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 Pexp of (sub-Poissonian) probability measures on Γ, we prove the existence of a unique family {Pt,µ : t ≥ 0, µ ∈ Pexp} 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.