Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Infinite systems of competing Brownian particles

Andrey Sarantsev

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Abstract

Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (Ann. Appl. Probab. 18 (2008) 2179–2207). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (Ann. Inst. H. Poincare 52 (2016) 323–354).

Résumé

Nous considérons un système infini de particules browniennes sur la droite réelle. À tout moment ces particules peuvent être ordonnées de façon croissante. Chaque particule se déplace suivant un mouvement brownien dont les coefficients de dérive et de diffusion dépendent du rang de la particule. Les distances entre les particules successives forment le processus (infini dimensionnel) des écarts. Nous trouvons une mesure stationnaire du processus des écarts. Nous montrons aussi que si la distribution initiale du processus des écarts domine stochastiquement la distribution stationnaire, le processus converge vers cette distribution en grand temps. Ce travail poursuit donc l’étude de Pal et Pitman (Ann. Appl. Probab. 18 (2008) 2179–2207). Il inclut aussi le cas des systèmes infinis avec collisions asymétriques, similaire au cas fini de Karatzas et Shkolnikov (Ann. Inst. H. Poincare 52 (2016) 323–354).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2279-2315.

Dates
Received: 8 September 2015
Revised: 30 August 2016
Accepted: 3 September 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773746

Digital Object Identifier
doi:10.1214/16-AIHP791

Mathematical Reviews number (MathSciNet)
MR3729655

Zentralblatt MATH identifier
1382.60123

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05] 91B26: Market models (auctions, bargaining, bidding, selling, etc.)

Keywords
Reflected Brownian motion Competing Brownian particles Asymmetric collisions Interacting particle systems Weak convergence Stochastic comparison Triple collisions Stationary distribution

Citation

Sarantsev, Andrey. Infinite systems of competing Brownian particles. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2279--2315. doi:10.1214/16-AIHP791. https://projecteuclid.org/euclid.aihp/1511773746


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