A Markov process for a continuum infinite particle system with attraction

Abstract

An infinite system of point particles placed in ${ℝ}^d$ is studied. The particles are of two types; they perform random walks in the course of which those of distinct type 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 ${ℝ}^d$. The set of such states ${\mathrm{\Gamma }}^2$ is equipped with the vague topology and the corresponding Borel σ-field. For a special class ${\mathcal{P}}_{\exp }$ of probability measures defined on ${\mathrm{\Gamma }}^2$, we prove the existence of a family $\{P_{t,\mathrm{\mu }}:t\ge 0,\mathrm{\mu }\in {\mathcal{P}}_{\exp }\}$ of probability measures defined on the space of càdlàg paths with values in ${\mathrm{\Gamma }}^2$, which is a unique solution of the restricted martingale problem for the mentioned stochastic dynamics. Thereby, the corresponding Markov process is specified.