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
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 journal › Article
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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 -