Changing the branching mechanism of a continuous state branching process using immigration

Romain Abraham; Jean-François Delmas

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 226-238
  • ISSN: 0246-0203

Abstract

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We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.

How to cite

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Abraham, Romain, and Delmas, Jean-François. "Changing the branching mechanism of a continuous state branching process using immigration." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 226-238. <http://eudml.org/doc/78017>.

@article{Abraham2009,
abstract = {We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.},
author = {Abraham, Romain, Delmas, Jean-François},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {continuous state branching processes; immigration process; multitype populations},
language = {eng},
number = {1},
pages = {226-238},
publisher = {Gauthier-Villars},
title = {Changing the branching mechanism of a continuous state branching process using immigration},
url = {http://eudml.org/doc/78017},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Abraham, Romain
AU - Delmas, Jean-François
TI - Changing the branching mechanism of a continuous state branching process using immigration
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 226
EP - 238
AB - We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.
LA - eng
KW - continuous state branching processes; immigration process; multitype populations
UR - http://eudml.org/doc/78017
ER -

References

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  11. [11] Z.-H. Li. Branching processes with immigration and related topics. Front. Math. China 1 (2006) 73–97. Zbl1222.60064MR2225400
  12. [12] J. Pitman and M. Yor. A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 (1982) 425–457. Zbl0484.60062MR656509
  13. [13] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Heidelberg, 1991. Zbl0731.60002MR1083357
  14. [14] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, 1999. Zbl0973.60001MR1739520
  15. [15] L. Serlet. Creation or deletion of a drift on a Brownian trajectory. In Séminaire de Probabilités, XLI 215–232. Zbl1157.60077MR2483734
  16. [16] J. Warren. Branching processes, the Ray–Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités, XXXI 1–15. Lecture Notes in Math. 1655. Springer, Berlin, 1997. Zbl0884.60081MR1478711

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