Cluster point processes on manifolds

JOURNAL OF GEOMETRY AND PHYSICS

卷:63

Article

Bogachev, Leonid²  Daletskii, Alexei¹ 

[1] Univ York, Dept Math, York YO10 5DD, N Yorkshire, England

[2] Univ Leeds, Dept Stat, Leeds LS2 9JT, W Yorkshire, England

关键词: Cluster point process;    Configuration space;    Riemannian manifold;    Quasi-invariance;    Integration by parts;    Stochastic dynamics;   

DOI  :  10.1016/j.geomphys.2012.09.007

来源: Elsevier

【 摘 要 】

The probability distribution mu(cl) of a general cluster point process in a Riemannian manifold X (with independent random clusters attached to points of a configuration with distribution mu) is studied via the projection of an auxiliary measure (mu) over cap in the space of configurations (gamma) over cap = {(x, (y) over bar)} subset of X x < sic >, where x is an element of X indicates a cluster centre and (y) over bar is an element of < sic > := coproduct(n) X-n represents a corresponding cluster relative to x. We show that the measure mu(cl) is quasi-invariant with respect to the group Diff(0)(X) of compactly supported diffeomorphisms of X, and prove an integration-by-parts formula for mu(cl). 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. (c) 2012 Elsevier B.V. All rights reserved.

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