Poisson Cluster Measures: Quasi-invariance, Integration by Parts and Equilibrium Stochastic Dynamics

TL;DR

This paper investigates Poisson cluster measures in Euclidean space, establishing their quasi-invariance, deriving an integration-by-parts formula, and constructing associated equilibrium stochastic dynamics using Dirichlet forms.

Contribution

It introduces a novel approach to analyze Poisson cluster measures via an auxiliary Poisson measure and constructs equilibrium dynamics through Dirichlet forms.

Findings

Proves quasi-invariance of Poisson cluster measures under diffeomorphisms.

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

Constructs equilibrium stochastic dynamics using Dirichlet forms.

Abstract

The distribution $\mu_{cl}$ of a Poisson cluster process in $X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure $\mu_{cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $\mu_{cl}$. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.