The Annals of Probability

Coupling and ergodic theorems for Fleming-Viot processes

S. N. Ethier and Thomas G. Kurtz

Full-text: Open access

Abstract

Fleming-Viot processes are probability-measure-valued diffusion processes that can be used as stochastic models in population genetics. Here we use duality methods to prove ergodic theorems for Fleming-Viot processes, including those with recombination. Coupling methods are also used to establish ergodicity of Fleming-Viot processes, first without and then with selection. A special type of selection known as symmetric overdominance is treated by other methods.

Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 533-561.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855643

Digital Object Identifier
doi:10.1214/aop/1022855643

Mathematical Reviews number (MathSciNet)
MR1626158

Zentralblatt MATH identifier
0940.60045

Subjects
Primary: 60F99: None of the above, but in this section 60J60: Diffusion processes [See also 58J65]
Secondary: 60G10: Stationary processes 60G57: Random measures 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Ergodicity stationary distribution duality coupling measure-valued diffusion population genetics recombination selection

Citation

Ethier, S. N.; Kurtz, Thomas G. Coupling and ergodic theorems for Fleming-Viot processes. Ann. Probab. 26 (1998), no. 2, 533--561. doi:10.1214/aop/1022855643. https://projecteuclid.org/euclid.aop/1022855643


Export citation

References

  • ATHREYA, K. and NEY, P. E. 1978. A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493 501.
  • ATHREYA, K., MCDONALD, D. and NEY, P. E. 1978. Coupling and the renewal theorem. Amer. Math. Monthly 85 809 814.
  • DAWSON, D. 1978. Geostochastic calculus. Canad. J. Statist. 6 143 168.
  • DAWSON, D. A. and HOCHBERG, K. J. 1982. Wandering random measures in the Fleming Viot model. Ann. Probab. 10 554 580. ´
  • DOEBLIN, W. 1940. Elements d'une theorie generale des chaines simple constantes de Markoff. ´ ´ ´ ´ ´Ann. Sci. Ecole Norm. Sup. 3 57 61 111. Z.
  • DYNKIN, E. B. 1978. Sufficient statistics and extreme points. Ann. Probab. 6 705 730.
  • ETHIER, S. N. 1992. Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Probab. 29 487 498.
  • ETHIER, S. N. and GRIFFITHS, R. C. 1990. The neutral two-locus model as a measure-valued diffusion. Adv. in Appl. Probab. 22 773 786.
  • ETHIER, S. N. and GRIFFITHS, R. C. 1993. The transition function of a Fleming Viot process. Ann. Probab. 21 1571 1590.
  • ETHIER, S. N. and KURTZ, T. G. 1981. On the infinitely-many-neutral-alleles diffusion model. Adv. in Appl. Probab. 13 429 452.
  • ETHIER, S. N. and KURTZ, T. G. 1986. Markov Processes: Characterization and Convergence. Wiley, New York.
  • ETHIER, S. N. and KURTZ, T. G. 1987. The infinitely-many-alleles model with selection as a measure-valued diffusion. Stochastic Models in Biology. Lecture Notes in Biomath. 70 72 86. Springer, Berlin.
  • ETHIER, S. N. and KURTZ, T. G. 1993. Fleming Viot processes in population genetics. SIAM J. Control Optim. 31 345 386.
  • ETHIER, S. N. and KURTZ, T. G. 1994. Convergence to Fleming Viot processes in the weak atomic topology. Stochastic Process. Appl. 54 1 27.
  • FUKUSHIMA, M. and STROOCK, D. W. 1986. Reversibility of solutions to martingale problems. In Probability, Statistical Mechanics, and Number Theory. Advances in MathematicsSupplementary Studies G.-C. Rota, ed. 9 107 123. Academic Press, Orlando, FL. Z.
  • GRIFFEATH, D. 1976. Partial coupling and loss of memory for Markov chains. Ann. Probab. 4 850 858.
  • GRIFFEATH, D. 1978. Coupling methods for Markov processes. In Studies in Probability and Ergodic Theory 1 43. Academic Press, New York.
  • GRIFFITHS, R. C. 1979. A transition density expansion for a multi-allele diffusion model. Adv. in Appl. Probab. 11 310 325.
  • KINGMAN, J. F. C. 1975. Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37 1 22.
  • LIGGETT, T. M. 1985. Interacting Particle Systems. Springer, Berlin.
  • LINDVALL, T. 1983. On coupling diffusion processes. J. Appl. Probab. 20 82 93.
  • LINDVALL, T. and ROGERS, L. C. G. 1986. Coupling of multidimensional diffusions by reflection. Ann. Probab. 14 860 872.
  • NUMMELIN, E. 1978. A splitting technique for Harris recurrent Markov chains.Wahrsch. Verw. Gebiete 43 309 318. Z.
  • NUMMELIN, E. 1984. General Irreducible Markov Chains and Non-negative Operators. Cambridge Univ. Press.
  • SHIGA, T. 1981. Diffusion processes in population genetics. J. Math. Kyoto Univ. 21 133 151.
  • SHIGA, T. 1982. Wandering phenomena in infinite-allelic diffusion models. Adv. in Appl. Probab. 14 457 483.
  • 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.
  • VASERSHTEIN, L. N. 1969. Markov processes over denumerable products of spaces describing large systems of automata. Probablemy Peredachi Informatsii 5 64 73.
  • WATTERSON, G. A. 1977. Heterosis or neutrality? Genetics 85 789 814.
  • SALT LAKE CITY, UTAH 84112 MADISON, WISCONSIN 53706 E-MAIL: ethier@math.utah.edu E-MAIL: kurtz@math.wisc.edu