Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason
Critical branching brownian motion with absorption: Particle configurations
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4 , p. 1215-1250
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MR 3414446
doi : 10.1214/14-AIHP613
URL stable : http://www.numdam.org/item?id=AIHPB_2015__51_4_1215_0

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 -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.
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 -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.

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