Electronic Communications in Probability

Survival probabilities for branching Brownian motion with absorption

John Harris and Simon Harris

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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 $A\in F_s$.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 10, 81-92.

Dates
Accepted: 7 April 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224953

Digital Object Identifier
doi:10.1214/ECP.v12-1259

Mathematical Reviews number (MathSciNet)
MR2300218

Zentralblatt MATH identifier
1132.60059

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching Brownian motion with absorption spine constructions additive martingales

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Harris, John; Harris, Simon. Survival probabilities for branching Brownian motion with absorption. Electron. Commun. Probab. 12 (2007), paper no. 10, 81--92. doi:10.1214/ECP.v12-1259. https://projecteuclid.org/euclid.ecp/1465224953


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