Random sets of isomorphism of linear operators on Hilbert space

Abstract

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 $\hat{\mu}$ in the space of configurations $\hat{\gamma}=\{(x,\bar{y})\}\subset X\times\mathfrak{X}$, where $x\in X$ indicates a cluster "centre" and $\bar{y}\in\mathfrak{X}:=\bigsqcup_{n} X^n$ 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 nonpositive curvature and metric spaces. The paper is an extension of our earlier results for Poisson cluster measures [J. Funct. Analysis 256 (2009) 432-478] and for Gibbs cluster measures [arxiv:1007.3148], where different projection constructions were utilised.