Branching Brownian motion with an inhomogeneous breeding potential

JW Harris, SC Harris

Research output: Contribution to journalArticle (Academic Journal)peer-review

10 Citations (Scopus)

Abstract

This article concerns branching Brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β>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, Rt, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of Rt as t→∞. In the case of constant breeding at rate β the linear asymptotic for Rt is long established. Here, we find asymptotic results for Rt in the case p∈(0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p↑2, and a non-trivial limit for lnRt when p=2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.
Translated title of the contributionBranching Brownian motion with an inhomogeneous breeding potential
Original languageEnglish
Pages (from-to)793 - 801
Number of pages9
JournalAnnales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Volume45, number 3
DOIs
Publication statusPublished - Aug 2009

Bibliographical note

Publisher: IMS

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