Consider a (space-time) realization $\omega$ of a critical or subcritical one-dimensional branching Brownian motion. Let $Z_x(\omega)$ for $x \geq 0$ be the number of particles which are located for the first time on the vertical line through $(x, 0)$ and which do not have an ancestor on this line. In this note we study the process $Z = \{Z_x; x \geq 0\}$. We show that $Z$ is a continuous-time Galton-Watson process and compute its creation rate and offspring distribution. Here we use ideas of Neveu, who considered a similar problem in a supercritical case. Moreover, in the critical case we characterize the continuous state branching processes obtained as weak limits of the processes $Z$ under rescaling.
Publié le : 1993-02-14
Classification:
Branching Brownian motion,
Galton-Watson process,
Levy process,
super-Brownian motion,
weak convergence,
first passage,
60J65,
60J80
@article{1177005513,
author = {Kaj, Ingemar and Salminen, Paavo},
title = {On a First Passage Problem for Branching Brownian Motions},
journal = {Ann. Appl. Probab.},
volume = {3},
number = {4},
year = {1993},
pages = { 173-185},
language = {en},
url = {http://dml.mathdoc.fr/item/1177005513}
}
Kaj, Ingemar; Salminen, Paavo. On a First Passage Problem for Branching Brownian Motions. Ann. Appl. Probab., Tome 3 (1993) no. 4, pp. 173-185. http://gdmltest.u-ga.fr/item/1177005513/