%0 Journal Article %1 kaj1993first %A Kaj, Ingemar %A Salminen, Paavo %D 1993 %I The Institute of Mathematical Statistics %J Ann. Appl. Probab. %K branching_Brownian_motion branching_processes continuous-state_branching_processes decomposition space-time_processes subcritical_branching %N 1 %P 173--185 %R 10.1214/aoap/1177005513 %T On a First Passage Problem for Branching Brownian Motions %U http://dx.doi.org/10.1214/aoap/1177005513 %V 3 %X Consider a (space-time) realization $ømega$ of a critical or subcritical one-dimensional branching Brownian motion. Let $Z_x(ømega)$ 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 = \Z_x; x0\$. 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 simliar problem in the supercritical case. Moreover, in the critical case we characterize the continuous state branching process obtained as weak limits of the processes $Z$ under rescaling.

@article{kaj1993first, abstract = {Consider a (space-time) realization $\omega$ of a critical or subcritical one-dimensional branching Brownian motion. Let $Z_x(\omega)$ for $x \ge 0$ 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 $Z = \{Z_x; x\ge 0\}$. 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 simliar problem in the supercritical case. Moreover, in the critical case we characterize the continuous state branching process obtained as weak limits of the processes $Z$ under rescaling.}, added-at = {2015-08-30T08:13:01.000+0200}, author = {Kaj, Ingemar and Salminen, Paavo}, biburl = {https://www.bibsonomy.org/bibtex/2ec22fed59812809815b1bfd52dbe3d30/peter.ralph}, doi = {10.1214/aoap/1177005513}, fjournal = {The Annals of Applied Probability}, interhash = {3ac87b3a949035bb71c5ffbabb1c5d5e}, intrahash = {ec22fed59812809815b1bfd52dbe3d30}, journal = {Ann. Appl. Probab.}, keywords = {branching_Brownian_motion branching_processes continuous-state_branching_processes decomposition space-time_processes subcritical_branching}, month = {02}, number = 1, pages = {173--185}, publisher = {The Institute of Mathematical Statistics}, timestamp = {2015-08-30T08:13:01.000+0200}, title = {On a First Passage Problem for Branching Brownian Motions}, url = {http://dx.doi.org/10.1214/aoap/1177005513}, volume = 3, year = 1993 }