Branching Brownian motion with an inhomogeneous breeding potential

Reference:

Harris, J. W. and Harris, S. C., 2009. Branching Brownian motion with an inhomogeneous breeding potential. Annales De L Institut Henri Poincare-Probabilites Et Statistiques, 45 (3), pp. 793-801.

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Official URL:

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

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.

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