Critical branching brownian motion with absorption: Particle configurations

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1215-1250.

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.

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.

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