Nous considérons un mouvement brownien branchant avec absorption critique, issu d’une particule en , dans lequel les particules se déplacent selon des mouvement browniens réels indépendants avec une dérive critique de , et sont absorbées en zero. Nous obtenons des résultats asymptotiques sur le comportement de ce processus avant son extinction, quand la position 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 , particles move according to independent one-dimensional Brownian motions with the critical drift of , 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 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.
@article{AIHPB_2015__51_4_1215_0, author = {Berestycki, Julien and Berestycki, Nathana\"el and Schweinsberg, Jason}, title = {Critical branching brownian motion with absorption: {Particle} configurations}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1215--1250}, publisher = {Gauthier-Villars}, volume = {51}, number = {4}, year = {2015}, doi = {10.1214/14-AIHP613}, mrnumber = {3414446}, zbl = {1329.60300}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP613/} }
TY - JOUR AU - Berestycki, Julien AU - Berestycki, Nathanaël AU - Schweinsberg, Jason TI - Critical branching brownian motion with absorption: Particle configurations JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 1215 EP - 1250 VL - 51 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP613/ DO - 10.1214/14-AIHP613 LA - en ID - AIHPB_2015__51_4_1215_0 ER -
%0 Journal Article %A Berestycki, Julien %A Berestycki, Nathanaël %A Schweinsberg, Jason %T Critical branching brownian motion with absorption: Particle configurations %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 1215-1250 %V 51 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP613/ %R 10.1214/14-AIHP613 %G en %F AIHPB_2015__51_4_1215_0
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, pp. 1215-1250. doi : 10.1214/14-AIHP613. http://www.numdam.org/articles/10.1214/14-AIHP613/
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