On a First Passage Problem for Branching Brownian Motions

Consider a (space-time) realization . of a critical or subcritical one-dimensional branching Brownian motion. Let Zx(.) for x.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={Zx;x.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.