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2021 A Markov process for an infinite interacting particle system in the continuum
Yuri Kozitsky, Michael Röckner
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Electron. J. Probab. 26: 1-53 (2021). DOI: 10.1214/21-EJP631


An infinite system of point particles placed in 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 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:t0,μPexp} 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.

Funding Statement

This work was supported by the Deutsche Forschungsgemeinschaft through SFB 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” that is acknowledged by the authors.


Yuri Kozitsky thanks Lucian Beznea, Oleh Lopushansky and Yuri Tomilov for discussing some of its aspects. He thanks also Lucian Beznea and BIT DEFENDER for warm hospitality and financial support during his stay in Bucharest in April 2019, where a part of this work was done. Last but not least, the authors are cordially grateful to the referee for valuable and favorable suggestion that helped to improve the quality of the final version of this work.


Download Citation

Yuri Kozitsky. Michael Röckner. "A Markov process for an infinite interacting particle system in the continuum." Electron. J. Probab. 26 1 - 53, 2021.


Received: 10 February 2020; Accepted: 27 April 2021; Published: 2021
First available in Project Euclid: 25 May 2021

Digital Object Identifier: 10.1214/21-EJP631

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

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


Vol.26 • 2021
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