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

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 ${\mathcal{P}}_{\exp }$ of (sub-Poissonian) probability measures on $\Gamma$, we prove the existence of a unique family $\{P_{t,\mu }:t\ge 0,\mu \in {\mathcal{P}}_{\exp }\}$ 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.

44 pages

References in corpus (3)

• Spatial birth and death processes as solutions of stochastic equations
• Existence and spatial limit theorems for lattice and continuum particle systems
• A Phase Transition in a Widom-Rowlinson Model with Curie-Weiss Interaction