Changing the branching mechanism of a continuous state branching process using immigration
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 226-238.

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.

Nous considérons une population initiale dont la taille évolue selon un processus de branchement continu. Nous ajoutons ensuite à ce processus une population migrante (qui évolue selon le même mécanisme de branchement que la population initiale), avec un taux d'immigration propotionnel à la taille de la population totale. Nous montrons que ce processus de branchement continu avec immgration proportionnelle à sa taille est encore un processus de branchement continu. En voyant cette immigration comme l'apparition d'un nouveau type, cette construction est un moyen naturel de modéliser des mutations, neutres vis à vis de l'évolution. Elle peut être également vue comme la construction duale de l'élagage aux noeuds de l'arbre généalogique associé à la population totale, introduit par les auteurs dans un article précédent. Lorsque le mécanisme de branchement est quadratique et critique ou sous-critique, il est possible de calculer explicitement certaines quantités intéressantes. Par example, nous calculons la transformée de Laplace de la taille de la population initiale conditionnellement à la non-extinction de la population totale. Nous en déduisons également la probabilité d'extinction simultanée de la population initiale et de la population totale.

DOI: 10.1214/07-AIHP165
Classification: 60G55,  60J25,  60J80,  60J85
Keywords: continuous state branching processes, immigration process, multitype populations
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Abraham, Romain; Delmas, Jean-François. Changing the branching mechanism of a continuous state branching process using immigration. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 226-238. doi : 10.1214/07-AIHP165. http://www.numdam.org/articles/10.1214/07-AIHP165/

[1] R. Abraham and J.-F. Delmas. Williams' decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic. Process. Appl. DOI: 10.1016/j.spa.2008.06.001. | Zbl

[2] R. Abraham and J.-F. Delmas. Fragmentation associated with Lévy processes using snake. Probab. Theory Related Fields 141 (2008) 113-154. | MR | Zbl

[3] R. Abraham, J.-F. Delmas and G. Voisin. Pruning a Lévy continuum random tree. Preprint. Available at arXiv: 0804.1027.

[4] R. Abraham and L. Serlet. Poisson snake and fragmentation. Electron. J. Probab. 7 (2002). | MR | Zbl

[5] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002). | MR | Zbl

[6] E. Dynkin. Branching particle systems and superprocesses. Ann. Probab. 19 (1991) 1157-1194. | MR | Zbl

[7] K. Kawazu and S. Watanabe. Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16 (1971) 34-51. | MR | Zbl

[8] A. Lambert. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 (2002) 42-70. | MR | Zbl

[9] J. Lamperti. Continuous state branching process. Bull. Amer. Math. Soc. 73 (1967) 382-386. | MR | Zbl

[10] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. | MR | Zbl

[11] Z.-H. Li. Branching processes with immigration and related topics. Front. Math. China 1 (2006) 73-97. | MR

[12] J. Pitman and M. Yor. A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 (1982) 425-457. | MR | Zbl

[13] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Heidelberg, 1991. | MR | Zbl

[14] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, 1999. | MR | Zbl

[15] L. Serlet. Creation or deletion of a drift on a Brownian trajectory. In Séminaire de Probabilités, XLI 215-232. | MR | Zbl

[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. | Numdam | MR | Zbl

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