Electronic Journal of Probability

Hierarchically Interacting Fleming-Viot Processes With Selection and Mutation: Multiple Space Time Scale Analysis and Quasi-Equilibria

Donald Dawson and Andreas Greven

Full-text: Open access


Genetic models incorporating resampling and migration are now fairly well-understood. Problems arise in the analysis, if both selection and mutation are incorporated. This paper addresses some aspects of this problem, in particular the analysis of the long-time behaviour before the equilibrium is reached (quasi-equilibrium, which is the time range of interest in most applications).

Article information

Electron. J. Probab., Volume 4 (1999), paper no. 4, 81 pp.

Accepted: 4 March 1999
First available in Project Euclid: 4 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Interacting Fleming-Viot processes Renormalization analysis Selection Mutation Recombination

This work is licensed under aCreative Commons Attribution 3.0 License.


Dawson, Donald; Greven, Andreas. Hierarchically Interacting Fleming-Viot Processes With Selection and Mutation: Multiple Space Time Scale Analysis and Quasi-Equilibria. Electron. J. Probab. 4 (1999), paper no. 4, 81 pp. doi:10.1214/EJP.v4-41. https://projecteuclid.org/euclid.ejp/1457125513

Export citation


  • J. T. Cox, K. Fleischmann, A. Greven (1996), Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Rel. Fields, 105, 513-528.
  • J. T. Cox, A. Greven, T. Shiga (1995), Finite and infinite systems of interacting diffusions. Probab. Theory Relat. Fields 103, 165-197.
  • D. A. Dawson (1993), Measure-valued Markov Processes. In: École d'Été de Probabilités de Saint Flour XXI, Lecture Notes in Mathematics 1541, 1-261, Springer-Verlag.
  • D. A. Dawson and A. Greven (1993a), Multiple time scale analysis of hierarchically interacting systems. In: A Festschrift to honor G. Kallianpur, 41-50, Springer-Verlag.
  • D. A. Dawson and A. Greven (1993b), Multiple time scale analysis of interacting diffusions. Probab. Theory Rel. Fields 95, 467-508.
  • D. A. Dawson and A. Greven (1993c), Hierarchical models of interacting diffusions: multiple time scale phenomena. Phase transition and pattern of cluster-formation. Probab. Theory Rel. Fields, 96, 435-473.
  • D. A. Dawson and A. Greven (1996), Hierarchically interacting Fleming-Viot processes with selection and mutation: Multiple space time scale analysis and quasi equilibria, Electronic Journal of Probability, Vol. 1, paper no. 14, pages 1-84.
  • D. A. Dawson, A. Greven, J. Vaillancourt (1995), Equilibria and Quasiequilibria for Infinite Collections of Interacting Fleming-Viot processes. Transactions of the American Math. Society, vol 347, No. 7, 2277-2360.
  • D. A. Dawson and P. March (1995), Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related Martingale problems, J. Funct. Anal., Vol. 132, 417-472.
  • S. N. Ethier (1990), The distribution of frequencies of age-ordered alleles in a diffusion model, Adv. Appl. Probab. 22, 519-532.
  • S. N. Ethier and T.G. Kurtz (1992), On the stationary distribution of the neutral one-locus diffusion model in population genetics, Ann. Appl. Probab. 2, 24-35.
  • S. N. Ethier (1992), Equivalence of two descriptions of the ages of alleles, J. Appl. Probab. 29, 185-189.
  • G. M. Edelmann (1987), Neural Darwinism, Basic Books. New York.
  • S. N. Ethier and B. Griffiths (1981), The infinitely-many-sites model as a measure-valued diffusion, Ann. Probab. 15, 515-545.
  • S. N. Ethier and T. G. Kurtz (1981), The infinitely-many-neutral-alleles diffusion model, Adv. Appl. Probab. 13, 429-452.
  • S. N. Ethier and T. G. Kurtz (1986/93), Markov processes, characterization and convergence, Wiley, New York.
  • S. N. Ethier and T. G. Kurtz (1987), The infinitely-many-alleles-model with selection as a measure-valued diffusion, Lecture Notes in Biomath., vol. 70, Springer-Verlag, 72-86.
  • S. N. Ethier and T. G. Kurtz (1994), Convergence to Fleming-Viot processes in the weak atomic topology, Stochastic Process, Appl. 54, 1-27.
  • S. N. Ethier and T.G. Kurtz (1998), Coupling and ergodic theorems for Fleming-Viot processes, Ann. Probab., vol 25,No.2, 533/561.
  • J. Holland (1992), Adaptation in natural and artificial systems, MIT Press.
  • A. Joffe and M. Métivier (1986), Weak convergence of sequences of semi-martingales with application to multitype branching processes. Adv. Appl. Probab. 18, 20-65.
  • H. C. Kang; S. M. Krone; C. Neuhauser (1995): Stepping-Stone models with extinction and recolonization. Ann. Appl. Prob., Vol. 5, No. 4, 1025-1060
  • L. Overbeck, M. Röckner, J. Schmuland (1995): An analytic approach to Fleming-Viot processes, Ann. Prob. 23, 1-36.
  • G. P. Patil and C. Taillie (1977), Diversity as a concept and its implications for random communities, Bull. Internat. Statist, Inst. 47, 497-515.
  • S. Sawyer and J. Felsenstein (1983). Isolation by distance in a hierarchically clustered population, J. Appl. Prob. 20, 1-10.
  • T. Shiga (1982), Wandering phenomena in infinite allelic diffusion models, Adv. Appl. Probab. 14, 457-483.
  • T. Shiga (1982), Continuous time multi-allelic stepping stone models in population genetics, J. Math. Kyoto Univ. 22, 1-40.
  • T. Shiga, K. Uchiyama (1986), Stationary states and the stability of the stepping stone model involving mutation and selection, Prob. Th. Rel. Fields 73, 87-117.
  • J. Maynard Smith (1982): Evolution of the theory of games? Cambridge University Press
  • J. Vaillancourt (1990), Interacting Fleming-Viot processes, Stochastic Process Appl. 36, 45-57.
  • M. J. Wade (1992), Sewall Wright: gene interaction and the shifting balance theory, Oxford Surveys Evol. Biol. 8, 35-62.