Branching Brownian motion with an inhomogeneous breeding potential

J W Harris, Simon C Harris

Research output: Contribution to journalArticle

8 Citations (Scopus)
78 Downloads (Pure)

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 languageEnglish
Pages (from-to)793-801
Number of pages9
JournalAnnales de l'Institut Henri Poincaré: Probabilités et Statistiques
Volume45
Issue number3
DOIs
Publication statusPublished - Aug 2009

Fingerprint

Branching Brownian Motion
Blow-up
Vertical
Change of Measure
Spine
Supremum
Martingale
Breeding
Brownian motion
Branching
Infinity
Higher Order
Polynomial

Keywords

  • spine constructions
  • additive martingales
  • Branching Brownian motion

Cite this

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

In: Annales de l'Institut Henri Poincaré: Probabilités et Statistiques, Vol. 45, No. 3, 08.2009, p. 793-801.

Research output: Contribution to journalArticle

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