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July 2022 Domains of attraction of invariant distributions of the infinite Atlas model
Sayan Banerjee, Amarjit Budhiraja
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Ann. Probab. 50(4): 1610-1646 (July 2022). DOI: 10.1214/22-AOP1570

Abstract

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every a0, πa:=i=1Exp(2+ia) is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure πa if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to πa weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of πa for each a0. The cases a=0 and a>0 are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions.

Funding Statement

The research of SB was supported in part by the NSF CAREER award DMS-2141621. The research of AB was supported in part by the NSF (DMS-1814894 and DMS-1853968).

Acknowledgments

We thank two anonymous referees whose valuable inputs significantly improved the article.

Citation

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Sayan Banerjee. Amarjit Budhiraja. "Domains of attraction of invariant distributions of the infinite Atlas model." Ann. Probab. 50 (4) 1610 - 1646, July 2022. https://doi.org/10.1214/22-AOP1570

Information

Received: 1 March 2021; Revised: 1 October 2021; Published: July 2022
First available in Project Euclid: 11 May 2022

Digital Object Identifier: 10.1214/22-AOP1570

Subjects:
Primary: 60J60 , 60K35 , 82C22
Secondary: 60J55

Keywords: atlas model , ergodicity , infinite dimensional diffusions , Interacting diffusions , Local time , synchronous couplings , time averaged occupancy measures

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 4 • July 2022
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