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Branching brownian motion with an inhomogeneous breeding potential

J. W. Harris;  S. C. Harris — 2009

Annales de l'I.H.P. Probabilités et statistiques

This article concerns branching brownian motion (BBM) with dyadic branching at rate || for a particle with spatial position ∈ℝ, where >0. It is known that for >2 the number of particles blows up almost surely in finite time, while for =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, , to be the supremum of the spatial positions of the...

Strong law of large numbers for fragmentation processes

S. C. Harris;  R. Knobloch;  A. E. Kyprianou — 2010

Annales de l'I.H.P. Probabilités et statistiques

In the spirit of a classical result for Crump–Mode–Jagers processes, we prove a strong law of large numbers for fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than for 1≥>0.

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