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 Electronic Communications in Probability > Vol. 12 (2007) > Paper 10 open journal systems 


Survival probabilities for branching Brownian motion with absorption

John William Harris, University of Bristol
Simon C Harris, University of Bath


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 $rho>sqrt{2beta}$ 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 $rho>sqrt{2beta}$, 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 $rho>sqrt{2beta}$, in the sense that it is the large $t$-limit of the conditional probabilities $P^x(A|zeta>t+s)$, for $AinF_s$.


Full text: PDF

Pages: 81-92

Published on: April 7, 2007
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Electronic Communications in Probability. ISSN: 1083-589X