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Transactions of the American Mathematical Society

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Equilibria and quasiequilibria for infinite collections of interacting Fleming-Viot processes


Authors: Donald A. Dawson, Andreas Greven and Jean Vaillancourt
Journal: Trans. Amer. Math. Soc. 347 (1995), 2277-2360
MSC: Primary 60K35; Secondary 60J70
DOI: https://doi.org/10.1090/S0002-9947-1995-1297523-5
MathSciNet review: 1297523
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Abstract: In this paper of infinite systems of interacting measure-valued diffusions each with state space $ \mathcal{P}\left( {[0,1]} \right)$, the set of probability measures on [0, 1], is constructed and analysed (Fleming-Viot systems). These systems arise as diffusion limits of population genetics models with infinitely many possible types of individuals (labelled by [0, 1]), spatially distributed over a countable collection of sites and evolving as follows. Individuals can migrate between sites and after an exponential waiting time a colony replaces its population by a new generation where the types are assigned by resampling from the empirical distribution of types at this site.

It is proved that, depending on recurrence versus transience properties of the migration mechanism, the system either clusters as $ t \to \infty $, that is, converges in distribution to a law concentrated on the states in which all components are equal to some $ {\delta _u},\ u \in [0,1]$, or the system approaches a nontrivial equilibrium state. The properties of the equilibrium states, respectively the cluster formation, are studied by letting a parameter in the migration mechanism tend to infinity and explicitly identifying the limiting dynamics in a sequence of different space-time scales. These limiting dynamics have stationary states which are quasi-equilibria of the original system, that is, change only in longer time scales. Properties of these quasi-equilibria are derived and related to the global equilibrium process for large $ N$. Finally we establish that the Fleming-Viot systems are the unique dynamics which remain invariant under the associated space-time renormalization procedure.


