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October, 1992 Localization and Selection in a Mean Field Branching Random Walk in a Random Environment
Klaus Fleischmann, Andreas Greven
Ann. Probab. 20(4): 2141-2163 (October, 1992). DOI: 10.1214/aop/1176989543

Abstract

We consider a continuous time branching random walk on the finite set $\{1,2,\ldots, N\}$ with totally symmetric diffusion jumps and some site-dependent i.i.d. random birth rates which are unbounded. We study this process as the time $t$ and the space size $N$ tend to infinity simultaneously. In the classical law of large numbers setup for spatial branching models, the growth of the population obeys an exponential limit law due to the localization of the overwhelming portion of particles in the record point of the medium. This phenomenon is analyzed further: The historical path (in space) of a typical particle picked at time $t$ (selection) is of a rather simple and special nature and becomes in the limit singular (in distribution) to the path of the underlying mean field random walk. In general, the properties of the typical path depend on the relation in which $t$ and $N$ tend to infinity.

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Klaus Fleischmann. Andreas Greven. "Localization and Selection in a Mean Field Branching Random Walk in a Random Environment." Ann. Probab. 20 (4) 2141 - 2163, October, 1992. https://doi.org/10.1214/aop/1176989543

Information

Published: October, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0771.60095
MathSciNet: MR1188056
Digital Object Identifier: 10.1214/aop/1176989543

Subjects:
Primary: 60K35
Secondary: 60J80 , 82A42

Keywords: Branching random walk , Infinite particle system , random medium

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.20 • No. 4 • October, 1992
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