A Markov process for a continuum infinite particle system with attraction

TL;DR

This paper models an infinite particle system with two types of particles in continuous space, demonstrating the existence and uniqueness of a Markov process describing their stochastic dynamics with attraction effects.

Contribution

It introduces a new Markov process for a continuum particle system with attraction, proving existence and uniqueness for the associated stochastic dynamics.

Findings

Existence of a Markov process for the system.

Uniqueness of the solution to the martingale problem.

Characterization of the process dynamics.

Abstract

An infinite system of point particles placed in $\mathds{R}^d$ is studied. The particles are of two types; they perform random walks in the course of which those of distinct types repel each other. The interaction of this kind induces an effective multi-body attraction of the same type particles, which leads to the multiplicity of states of thermal equilibrium in such systems. The pure states of the system are locally finite counting measures on $\mathds{R}^d$. The set of such states $\Gamma^2$ is equipped with the vague topology and the corresponding Borel $\sigma$-field. For a special class $\mathcal{P}_{\rm exp}$ of probability measures defined on $\Gamma^2$, we prove the existence of a family $\{P_{t,\mu}: t\geq 0, \ \mu \in \mathcal{P}_{\rm exp}\}$ of probability measures defined on the space of c{\`a}dl{\`a}g paths with values in $\Gamma^2$, which is a unique solution of the restricted martingale problem for the mentioned stochastic dynamics. Thereby, the corresponding Markov process is specified.