February 2024 On the rate of convergence to coalescing Brownian motions
Konstantin Khanin, Liying Li
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 208-231 (February 2024). DOI: 10.1214/22-AIHP1348


In this paper we study the rate of convergence of the iterates of i.i.d. random piecewise constant monotone maps to the time-1 transport map for the process of coalescing Brownian motions. We prove that the rate of convergence is given by a power law. The time-1 map for the coalescing Brownian motions can be viewed as a fixed point for a natural renormalization transformation acting in the space of probability laws for random piecewise constant monotone maps. Our result implies that this fixed point is exponentially stable.

Dans cet article nous étudions le taux de convergence des itérées de fonctions aléatoires monotones et constantes par morceaux i.i.d. vers l’application de transport du mouvement Brownien coalescents au temps 1. Nous prouvons que le taux de convergence est donné par une loi de puissance. Cette application de transport peut être vue comme le point fixe d’une transformation de renormalisation naturelle agissant sur l’espace des lois de probabilité pour les fonctions aléatoires monotones et constantes par morceaux. Notre résultat implique que ce point fixe est exponentiellement stable.

Funding Statement

The first author is supported by the NSERC Discovery Grant RGPIN-2018-04510.


In memory of a dear friend Dima Ioffe


The authors would like to thank the anonymous referees for pointing out many small mistakes in the manuscript, and for informing us about many useful references on the coalescing Brownian motion and the study of the convergence of Harris flows.


Download Citation

Konstantin Khanin. Liying Li. "On the rate of convergence to coalescing Brownian motions." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 208 - 231, February 2024. https://doi.org/10.1214/22-AIHP1348


Received: 25 December 2021; Revised: 7 October 2022; Accepted: 11 November 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718379
Digital Object Identifier: 10.1214/22-AIHP1348

Primary: 37A25 , 37H30 , 60J05 , 60J65

Keywords: Coalescing Brownian motions , coupling , Random monotone maps , rate of convergence

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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