The Annals of Applied Probability

Genealogical processes for Fleming-Viot models with selection and recombination

Peter Donnelly and Thomas G. Kurtz

Full-text: Open access


Infinite population genetic models with general type space incorporating mutation, selection and recombination are considered. The Fleming-Viot measure-valued diffusion is represented in terms of a countably infinite-dimensional process. The complete genealogy of the population at each time can be recovered from the model. Results are given concerning the existence of stationary distributions and ergodicity and absolute continuity of the stationary distribution for a model with selection with respect to the stationary distribution for the corresponding neutral model.

Article information

Ann. Appl. Probab., Volume 9, Number 4 (1999), 1091-1148.

First available in Project Euclid: 21 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Genetic models recombination selection Fleming-Viot process particle representation measure-valued diffusion exchangeability genealogical processes coalescent


Donnelly, Peter; Kurtz, Thomas G. Genealogical processes for Fleming-Viot models with selection and recombination. Ann. Appl. Probab. 9 (1999), no. 4, 1091--1148. doi:10.1214/aoap/1029962866.

Export citation


  • Blackwell, D. and Dubins, L. E. (1983). An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 691-692.
  • Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming-Viot measurevalued diffusion. Ann. Probab. 24 698-742.
  • Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166-205.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Ethier, S. N. and Kurtz, T. G. (1993). Measure-valued diffusion processes in population genetics. SIAM J. Control Optim. 31 345-386.
  • Ethier, S. N. and Kurtz, T. G. (1998). Coupling and ergodic theorems for Fleming-Viot processes. Ann. Probab. 26 533-561.
  • Griffiths, R. C. and Majoram, P. (1997). An ancestral recombination graph. In Progress in Population Genetics and Human Evolution (P. Donnelly and S. Tavar´e, eds.) 87. Springer, New York.
  • Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235-248.
  • Krone, S. M. and Neuhauser, C. (1997). Ancestral processes with selection. Theoret. Population Biol. 51 210-237. Kurtz, T. G. (1998a). Martingale problems for conditional distributions of Markov processes. Electronic Journal of Probability 3. To appear. Kurtz, T. G. (1998b). Equivalence of martingale problems and stochastic differential equations. Unpublished manuscript.
  • Kurtz, T. G. and Stockbridge, R. H. (1998). Existence of Markov controls and characterization of optimal Markov controls. SIAM J. Control Optim. 36 609-653.
  • Mey n, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Neuhauser, C. (1998). The ancestral graph and gene genealogy under frequency-dependent selection. Preprint.
  • Neuhauser, C. and Krone, S. M. (1997). The genealogy of samples in models with selection. Genetics 154 519-534.
  • Tavar´e, S. (1984). Line-of-descent and genealogical processes, and their applications in genetics models. Theoret. Population Biol. 26 119-164.