Gibbs cluster measures on configuration spaces

Research group:
 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