Open Access
2023 A Markov process for a continuum infinite particle system with attraction
Yuri Kozitsky, Michael Röckner
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Electron. J. Probab. 28: 1-59 (2023). DOI: 10.1214/23-EJP952


An infinite system of point particles placed in 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 d. The set of such states Γ2 is equipped with the vague topology and the corresponding Borel σ-field. For a special class Pexp of probability measures defined on Γ2, we prove the existence of a family {Pt,μ:t0,μPexp} of probability measures defined on the space of càdlàg paths with values in Γ2, which is a unique solution of the restricted martingale problem for the mentioned stochastic dynamics. Thereby, the corresponding Markov process is specified.

Funding Statement

Supported by the Deutsche Forschungsgemeinschaft (DFG) through the SFB 1238 “Taming uncertainty and profiting from randomness and low regularity in analysis, statistics and their applications”.


The author are grateful to both referees for careful reading of the manuscript and making valuable remarks.


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Yuri Kozitsky. Michael Röckner. "A Markov process for a continuum infinite particle system with attraction." Electron. J. Probab. 28 1 - 59, 2023.


Received: 17 July 2022; Accepted: 1 May 2023; Published: 2023
First available in Project Euclid: 12 May 2023

MathSciNet: MR4587444
zbMATH: 07707068
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP952

Primary: 35Q84 , 60G55 , 60J25 , 60J75

Keywords: Fokker-Planck equation , martingale solution , Measure-valued Markov process , point process , stochastic semigroup

Vol.28 • 2023
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