Branching Brownian motion with an inhomogeneous breeding potential


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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    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.


    Item Type Articles
    CreatorsHarris, J. W.and Harris, S. C.
    Uncontrolled Keywordsspine constructions,additive martingales,branching brownian motion
    DepartmentsFaculty of Science > Mathematical Sciences
    ID Code21547


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