On explosions in heavy-tailed branching random walks

Details On explosions in heavy-tailed branching random walks On explosions in heavy-tailed branching random walks

2011-02-04

arxiv.org/abs/1102.0950v1

Mathematics

Probability

35 pages

Scientific paper

Consider a branching random walk on $\mathbb R$, with offspring distribution $Z$ and non-negative displacement distribution $W$. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution $Z$ is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs $(Z, W)$ for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur. We also investigate, under additional smoothness assumptions, the behavior of $M_n$, the position of the particle in generation $n$ closest to the origin, when explosion does not occur (and hence $\lim_{n \rightarrow \infty} M_n = \infty$).

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