometry and Physics, 63 (1). 45 - 79 (35). http://dx.doi.org/10.1016/j.geomphys.2012.09.007 White Rose Research Online
[email protected] Cluster Point Processes on Manifolds Leonid Bogacheva and Alexei Daletskiib aDepartment of Statistics, University of Leeds, Leeds LS2 9JT, UK. E-mail:
[email protected] bDepartment of Mathematics, University of York, York YO10 5DD, UK. E-mail:
[email protected] Abstract The probability distribution µcl 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 µˆ in the space of configurations γˆ = {(x, y¯)} ⊂ X × X, where x ∈ X indicates a cluster “cen- F n tre” and y¯ ∈ X := n X represents a corresponding cluster relative to x. We show that the measure µcl 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 curva- ture and metric spaces.