×
Labbé, Cyril

Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property. (English. French summary) Zbl 1308.60099

Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 3, 732-769 (2014).
The branching mechanism \(\Psi\) of a continuous-state branching process (CSBP) \((Z_t)_{t\geq 0}\) is said to satisfy the Eve property if there exists a random variable \(e\) with values in \([0,1]\) such that \(\lim_{t\uparrow T}\,m_t/m_t([0,1])= \delta_e\) a.s.in the sense of weak convergence of probability measures. Here, \(T:=\inf\{t\geq 0: Z_t\notin (0,\infty)\}\) and \((m_t)_{t\in[0,T)}\) is a measure-valued process starting at the Lebesgue measure on \([0,1]\) and such that \((m_t([0,x]))_{t\geq 0}\) and \((m_t((x,1]))_{t\geq 0}\) are two independent CSBPs with the branching mechanism \(\Psi\). The author shows that \(e\) necessarily has the uniform \([0,1]\) law and investigates the Eve property in connection with the genealogy of the CSBP. To this end, a new object, called a stochastic flow of partitions associated with a branching mechanism, is introduced which unifies the flow of subordinators of J. Bertoin and J.-F.Le Gall [Probab. Theory Relat. Fields 117, No. 2, 249–266 (2000; Zbl 0963.60086)] and the lookdown representation of P. Donnelly and T. G. Kurtz [Ann. Probab. 27, No. 1, 166–205 (1999; Zbl 0956.60081)] when the Eve property holds. For the latter to be true, a necessary and sufficient condition is given.
Reviewer: Aleksander Iksanov (Kiev)

Cited in 4 Documents

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

continuous-state branching process; measure-valued process; Eve property; genealogy; partition; stochastic flow; lookdown process; subordinator

Citations:

Zbl 0963.60086; Zbl 0956.60081
Cite Review PDF
Full Text: DOI arXiv Euclid

References:

[1] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1991) 1-28. · Zbl 0722.60013 · doi:10.1214/aop/1176990534
[2] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102 . Cambridge Univ. Press, Cambridge, 2006. · Zbl 1107.60002 · doi:10.1017/CBO9780511617768
[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. · Zbl 1154.60066 · doi:10.1239/jap/1222441825
[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. · Zbl 0963.60086 · doi:10.1007/s004400000053
[5] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261-288. · Zbl 1023.92018 · doi:10.1007/s00440-003-0264-4
[6] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006) 147-181. · 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. · Zbl 1066.60072 · doi:10.1214/EJP.v10-241
[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. · Zbl 1194.60053 · doi:10.1214/09-PS154
[9] D. A. Dawson. Measure-Valued Markov Processes. Lecture Notes in Math. 1541 . Springer, Berlin, 1993. · Zbl 0799.60080
[10] D. A. Dawson and E. A. Perkins. Historical processes. Mem. Amer. Math. Soc. 93 (1991) iv+179. · Zbl 0754.60062 · doi:10.1090/memo/0454
[11] P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166-205. · Zbl 0956.60081 · doi:10.1214/aop/1022677258
[12] T. Duquesne and C. Labbé. On the Eve property for CSBP. Preprint, 2013. Available at . 1305.6502 · Zbl 1287.60100 · doi:10.1214/EJP.v19-2831
[13] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. · Zbl 1037.60074
[14] T. Duquesne and M. Winkel. Growth of Lévy trees. Probab. Theory Related Fields 139 (2007) 313-371. · Zbl 1126.60068 · doi:10.1007/s00440-007-0064-3
[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. · Zbl 0743.60081 · doi:10.1016/0304-4149(91)90093-R
[16] A. Greven, P. Pfaffelhuber and A. Winter. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013) 789-838. · Zbl 1379.60099 · doi:10.1007/s00440-012-0413-8
[17] A. Greven, L. Popovic and A. Winter. Genealogy of catalytic branching models. Ann. Appl. Probab. 19 (2009) 1232-1272. · Zbl 1178.60057 · doi:10.1214/08-AAP574
[18] D. R. Grey. Asymptotic behaviour of continuous time, continuous state-space branching processes. J. App. Probab. 11 (1974) 669-677. · Zbl 0301.60060 · doi:10.2307/3212550
[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. · Zbl 1018.60002
[20] M. Jiřina. Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (1958) 292-313. · Zbl 0168.38602
[21] O. Kallenberg. Foundations of Modern Probability , 2nd edition. Probability and Its Applications (New York) . Springer-Verlag, New York, 2002. · Zbl 0996.60001
[22] C. Labbé. From flows of Lambda Fleming-Viot processes to lookdown processes via flows of partitions. Preprint, 2011. Available at . 1107.3419 · Zbl 1334.60152
[23] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. · Zbl 0948.60071 · doi:10.1214/aop/1022855417
[24] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. · Zbl 0963.60079 · doi:10.1214/aop/1022677552
[25] M. Silverstein. A new approach to local times. J. Math. Mech. 17 (1968) 1023-1054. · Zbl 0184.41101
[26] R. Tribe. The behavior of superprocesses near extinction. Ann. Probab. 20 (1992) 286-311. · Zbl 0749.60046 · doi:10.1214/aop/1176989927
[27] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 (1968) 141-167. · Zbl 0159.46201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.