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February, 1993 On a First Passage Problem for Branching Brownian Motions
Ingemar Kaj, Paavo Salminen
Ann. Appl. Probab. 3(1): 173-185 (February, 1993). DOI: 10.1214/aoap/1177005513

Abstract

Consider a (space-time) realization $\omega$ of a critical or subcritical one-dimensional branching Brownian motion. Let $Z_x(\omega)$ for $x \geq 0$ be the number of particles which are located for the first time on the vertical line through $(x, 0)$ and which do not have an ancestor on this line. In this note we study the process $Z = \{Z_x; x \geq 0\}$. We show that $Z$ is a continuous-time Galton-Watson process and compute its creation rate and offspring distribution. Here we use ideas of Neveu, who considered a similar problem in a supercritical case. Moreover, in the critical case we characterize the continuous state branching processes obtained as weak limits of the processes $Z$ under rescaling.

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Ingemar Kaj. Paavo Salminen. "On a First Passage Problem for Branching Brownian Motions." Ann. Appl. Probab. 3 (1) 173 - 185, February, 1993. https://doi.org/10.1214/aoap/1177005513

Information

Published: February, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0771.60058
MathSciNet: MR1202521
Digital Object Identifier: 10.1214/aoap/1177005513

Subjects:
Primary: 60J65
Secondary: 60J80

Keywords: Branching Brownian motion , first passage , Galton-Watson process , Levy process , Super-Brownian motion , weak convergence

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.3 • No. 1 • February, 1993
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