Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The infinite Atlas process: Convergence to equilibrium

Amir Dembo, Milton Jara, and Stefano Olla

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The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.


Let modèle de Atlas demi-infini est un système unidimensional des particules Browniens, où seulement la particule plus à gauche a une vitesse positive. Le processus d’incréments correspondant a une infinité des mesures invariantes, avec une mesure distinguée, reversible et invariante par translations. On montre que cette mesure attire une grande classe des configurations initiales avec densité bornée ou à croissance modérée. Central à notre preuve est une nouvelle estimation de la vitesse de convergence vers l’équilibre du processus d’incréments dans un tableau triangulaire de systèmes finis et de grande taille.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 607-619.

Received: 11 September 2017
Revised: 15 November 2017
Accepted: 16 November 2017
First available in Project Euclid: 14 May 2019

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Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes [See also 58J65]

Interacting particles Reflecting Brownian motions Non-equilibrium hydrodynamics Infinite Atlas process


Dembo, Amir; Jara, Milton; Olla, Stefano. The infinite Atlas process: Convergence to equilibrium. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 607--619. doi:10.1214/17-AIHP875.

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