Gibbs states over the cone of discrete measures

TL;DR

This paper develops a framework for constructing Gibbs measures over the cone of discrete Radon measures, extending to general Lévy processes, with applications in modeling densely distributed particle systems.

Contribution

It introduces a new definition of Gibbs measures over the cone of discrete measures and analyzes their existence for Gamma and Lévy processes, including moment estimates and inequalities.

Findings

Established existence conditions for Gibbs measures over the cone.

Derived uniform moment estimates for Gibbs distributions.

Proved a Mecke type characterization and FKG inequality for Gamma measures.

Abstract

We construct Gibbs perturbations of the Gamma process on $\mathbbm{R}^d$, which may be used in applications to model systems of densely distributed particles. First we propose a definition of Gibbs measures over the cone of discrete Radon measures on $\mathbbm{R}^d$ and then analyze conditions for their existence. Our approach works also for general L\'evy processes instead of Gamma measures. To this end, we need only the assumption that the first two moments of the involved L\'evy intensity measures are finite. Also uniform moment estimates for the Gibbs distributions are obtained, which are essential for the construction of related diffusions. Moreover, we prove a Mecke type characterization for the Gamma measures on the cone and an FKG inequality for them.