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

Cyril Labbé

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 732-769
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

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 50.3 (2014): 732-769. <http://eudml.org/doc/272077>.

@article{Labbé2014,
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.},
author = {Labbé, Cyril},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {continuous-state branching process; measure-valued process; genealogy; partition; stochastic flow; lookdown process; subordinator; EVE; Eve property},
language = {eng},
number = {3},
pages = {732-769},
publisher = {Gauthier-Villars},
title = {Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property},
url = {http://eudml.org/doc/272077},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Labbé, Cyril
TI - Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 732
EP - 769
AB - 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.
LA - eng
KW - continuous-state branching process; measure-valued process; genealogy; partition; stochastic flow; lookdown process; subordinator; EVE; Eve property
UR - http://eudml.org/doc/272077
ER -

References

top
  1. [1] D. Aldous. The continuum random tree. I. Ann. Probab.19 (1991) 1–28. Zbl0722.60013MR1085326
  2. [2] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. Zbl1107.60002MR2253162
  3. [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. Zbl1154.60066MR2455180
  4. [4] J. Bertoin and J.-F. Le Gall. The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields117 (2000) 249–266. Zbl0963.60086MR1771663
  5. [5] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields126 (2003) 261–288. Zbl1023.92018MR1990057
  6. [6] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006) 147–181. Zbl1110.60026MR2247827
  7. [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. Zbl1066.60072MR2120246
  8. [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. Zbl1194.60053MR2592395
  9. [9] D. A. Dawson. Measure-Valued Markov Processes. Lecture Notes in Math. 1541. Springer, Berlin, 1993. Zbl0799.60080MR1242575
  10. [10] D. A. Dawson and E. A. Perkins. Historical processes. Mem. Amer. Math. Soc. 93 (1991) iv+179. Zbl0754.60062MR1079034
  11. [11] P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab.27 (1999) 166–205. Zbl0956.60081MR1681126
  12. [12] T. Duquesne and C. Labbé. On the Eve property for CSBP. Preprint, 2013. Available at arXiv:1305.6502. Zbl1287.60100MR3164759
  13. [13] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. Zbl1037.60074MR1954248
  14. [14] T. Duquesne and M. Winkel. Growth of Lévy trees. Probab. Theory Related Fields139 (2007) 313–371. Zbl1126.60068MR2322700
  15. [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. Zbl0743.60081MR1119983
  16. [16] A. Greven, P. Pfaffelhuber and A. Winter. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields155 (2013) 789–838. Zbl06162381MR3034793
  17. [17] A. Greven, L. Popovic and A. Winter. Genealogy of catalytic branching models. Ann. Appl. Probab.19 (2009) 1232–1272. Zbl1178.60057MR2537365
  18. [18] D. R. Grey. Asymptotic behaviour of continuous time, continuous state-space branching processes. J. App. Probab.11 (1974) 669–677. Zbl0301.60060MR408016
  19. [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. Zbl1018.60002MR1943877
  20. [20] M. Jiřina. Stochastic branching processes with continuous state space. Czechoslovak Math. J.8 (1958) 292–313. Zbl0168.38602MR101554
  21. [21] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer-Verlag, New York, 2002. Zbl0892.60001MR1876169
  22. [22] C. Labbé. From flows of Lambda Fleming–Viot processes to lookdown processes via flows of partitions. Preprint, 2011. Available at arXiv:1107.3419. Zbl1334.60152
  23. [23] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab.26 (1998) 213–252. Zbl0948.60071MR1617047
  24. [24] J. Pitman. Coalescents with multiple collisions. Ann. Probab.27 (1999) 1870–1902. Zbl0963.60079MR1742892
  25. [25] M. Silverstein. A new approach to local times. J. Math. Mech.17 (1968) 1023–1054. Zbl0184.41101MR226734
  26. [26] R. Tribe. The behavior of superprocesses near extinction. Ann. Probab.20 (1992) 286–311. Zbl0749.60046MR1143421
  27. [27] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ.8 (1968) 141–167. Zbl0159.46201MR237008

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.