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  • [D] J. Aldous, Exchangeability and related topics, École d'Été de Probabilités de Saint Flour XIII, Lecture Notes in Math., vol. 1117, Springer-Verlag, 1985, pp. 1-198. MR 883646 (88d:60107)
  • [J] Baillon, P. Clement, A. Greven, and F. den Hollander, On the attracting orbit of a nonlinear transformation arising from renormalization of hierarchically interacting diffusions, Canad. J. Math. (1995) (to appear).
  • [J] T. Cox, A. Greven, and T. Shiga, Finite and infinite systems of interacting diffusions, preprint, 1993. MR 1355055 (96i:60105)
  • [D] A. Dawson, Measure-valued Markov processes, École d'Été de Probabilités de Saint Flour XXI, Lecture Notes in Math., vol. 1541, Springer-Verlag, 1993, pp. 1-261. MR 1242575 (94m:60101)
  • [D] A. Dawson and A. Greven, Multiple time scale analysis of hierarchically interacting systems, A Festschrift to Honor G. Kallianpur, Springer-Verlag, 1993a, pp. 41-50. MR 1427299 (97j:60181)
  • [D] A. Dawson and A. Greven, Multiple time scale analysis of interacting diffusions, Probab. Theory Related Fields 95 (1993b), 457-508. MR 1217447 (94i:60122)
  • [D] A. Dawson and A. Greven, Hierarchical models of interacting diffusions: multiple time scale phenomena, phase transition and pattern of cluster-formation, Probab. Theory Related Fields 96 (1993c), 435-473. MR 1234619 (94k:60155)
  • [D] A. Dawson and P. March, Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems, J. Funct. Anal. (1995) (to appear). MR 1347357 (97a:60105)
  • [P] Donnelly and T. G. Kurtz, A countable representation of the Fleming-Viot measure-valued diffusion, preprint, 1991. MR 1404525 (98f:60162)
  • [R] M. Dudley, Real analysis and probability, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1989. MR 982264 (91g:60001)
  • [S] N. Ethier, The distribution of frequencies of age-ordered alleles in a diffusion model, Adv. Appl. Probab. 22 (1990a), 519-532. MR 1066961 (91m:92018)
  • [S] N. Ethier, On the stationary distribution of the neutral one-locus diffusion model in population genetics, Ann. Appl. Probab. 2 (1990b), 24-35.
  • [S] N. Ethier, Equivalence of two descriptions of the ages of alleles, J. Appl. Probab. 29 (1992), 185-189. MR 1147778 (93f:60124)
  • [S] N. Ethier and Griffiths, The infinitely-many-sites model as a measure-valued diffusion, Ann. Probab. 15 (1987), 515-545. MR 885129 (89a:60130)
  • [S] N. Ethier and T. G. Kurtz, The infinitely-many-neutral-alleles diffusion model, Adv. Appl. Probab. 13 (1981), 429-452. MR 615945 (82j:60143)
  • [S] N. Ethier and T. G. Kurtz, Markov processes, characterization and convergence, Wiley, New York, 1986. MR 838085 (88a:60130)
  • [S] N. Ethier and T. G. Kurtz, The infinitely-many-alleles-model with selection as a measure-valued diffusion, Lecture Notes in Biomath., vol. 70, Springer-Verlag, 1987, pp. 72-86. MR 893637 (89c:92037)
  • [S] N. Ethier and T. G. Kurtz, Convergence to Fleming-Viot processes in the weak atomic topology, Stochastic Process. Appl. 54 (1995), 1-27. MR 1302692 (95m:60075)
  • [K] Fleischmann, Mixing properties of cluster invariant distributions, Litovsk. Mat. Sb. 18 (1978), 191-199. MR 0494542 (58:13383)
  • [K] Fleischmann and A. Greven, Diffusive clustering in an infinite system of hierarchically interacting diffusions, Probab. Theory Related Fields. 98 (1994), 517-566. MR 1271108 (95j:60163)
  • [A] Greven, Couplings of Markov chains by randomized stopping times. Part I: Couplings, harmonic functions and the Poisson equation, Probab. Theory Related Fields 75 (1987), 195-212. MR 885462 (88g:60185)
  • [K] Handa, A measure-valued diffusion process describing the stepping stone model with infinitely many alleles, Stochastic Process. Appl. 36 (1990), 269-296. MR 1084980 (92e:60160)
  • [A] Joffe and M. Métivier, Weak convergence of sequences of semimartingales with application to multitype branching processes, Adv. Appl. Probab. 18 (1986), 20-65. MR 827331 (88c:60097)
  • [J] F. C. Kingman, Random discrete distributions, J. Roy. Statist. Soc. Ser. B 37 (1975), 1-22. MR 0368264 (51:4505)
  • [G] P. Patil and C. Taillie, Diversity as a concept and its implications for random communities, Bull. Internat. Statist. Inst. 47 (1977), 497-515. MR 617593 (82k:92066)
  • [D] Ruelle, Statistical mechanics: Rigorous results, Benjamin, 1969. MR 0289084 (44:6279)
  • [T] Shiga, Wandering phenomena in infinite allelic diffusion models, Adv. Appl. Probab. 14 (1982), 457-483. MR 665289 (84b:92036)
  • [T] Shiga, Continuous time multi-allelic stepping stone models in population genetics, J. Math. Kyoto Univ. 22 (1982), 1-40. MR 648554 (84h:60138)
  • [F] Spitzer, Principles of random walk, Van Nostrand, Princeton, NJ, 1964. MR 0171290 (30:1521)
  • [J] Vaillancourt, Interacting Fleming-Viot processes, Stochastic Process. Appl. 36 (1990), 45-57. MR 1075600 (92i:60170)
  • [G] A. Watterson, The stationary distribution of the infinitely many neutral alleles diffusion model, J. Appl. Probab. 13 (1976), 639-651. MR 0504014 (58:20594a)
  • [K] Yosida, Functional analysis, 6th ed., Springer-Verlag, 1980. MR 617913 (82i:46002)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1297523-5
Keywords: Fleming-Viot process, hierarchical group, stability-clustering dichotomy, renormalization
Article copyright: © Copyright 1995 American Mathematical Society

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