On a First Passage Problem for Branching Brownian Motions

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.