### Abstract

This article concerns branching Brownian motion (BBM) with dyadic branching at rate beta vertical bar y vertical bar(p) for a particle with spatial position y is an element of R, where beta > 0. It is known that for p > 2 the number of particles blows up almost surely in finite time, while for p = 2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, R-t, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of R-t as t -> infinity. In the case of constant breeding at rate beta the linear asymptotic for R-t is long established. Here, we find asymptotic results for R-t in the case p is an element of (0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p up arrow 2, and a non-trivial limit for In R-t when p = 2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.

Original language | English |
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Pages (from-to) | 793-801 |

Number of pages | 9 |

Journal | Annales de l'Institut Henri Poincaré: Probabilités et Statistiques |

Volume | 45 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 2009 |

### Keywords

- spine constructions
- additive martingales
- Branching Brownian motion

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## Cite this

Harris, J. W., & Harris, S. C. (2009). Branching Brownian motion with an inhomogeneous breeding potential.

*Annales de l'Institut Henri Poincaré: Probabilités et Statistiques*,*45*(3), 793-801. https://doi.org/10.1214/08-aihp300