Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration

Clément Foucart (Université Pierre et Marie Curie Paris 6)
Olivier Hénard (École des Ponts, Université Paris-Est)


Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the $\alpha$-stable continuous state branching process with the $Beta(2-\alpha,\alpha)$-generalized Fleming-Viot process. In a recent work, a new class of probability-measure valued processes, called $M$-generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called $M$ coalescents. The purpose of this article is to investigate the links between this new class of processes and the continuous-state branching processes with immigration. In the specific case of the $\alpha$-stable branching process conditioned to be never extinct, we get that its genealogy is given, up to a random time change, by a $Beta(2-\alpha, \alpha-1)$-coalescent.

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Pages: 1-21

Publication Date: February 12, 2013

DOI: 10.1214/EJP.v18-2024


  • Berestycki, Julien; Berestycki, Nathanaël. Kingman's coalescent and Brownian motion. ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 239--259. MR2534485
  • Berestycki, Nathanaël. Recent progress in coalescent theory. Ensaios Matemáticos [Mathematical Surveys], 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. 193 pp. ISBN: 978-85-85818-40-1 MR2574323
  • Bertoin, Jean. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2 MR2253162
  • 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
  • Bertoin, Jean; Le Gall, Jean-Francois. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006), no. 1-4, 147--181 (electronic). MR2247827
  • Birkner, Matthias; Blath, Jochen; Capaldo, Marcella; Etheridge, Alison; Möhle, Martin; Schweinsberg, Jason; Wakolbinger, Anton. Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 (2005), no. 9, 303--325 (electronic). MR2120246
  • Dawson, Donald A. Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI—1991, 1--260, Lecture Notes in Math., 1541, Springer, Berlin, 1993. MR1242575
  • Duquesne, Thomas. Continuum random trees and branching processes with immigration. Stochastic Process. Appl. 119 (2009), no. 1, 99--129. MR2485021
  • 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
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085
  • Fitzsimmons, P. J.; Fristedt, Bert; Shepp, L. A. The set of real numbers left uncovered by random covering intervals. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 2, 175--189. MR0799145
  • Foucart, Clément. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration. Adv. in Appl. Probab. 43 (2011), no. 2, 348--374. MR2848380
  • C. Foucart. Generalized Fleming-Viot processes with immigration via stochastic flows of partitions. Hal preprint-to appear, 2012.
  • K. Handa. Stationary distributions for a class of generalized Fleming-Viot processes. ArXiv e-prints, May 2012.
  • Kawazu, Kiyoshi; Watanabe, Shinzo. Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16 1971 34--51. MR0290475
  • Kyprianou, A. E.; Pardo, J. C. Continuous-state branching processes and self-similarity. J. Appl. Probab. 45 (2008), no. 4, 1140--1160. MR2484167
  • Lambert, Amaury. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 (2002), no. 1, 42--70. MR1883717
  • Lambert, Amaury. Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007), no. 14, 420--446. MR2299923
  • 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
  • Perkins, Edwin A. Conditional Dawson-Watanabe processes and Fleming-Viot processes. Seminar on Stochastic Processes, 1991 (Los Angeles, CA, 1991), 143--156, Progr. Probab., 29, Birkhäuser Boston, Boston, MA, 1992. MR1172149
  • Sharpe, Michael. General theory of Markov processes. Pure and Applied Mathematics, 133. Academic Press, Inc., Boston, MA, 1988. xii+419 pp. ISBN: 0-12-639060-6 MR0958914
  • Shiga, Tokuzo. A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30 (1990), no. 2, 245--279. MR1068791
  • Volkonskiĭ, V. A. Random substitution of time in strong Markov processes. (Russian) Teor. Veroyatnost. i Primenen 3 1958 332--350. MR0100919

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