May 2023 Coalescing-fragmentating Wasserstein dynamics: Particle approach
Vitalii Konarovskyi
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(2): 983-1028 (May 2023). DOI: 10.1214/22-AIHP1271


We construct a family of semimartingales that describes the behavior of a particle system with sticky-reflecting interaction. The model is a physical improvement of the Howitt–Warren flow (Ann. Probab. 37 (2009) 1237–1272), an infinite system of diffusion particles on the real line that sticky-reflect from each other. But now particles have masses obeying the conservation law and the diffusion rate of each particle depends on its mass. The equation which describes the evolution of the particle system is a new type of equations in infinite-dimensional space and can be interpreted as an infinite-dimensional analog of the equation for sticky-reflected Brownian motion. The particle model appears as a particular solution to the corrected version of the Dean–Kawasaki equation.

Nous construisons une famille de semimartingales décrivant le comportement d’un système de particules avec interactions à effet réflectif et adhésif. Ce modèle est un amélioration plus physique du flot de Howitt–Warren (Ann. Probab. 37 (2009) 1237–1272), un système infini de particules diffusives sur la droite réelle interagissant avec effet réflectif et adhésif. Dans cet article, les particules ont désormais des masses qui satisfont à la loi de conservation, et le coefficient de diffusion de chaque particule dépend de sa masse. L’équation décrivant l’évolution du système de particules est un nouveau type d’équation sur un espace de dimension infinie et peut être interprétée comme un analogue infini-dimensionnel de l’équation satisfaite par le mouvement brownien à comportement réflectif et adhésif. Le modèle particulaire apparaît comme une solution particulière d’une version corrigée de l’équation de Dean–Kawasaki.

Funding Statement

The research was partly supported by Alexander von Humboldt Foundation and partly supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—SFB 1283/2 2021–317210226.


The author is grateful to Max von Renesse for useful discussions and suggestions. The author also thanks the anonymous referees for their careful reading of the manuscript and many valuable comments which improved the presentation of the results.


Download Citation

Vitalii Konarovskyi. "Coalescing-fragmentating Wasserstein dynamics: Particle approach." Ann. Inst. H. Poincaré Probab. Statist. 59 (2) 983 - 1028, May 2023.


Received: 20 August 2019; Revised: 17 March 2022; Accepted: 30 March 2022; Published: May 2023
First available in Project Euclid: 12 April 2023

MathSciNet: MR4575023
zbMATH: 07699948
Digital Object Identifier: 10.1214/22-AIHP1271

Primary: 60B12 , 60K35
Secondary: 60G44 , 60J60 , 82B21

Keywords: Howitt–Warren flow , Infinite-dimensional SDE with discontinuous coefficients , modified massive Arratia flow , Sticky-reflected Brownian motion , Wasserstein diffusion

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré


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Vol.59 • No. 2 • May 2023
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