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	508-550
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 quasi-invariant with respect to the group $Diff_0(X)$ of compactly supported diffeomorphisms of $X$, and prove an integration-by-parts 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

Department of Mathematics