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

Abstract

An infinite system of point particles placed in ${ℝ}^d$ is studied. Its constituents perform random jumps (walks) 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 ${ℝ}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures Γ is equipped with the vague topology and the corresponding Borel σ-field. For a special class ${\mathcal{P}}_{\exp }$ of (sub-Poissonian) probability measures on Γ, we prove the existence of a unique family $\{P_{t,\mathrm{\mu }}:t\ge 0,\mathrm{\mu }\in {\mathcal{P}}_{\exp }\}$ of probability measures on the space of cadlag paths with values in Γ 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.