### Abstract

Original language | English |
---|---|

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 |

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### Keywords

- spine constructions
- additive martingales
- Branching Brownian motion

### Cite this

*Annales de l'Institut Henri Poincaré: Probabilités et Statistiques*,

*45*(3), 793-801. https://doi.org/10.1214/08-aihp300

**Branching Brownian motion with an inhomogeneous breeding potential.** / Harris, J W; Harris, Simon C.

Research output: Contribution to journal › Article

*Annales de l'Institut Henri Poincaré: Probabilités et Statistiques*, vol. 45, no. 3, pp. 793-801. https://doi.org/10.1214/08-aihp300

}

TY - JOUR

T1 - Branching Brownian motion with an inhomogeneous breeding potential

AU - Harris, J W

AU - Harris, Simon C

PY - 2009/8

Y1 - 2009/8

N2 - 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.

AB - 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.

KW - spine constructions

KW - additive martingales

KW - Branching Brownian motion

UR - http://www.scopus.com/inward/record.url?scp=77952070408&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1214/08-aihp300

U2 - 10.1214/08-aihp300

DO - 10.1214/08-aihp300

M3 - Article

VL - 45

SP - 793

EP - 801

JO - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

SN - 0246-0203

IS - 3

ER -