Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property

Cyril Labbé

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Abstract

We encode the genealogy of a continuous-state branching process associated with a branching mechanism $\varPsi$ – or $\varPsi\mbox{-CSBP}$ in short – using a stochastic flow of partitions. This encoding holds for all branching mechanisms and appears as a very tractable object to deal with asymptotic behaviours and convergences. In particular we study the so-called Eve property – the existence of an ancestor from which the entire population descends asymptotically – and give a necessary and sufficient condition on the $\varPsi\mbox{-CSBP}$ for this property to hold. Finally, we show that the flow of partitions unifies the lookdown representation and the flow of subordinators when the Eve property holds.

Résumé

Nous construisons la généalogie d’un processus de branchement à espace d’états et temps continus associé à un mécanisme de branchement $\varPsi$ – ou $\varPsi\mbox{-CSBP}$ – à l’aide d’un flot stochastique de partitions. Cette construction est valable quel que soit le mécanisme de branchement et permet de définir un objet remarquablement efficace pour étudier les comportements asymptotiques et les convergences. En particulier, nous étudions la propriété d’Eve – l’existence d’un ancêtre dont descend asymptotiquement toute la population – et donnons une condition nécessaire et suffisante sur le $\varPsi\mbox{-CSBP}$ pour que cette propriété soit vérifiée. Finalement, nous montrons que le flot de partitions unifie la représentation lookdown et le flot de subordinateurs lorsque la propriété d’Eve est vérifiée.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 3 (2014), 732-769.

Dates
First available in Project Euclid: 20 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1403276997

Digital Object Identifier
doi:10.1214/13-AIHP542

Mathematical Reviews number (MathSciNet)
MR3224288

Zentralblatt MATH identifier
1308.60099

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G09: Exchangeability 60J25: Continuous-time Markov processes on general state spaces

Keywords
Continuous-state branching process Measure-valued process Genealogy Partition Stochastic flow Lookdown process Subordinator Eve

Citation

Labbé, Cyril. Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 3, 732--769. doi:10.1214/13-AIHP542. https://projecteuclid.org/euclid.aihp/1403276997


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References

  • [1] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1991) 1–28.
  • [2] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006.
  • [3] J. Bertoin, J. Fontbona and S. Martínez. On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 (2008) 714–726.
  • [4] J. Bertoin and J.-F. Le Gall. The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000) 249–266.
  • [5] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261–288.
  • [6] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006) 147–181.
  • [7] M. Birkner, J. Blath, M. Capaldo, A. M. Etheridge, M. Möhle, J. Schweinsberg and A. Wakolbinger. Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 (2005) 303–325.
  • [8] M.-E. Caballero, A. Lambert and G. Uribe Bravo. Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6 (2009) 62–89.
  • [9] D. A. Dawson. Measure-Valued Markov Processes. Lecture Notes in Math. 1541. Springer, Berlin, 1993.
  • [10] D. A. Dawson and E. A. Perkins. Historical processes. Mem. Amer. Math. Soc. 93 (1991) iv+179.
  • [11] P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166–205.
  • [12] T. Duquesne and C. Labbé. On the Eve property for CSBP. Preprint, 2013. Available at arXiv:1305.6502.
  • [13] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147.
  • [14] T. Duquesne and M. Winkel. Growth of Lévy trees. Probab. Theory Related Fields 139 (2007) 313–371.
  • [15] N. El Karoui and S. Roelly. Propriétés de martingales, explosion et représentation de Lévy-Khintchine d’une classe de processus de branchement à valeurs mesures. Stochastic Process. Appl. 38 (1991) 239–266.
  • [16] A. Greven, P. Pfaffelhuber and A. Winter. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013) 789–838.
  • [17] A. Greven, L. Popovic and A. Winter. Genealogy of catalytic branching models. Ann. Appl. Probab. 19 (2009) 1232–1272.
  • [18] D. R. Grey. Asymptotic behaviour of continuous time, continuous state-space branching processes. J. App. Probab. 11 (1974) 669–677.
  • [19] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer-Verlag, Berlin, 2003.
  • [20] M. Jiřina. Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (1958) 292–313.
  • [21] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer-Verlag, New York, 2002.
  • [22] C. Labbé. From flows of Lambda Fleming–Viot processes to lookdown processes via flows of partitions. Preprint, 2011. Available at arXiv:1107.3419.
  • [23] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213–252.
  • [24] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870–1902.
  • [25] M. Silverstein. A new approach to local times. J. Math. Mech. 17 (1968) 1023–1054.
  • [26] R. Tribe. The behavior of superprocesses near extinction. Ann. Probab. 20 (1992) 286–311.
  • [27] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 (1968) 141–167.