Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, p. 732-769

We encode the genealogy of a continuous-state branching process associated with a branching mechanism 𝛹 - or 𝛹-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 𝛹-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.

Nous construisons la généalogie d’un processus de branchement à espace d’états et temps continus associé à un mécanisme de branchement 𝛹 - ou 𝛹-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 𝛹-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.

DOI : https://doi.org/10.1214/13-AIHP542
Classification:  60J80,  60G09,  60J25
Keywords: continuous-state branching process, measure-valued process, genealogy, partition, stochastic flow, lookdown process, subordinator, EVE
@article{AIHPB_2014__50_3_732_0,
     author = {Labb\'e, Cyril},
     title = {Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     pages = {732-769},
     doi = {10.1214/13-AIHP542},
     zbl = {06340407},
     mrnumber = {3224288},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_3_732_0}
}
Labbé, Cyril. Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 732-769. doi : 10.1214/13-AIHP542. http://www.numdam.org/item/AIHPB_2014__50_3_732_0/

[1] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1991) 1-28. | MR 1085326 | Zbl 0722.60013

[2] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. | MR 2253162 | Zbl 1107.60002

[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. | MR 2455180 | Zbl 1154.60066

[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. | MR 1771663 | Zbl 0963.60086

[5] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261-288. | MR 1990057 | Zbl 1023.92018

[6] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006) 147-181. | MR 2247827 | Zbl 1110.60026

[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. | MR 2120246 | Zbl 1066.60072

[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. | MR 2592395 | Zbl 1194.60053

[9] D. A. Dawson. Measure-Valued Markov Processes. Lecture Notes in Math. 1541. Springer, Berlin, 1993. | MR 1242575 | Zbl 0799.60080

[10] D. A. Dawson and E. A. Perkins. Historical processes. Mem. Amer. Math. Soc. 93 (1991) iv+179. | MR 1079034 | Zbl 0754.60062

[11] P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166-205. | MR 1681126 | Zbl 0956.60081

[12] T. Duquesne and C. Labbé. On the Eve property for CSBP. Preprint, 2013. Available at arXiv:1305.6502. | MR 3164759 | Zbl 1287.60100

[13] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. | MR 1954248 | Zbl 1037.60074

[14] T. Duquesne and M. Winkel. Growth of Lévy trees. Probab. Theory Related Fields 139 (2007) 313-371. | MR 2322700 | Zbl 1126.60068

[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. | MR 1119983 | Zbl 0743.60081

[16] A. Greven, P. Pfaffelhuber and A. Winter. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013) 789-838. | MR 3034793 | Zbl pre06162381

[17] A. Greven, L. Popovic and A. Winter. Genealogy of catalytic branching models. Ann. Appl. Probab. 19 (2009) 1232-1272. | MR 2537365 | Zbl 1178.60057

[18] D. R. Grey. Asymptotic behaviour of continuous time, continuous state-space branching processes. J. App. Probab. 11 (1974) 669-677. | MR 408016 | Zbl 0301.60060

[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. | MR 1943877 | Zbl 1018.60002

[20] M. Jiřina. Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (1958) 292-313. | MR 101554 | Zbl 0168.38602

[21] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer-Verlag, New York, 2002. | MR 1876169 | Zbl 0892.60001

[22] C. Labbé. From flows of Lambda Fleming-Viot processes to lookdown processes via flows of partitions. Preprint, 2011. Available at arXiv:1107.3419. | Zbl pre06309119

[23] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. | MR 1617047 | Zbl 0948.60071

[24] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. | MR 1742892 | Zbl 0963.60079

[25] M. Silverstein. A new approach to local times. J. Math. Mech. 17 (1968) 1023-1054. | MR 226734 | Zbl 0184.41101

[26] R. Tribe. The behavior of superprocesses near extinction. Ann. Probab. 20 (1992) 286-311. | MR 1143421 | Zbl 0749.60046

[27] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 (1968) 141-167. | MR 237008 | Zbl 0159.46201