Consider a branching Brownian motion for which the instantaneous branching rate of a particle at position $x$ is given by $\beta(x)$. We assume that $\beta$ is an integrable continuous function converging to 0 as $x \rightarrow \pm \infty$. Let $R(t)$ be the position of the rightmost descendant at the time $t$ of a simple particle starting from position 0 at time 0. We show that there exists a constant $\lambda_0 > 0$ such that $R(t) - \sqrt{\lambda_0/2} t$ converges in distribution as $t \rightarrow \infty$ to a location mixture of the extreme value distribution $\exp (e^{-\sqrt{2\lambda_0 x}})$.
Publié le : 1988-07-14
Classification:
Inhomogeneous branching Brownian motion,
traveling wave,
extreme value distribution,
60J80,
60G55,
60F05
@article{1176991677,
author = {Lalley, S. and Sellke, T.},
title = {Traveling Waves in Inhomogeneous Branching Brownian Motions. I},
journal = {Ann. Probab.},
volume = {16},
number = {4},
year = {1988},
pages = { 1051-1062},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991677}
}
Lalley, S.; Sellke, T. Traveling Waves in Inhomogeneous Branching Brownian Motions. I. Ann. Probab., Tome 16 (1988) no. 4, pp. 1051-1062. http://gdmltest.u-ga.fr/item/1176991677/