@article{ECP1259,
author = {John Harris and Simon Harris},
title = {Survival probabilities for branching Brownian motion with absorption},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {12},
year = {2007},
keywords = {Branching Brownian motion with absorption; spine constructions; additive martingales.},
abstract = {We study a branching Brownian motion (BBM) with absorption, in which particles move as Brownian motions with drift $-\rho$, undergo dyadic branching at rate $\beta>0$, and are killed on hitting the origin. In the case $\rho>\sqrt{2\beta}$ the extinction time for this process, $\zeta$, is known to be finite almost surely. The main result of this article is a large-time asymptotic formula for the survival probability $P^x(\zeta>t)$ in the case $\rho>\sqrt{2\beta}$, where $P^x$ is the law of the BBM with absorption started from a single particle at the position $x>0$. We also introduce an additive martingale, $V$, for the BBM with absorption, and then ascertain the convergence properties of $V$. Finally, we use $V$ in a `spine' change of measure and interpret this in terms of `conditioning the BBM to survive forever' when $\rho>\sqrt{2\beta}$, in the sense that it is the large $t$-limit of the conditional probabilities $P^x(A\mid \zeta > t+s)$, for $A\in F_s$.},
pages = {no. 10, 81-92},
issn = {1083-589X},
doi = {10.1214/ECP.v12-1259},
url = {http://ecp.ejpecp.org/article/view/1259}}