Gibbs cluster measures on configuration spaces (Q1932194)

Let \(X\) be a topological space, and let \(\Gamma_X=\{\gamma\}\) be the configuration space over \(X\), that is, the space of countable subsets (called configurations) \(\gamma\subset X\) without accumulation points. In [J. Funct. Anal. 154, No. 2, 444--500 (1998; Zbl 0914.58028)] and (Gibbsian case) [J. Funct. Anal. 157, No. 1, 242--291 (1998; Zbl 0931.58019)], \textit{S. Albeverio, Yu. G. Kondratiev} and \textit{M. Röckner} proposed an approach to configuration spaces \(\Gamma_X\) as infinite-dimensional manifolds, based on the choice of a suitable probability measure \(\mu\) on \(\Gamma_X\). Provided that the measure \(\mu\) can be shown to satisfy an integration-by-parts formula, one can construct, using the theory of Dirichlet forms, an associated stochastic process on \(\Gamma_X\) such that \(\mu\) is its invariant measure. This general program has been first implemented in [Zbl 0914.58028] for the Poisson measure \(\mu\) on \(\Gamma_X\), and then extended in [Zbl 0931.58019] to a wider class of Gibbs measures. In the earlier papers of the authors, [``Equilibrium stochastic dynamics of Poisson cluster ensembles'', Condens. Matter Phys. 11, 261--273 (2008); J. Funct. Anal. 256, No. 2, 432--478 (2009; Zbl 1170.60022)], a similar analysis was developed for a different class of random spatial structures, namely Poisson cluster point processes, featuring spatial grouping of points around the background random (Poisson) configuration of invisible centers. The technique in [the authors, loc. cit.] was based on the representation of a given Poisson cluster measure on \(\Gamma_X\) as the projection image of an auxiliary Poisson measure on a more complex configuration space. The aim of this paper is to extend this approach to a more general class of Gibbs cluster measures on the configuration space \(\Gamma_X\), where the distribution of cluster centers is given by Gibbs measure \(g\) on \(\Gamma_X\) with reference measure \(\theta\) on \(X\) and an interaction potential \(\Phi\). The authors focus on Gibbs cluster processes in \(X=\mathbb{R}^d\) with independent random clusters of random size. Under some natural smoothness conditions on the reference measure \(\theta\) and the distribution \(\eta\) of the generic cluster, the authors prove the \(\operatorname{Diff}_0(X)\)-quasi-invariance of the corresponding Gibbs cluster measure \(g_{cl}\) (Section 3.2), establish the integration-by-parts formula (Section 3.3) and construct the associated Dirichlet operator, which leads to the existence of the equilibrium stochastic dynamics on \(\Gamma_X\) (Section 4), where \(\operatorname{Diff}_0(X)\) is the group of compactly supported diffeomorphisms of \(X\). These ideas are further developed in the authors' paper [J. Geom. Phys. 63, 45--79 (2013; Zbl 1268.58025)].