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International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 569081, 20 pages
http://dx.doi.org/10.1155/2012/569081
Research Article

Birth and Death Processes with Neutral Mutations

1IECN, Université de Lorraine, Campus Scientifique, B.P. 70239, 54506 Vandœuvre-lès-Nancy Cedex, France
2Inria, 54600 Villers-lès-Nancy, France
3Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 CNRS and UPMC Université Paris 06, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France
4CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Received 27 September 2012; Accepted 28 November 2012

Academic Editor: Fima Klebaner

Copyright © 2012 Nicolas Champagnat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at the birth of particles or at a constant rate during their lives. In both models, we study the allelic partition of the population at time . We give closed-form formulae for the expected frequency spectrum at and prove a pathwise convergence to an explicit limit, as , of the relative numbers of types younger than some given age and carried by a given number of particles (small families). We also provide the convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models.