A Markov process for an infinite interacting particle system in the continuum

TL;DR

This paper constructs a Markov process describing the stochastic dynamics of an infinite interacting particle system in continuum space, with particles performing jumps influenced by mutual repulsion and interaction potentials.

Contribution

It proves the existence and uniqueness of a Markov process for an infinite particle system with repulsive interactions, solving a martingale problem in continuum space.

Findings

Existence of a unique Markov process for the system.

The process describes particles performing jumps with mutual repulsion.

The process is well-defined on a space of locally finite configurations.

Abstract

An infinite system of point particles placed in $\mathds{R}^d$ is studied. Its constituents perform random jumps with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states of the system are locally finite subsets of $\mathds{R}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures $\Gamma$ is equipped with the vague topology and the corresponding Borel $\sigma$-field. For a special class $\mathcal{P}_{\rm exp}$ of (sub-Poissonian) probability measures on $\Gamma$, we prove the existence of a unique family $\{P_{t,\mu}: t\geq 0, \ \mu \in \mathcal{P}_{\rm exp}\}$ of probability measures on the space of cadlag paths with values in $\Gamma$ that solves a restricted initial-value martingale problem for the mentioned system. Thereby, a Markov process with cadlag paths is specified which describes the stochastic dynamics of this particle system.