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Gibbs cluster measures on configuration spaces
Title  Gibbs cluster measures on configuration spaces 
Publication Type  Journal Article 
Year of Publication  2013 
Authors  Bogachev L, Daletskii A 
Refereed Designation  Refereed 
Journal  Journal of Functional Analysis 
Volume  264 
Start Page  508 
Issue  2 
Pagination  508550 
Date Published  01/2013 
Abstract  The probability distribution $g_{cl}$ of a Gibbs cluster point process in a Euclidean space $X$(with
i.i.d. random clusters attached to points of a Gibbs configuration with distribution
g) is studied via the projection of an auxiliary Gibbs measure in the space of
configurations in $X\times X^\infty$.
We show that that the measure $g_{cl}$ is quasiinvariant with respect to the group $Diff_0(X)$ of compactly
supported diffeomorphisms of $X$, and prove an integrationbyparts formula
for gcl. The associated equilibrium stochastic dynamics is then constructed
using the method of Dirichlet forms. These results are quite general; in particular,
the uniqueness of the background Gibbs measure $g$ is not required. The paper is
an extension of the earlier results for Poisson cluster measures [J. Funct. Analysis
256 (2009) 432–478], where a different projection construction was utilized
specific to this “exactly soluble” case.

URL  http://arxiv.org/abs/1007.3148 
DOI  10.1016/j.jfa.2012.11.002 
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Edited 22 Jan 2013  12:15 by cac7
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