Cluster point processes on manifolds

TL;DR

This paper develops a mathematical framework for cluster point processes on Riemannian manifolds, establishing properties like quasi-invariance, integration-by-parts, and constructing associated stochastic dynamics, with applications to various geometric spaces.

Contribution

It introduces a new projection-based approach to analyze cluster point processes on manifolds, extending previous results for Poisson and Gibbs measures.

Findings

Proves quasi-invariance of cluster measures under diffeomorphisms.

Derives an integration-by-parts formula for the measures.

Constructs equilibrium stochastic dynamics via Dirichlet forms.

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.