On the Eve property for CSBP

Thomas Duquesne (Université Pierre et Marie Curie)
Cyril Labbé (Université Pierre et Marie Curie)

Abstract


We consider the population model associated to continuous state branching processes and we are interested in the so-called Eve property that asserts the existence of an ancestor with an overwhelming progeny at large times, and more generally, in the possible behaviours of the frequencies among the population at large times. In this paper, we classify all the possible behaviours according to the branching mechanism of the continuous state branching process.

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Pages: 1-31

Publication Date: January 12, 2014

DOI: 10.1214/EJP.v19-2831

References

  • Abraham, Romain; Delmas, Jean-François. Williams' decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 (2009), no. 4, 1124--1143. MR2508567
  • Bertoin, Jean. The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations. Ann. Probab. 37 (2009), no. 4, 1502--1523. MR2546753
  • Bertoin, Jean; Fontbona, Joaquin; Martínez, Servet. On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 (2008), no. 3, 714--726. MR2455180
  • Bertoin, Jean; Le Gall, Jean-François. The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000), no. 2, 249--266. MR1771663
  • Bertoin, Jean; Le Gall, Jean-François. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003), no. 2, 261--288. MR1990057
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
  • Bingham, N. H. Continuous branching processes and spectral positivity. Stochastic Processes Appl. 4 (1976), no. 3, 217--242. MR0410961
  • Bolthausen, E.; Sznitman, A.-S. On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998), no. 2, 247--276. MR1652734
  • Caballero, Ma. Emilia; Lambert, Amaury; Uribe Bravo, Gerónimo. Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6 (2009), 62--89. MR2592395
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166--205. MR1681126
  • Thomas Duquesne and Jean-Francois Le Gall, Random trees, Lévy processes and spatial branching processes, Astérisque (2002), no. 281, vi+147. 1954248
  • Dynkin, E. B.; Kuznetsov, S. E. $\Bbb N$-measures for branching exit Markov systems and their applications to differential equations. Probab. Theory Related Fields 130 (2004), no. 1, 135--150. MR2092876
  • Etheridge, Alison M. An introduction to superprocesses. University Lecture Series, 20. American Mathematical Society, Providence, RI, 2000. xii+187 pp. ISBN: 0-8218-2706-5 MR1779100
  • Grey, D. R. Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probability 11 (1974), 669--677. MR0408016
  • Grimvall, Anders. On the convergence of sequences of branching processes. Ann. Probability 2 (1974), 1027--1045. MR0362529
  • Helland, Inge S. Continuity of a class of random time transformations. Stochastic Processes Appl. 7 (1978), no. 1, 79--99. MR0488203
  • Heyde, C. C. Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist. 41 1970 739--742. MR0254929
  • Jiřina, Miloslav. Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (83) 1958 292--313. MR0101554
  • S. E. Kuznetsov, Construction of Markov processes with random birth and death times, Theor. Probability Appl. 18 (1974), no. 3, 571--575. MR0343376
  • Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7 MR2250061
  • Labbé, Cyril. From flows of Lambda Fleming-Viot processes to lookdown processes via flows of partitions, arXiv:1107.3419 (2011).
  • Labbé, Cyril. Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property, to appear in Annales de l'Institut Henri Poincare (2014).
  • Lamperti, John. Continuous state branching processes. Bull. Amer. Math. Soc. 73 1967 382--386. MR0208685
  • Lamperti, John. The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 1967 271--288. MR0217893
  • Lamperti, John. Limiting distributions for branching processes. 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2 pp. 225--241 Univ. California Press, Berkeley, Calif. MR0219148
  • Le Gall, Jean-François. Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1999. x+163 pp. ISBN: 3-7643-6126-3 MR1714707
  • Li, Zenghu. Skew convolution semigroups and related immigration processes. (Russian) Teor. Veroyatnost. i Primenen. 46 (2001), no. 2, 247--274; translation in Theory Probab. Appl. 46 (2003), no. 2, 274--296 MR1968685
  • Li, Zenghu. Measure-valued branching Markov processes. Probability and its Applications (New York). Springer, Heidelberg, 2011. xii+350 pp. ISBN: 978-3-642-15003-6 MR2760602
  • Li, Zenghu. Continuous-state branching processes, Lectures Notes, arXiv:1202.3223, Beijing Normal University, 2012.
  • Seneta, E. On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39 1968 2098--2102. MR0234530
  • Seneta, E. Functional equations and the Galton-Watson process. Advances in Appl. Probability 1 1969 1--42. MR0248917
  • Silverstein, M. L. A new approach to local times. J. Math. Mech. 17 1967/1968 1023--1054. MR0226734
  • Tribe, Roger. The behavior of superprocesses near extinction. Ann. Probab. 20 (1992), no. 1, 286--311. MR1143421
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