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Cluster point processes on manifolds

Bogachev, L and Daletskii, A (2011) Cluster point processes on manifolds. Journal of the European Mathematical Society. ISSN 1435-9855 (In Press)

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Abstract

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. The paper is an extension of our earlier results for Poisson cluster measures [J. Funct. Analysis 256 (2009) 432–478] and for Gibbs cluster measures [http://arxiv.org/abs/1007.3148], where different projection constructions were utilised.

Item Type: Article
Copyright, Publisher and Additional Information: © 2011 European Mathematical Society. This is an author produced version of a paper due to be published in Journal of the European Mathematical Society. Reproduced in accordance with the publisher's self-archiving policy.
Keywords: Cluster point process, Configuration space, Riemannian manifold, Poisson measure, Projection, Quasi-invariance, Integration by parts, Dirichlet form, Stochastic dynamics MSC 2010: P
Institution: The University of Leeds, The University of York
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds)
The University of York > Mathematics (York)
Depositing User: Symplectic Publications
Date Deposited: 17 Oct 2011 14:36
Last Modified: 17 Jun 2014 15:22
Status: In Press
Publisher: European Mathematical Society
URI: http://eprints.whiterose.ac.uk/id/eprint/43329

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