Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Critical branching Brownian motion with absorption: Particle configurations

Julien Berestycki, Nathanaël Berestycki, and Jason Schweinsberg

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Abstract

We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x>0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-\sqrt{2}$, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.

Résumé

Nous considérons un mouvement brownien branchant avec absorption critique, issu d’une particule en $x>0$, dans lequel les particules se déplacent selon des mouvement browniens réels indépendants avec une dérive critique de $-\sqrt{2}$, et sont absorbées en zero. Nous obtenons des résultats asymptotiques sur le comportement de ce processus avant son extinction, quand la position $x$ de la particule initiale tend vers l’infini. En particulier nous obtenons des éstimées sur le nombre de particules dans le système, la position de la particule la plus à droite, et la configuration des particules à un instant typique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1215-1250.

Dates
Received: 28 September 2013
Revised: 13 March 2014
Accepted: 14 March 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432041

Digital Object Identifier
doi:10.1214/14-AIHP613

Mathematical Reviews number (MathSciNet)
MR3414446

Zentralblatt MATH identifier
1329.60300

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces

Keywords
Branching Brownian motion Critical phenomena Yaglom limit laws

Citation

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Critical branching Brownian motion with absorption: Particle configurations. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1215--1250. doi:10.1214/14-AIHP613. https://projecteuclid.org/euclid.aihp/1445432041


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