The Annals of Probability

A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion

S. P. Lalley and T. Sellke

Full-text: Open access

Abstract

We prove a weak limit theorem which relates the large time behavior of the maximum of a branching Brownian motion to the limiting value of a certain associated martingale. This exhibits the minimal velocity travelling wave for the KPP-Fisher equation as a translation mixture of extreme-value distributions. We also show that every particle in a branching Brownian motion has a descendant at the frontier at some time. A final section states several conjectures concerning a hypothesized stationary "standing wave of particles" process and the relationship of this process to branching Brownian motion.

Article information

Source
Ann. Probab., Volume 15, Number 3 (1987), 1052-1061.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992080

Digital Object Identifier
doi:10.1214/aop/1176992080

Mathematical Reviews number (MathSciNet)
MR893913

Zentralblatt MATH identifier
0622.60085

JSTOR
links.jstor.org

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]

Keywords
Branching Brownian motion KPP-Fisher equation travelling wave extreme-value distribution

Citation

Lalley, S. P.; Sellke, T. A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion. Ann. Probab. 15 (1987), no. 3, 1052--1061. doi:10.1214/aop/1176992080. https://projecteuclid.org/euclid.aop/1176992080


Export citation