Gibbs point processes on path space: existence, cluster expansion and uniqueness

TL;DR

This paper establishes existence and uniqueness results for Gibbs point processes on path space with pair interactions, using entropy and cluster expansion methods in an infinite-dimensional diffusion context.

Contribution

It introduces new existence and uniqueness theorems for Gibbs point processes on path space with pair interactions, expanding the theoretical understanding of infinite-dimensional diffusions.

Findings

Proved existence of infinite-volume Gibbs point processes on path space.

Derived explicit activity domain for uniqueness using cluster expansion.

Applied entropy method to establish foundational results.

Abstract

We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space. More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to $\mathbb{R}^d$, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.