The Annals of Probability

Stationary distributions for a class of generalized Fleming–Viot processes

Kenji Handa

Full-text: Open access


We identify stationary distributions of generalized Fleming–Viot processes with jump mechanisms specified by certain beta laws together with a parameter measure. Each of these distributions is obtained from normalized stable random measures after a suitable biased transformation followed by mixing by the law of a Dirichlet random measure with the same parameter measure. The calculations are based primarily on the well-known relationship to measure-valued branching processes with immigration.

Article information

Ann. Probab., Volume 42, Number 3 (2014), 1257-1284.

First available in Project Euclid: 26 March 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75: Jump processes
Secondary: 60G57: Random measures

Generalized Fleming–Viot process measure-valued branching process stable random measure Dirichlet random measure


Handa, Kenji. Stationary distributions for a class of generalized Fleming–Viot processes. Ann. Probab. 42 (2014), no. 3, 1257--1284. doi:10.1214/12-AOP829.

Export citation


  • [1] Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261–288.
  • [2] Birkner, M. and Blath, J. (2009). Measure-valued diffusions, general coalescents and population genetic inference. In Trends in Stochastic Analysis 329–363. Cambridge Univ. Press, Cambridge.
  • [3] Birkner, M., Blath, J., Capaldo, M., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 303–325 (electronic).
  • [4] Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429–442.
  • [5] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • [6] Etheridge, A. and March, P. (1991). A note on superprocesses. Probab. Theory Related Fields 89 141–147.
  • [7] Ethier, S. N. (1990). The infinitely-many-neutral-alleles diffusion model with ages. Adv. in Appl. Probab. 22 1–24.
  • [8] Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming–Viot process. Ann. Probab. 21 1571–1590.
  • [9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [10] Ethier, S. N. and Kurtz, T. G. (1993). Fleming–Viot processes in population genetics. SIAM J. Control Optim. 31 345–386.
  • [11] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • [12] Foucart, C. (2011). Distinguished exchangeable coalescents and generalized Fleming–Viot processes with immigration. Adv. in Appl. Probab. 43 348–374.
  • [13] Foucart, C. and Hénard, O. (2013). Stable continuous-state branching processes with immigration and Beta–Fleming–Viot processes with immigration. Electron. J. Probab. 18 1–21.
  • [14] Hiraba, S. (2000). Jump-type Fleming–Viot processes. Adv. in Appl. Probab. 32 140–158.
  • [15] Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 36–54.
  • [16] Lebedev, N. N. (1972). Special Functions and Their Applications, Revised ed. Dover, New York.
  • [17] Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.
  • [18] Perkins, E. A. (1992). Conditional Dawson–Watanabe processes and Fleming–Viot processes. In Seminar on Stochastic Processes, 1991 (Los Angeles, CA, 1991). Progress in Probability 29 143–156. Birkhäuser, Boston, MA.
  • [19] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge.
  • [20] Shiga, T. (1990). A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30 245–279.
  • [21] Stannat, W. (2003). Spectral properties for a class of continuous state branching processes with immigration. J. Funct. Anal. 201 185–227.
  • [22] Vershik, A. M., Yor, M. and Tsilevich, N. V. (2004). The Markov–Kreĭn identity and the quasi-invariance of the gamma process. J. Math. Sci. 121 2303–2310.
  • [23] Yano, Y. (2006). On the occupation time on the half line of pinned diffusion processes. Publ. Res. Inst. Math. Sci. 42 787–802.