Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-30T03:35:20.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Reversibility of Interacting Fleming–Viot Processes with Mutation, Selection, and Recombination

Published online by Cambridge University Press:  20 November 2018

Shui Feng ,
Byron Schmuland ,
Jean Vaillancourt  and
Xiaowen Zhou
Shui Feng
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON email: shuifeng@mcmaster.ca
Byron Schmuland
Affiliation:
Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB email: schmu@stat.ualberta.ca
Jean Vaillancourt
Affiliation:
Université du Québec en Outaouais, Gatineau, QC email: jean.vaillancourt@uqo.ca
Xiaowen Zhou
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montreal, QC email: xzhou@mathstat.concordia.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

Keywords

60J6060J70

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 104 - 122
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Dawson, D. A., Greven, A., and Vaillancourt, J., Equilibria and quasiequilibria for infinite collections of interacting Fleming-Viot processes. Trans. Amer. Math. Soc. 347 (1995), no. 7, 2277-2360. doi:10.2307/2154827Google Scholar
[2] Dawson, D. A. and Greven, A., Hierarchically interacting Fleming-Viot processes with selection and mutation: multiple space time scale analysis and quasi-equilibria. Electron. J. Probab. 4 (1999), no. 4, 1-81.Google Scholar
[3] Ethier, S. N., The infinitely-many-neutral-alleles diffusion model with ages. Adv. Appl. Probab. 22 (1990), no. 1, 1-24. doi:10.2307/1427594Google Scholar
[4] Ethier, S. N. and Kurtz, T. G., Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics, JohnWiley ' Sons, New York, 1986.Google Scholar
[5] Ethier, S. N. and Kurtz, T. G., Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 (1993), no. 2, 345-386. doi:10.1137/0331019Google Scholar
[6] Ethier, S. N. and Kurtz, T. G., Convergence to Fleming-Viot processes in the weak atomic topology. Stochastic Process. Appl. 54 (1994), no. 1, 1-27. doi:10.1016/0304-4149(94)00006-9Google Scholar
[7] Handa, K., A measure-valued diffusion process describing the stepping stone model with infinitely many alleles. Stochastic Process. Appl. 36 (1990), no. 2, 269-296. doi:10.1016/0304-4149(90)90096-BGoogle Scholar
[8] Handa, K., Quasi-invariance and reversibility in the Fleming-Viot process. Probab. Theory Related Fields 122 (2002), no. 4, 545-566. doi:10.1007/s004400100178Google Scholar
[9] Kermany, A. R. R., Zhou, X., and Hickey, D. A., Joint stationary moments of a two-island diffusion model of population subdivision. Theoret. Pop. Biol. 74 (2008), 226-232.Google Scholar
[10] Li, Z. H., Shiga, T., and Yao, L., A reversibility problem for Fleming-Viot processes. Electron. Comm. Probab. 4 (1999), 65-76.Google Scholar
[11] Overbeck, L. and, Röckner, M., Geometric aspects of finite- and infinite-dimensional Fleming-Viot processes. Random Oper. Stochastic Equations 5 (1997), no. 1, 35-58. doi:10.1515/rose.1997.5.1.35Google Scholar
[12] Schmuland, B. and, Sun, W., A cocycle proof that reversible Fleming-Viot processes have uniform mutation. C. R. Math. Acad. Sci. Soc. R. Can. 24 , no. 3, 124-128.Google Scholar
[13] Shiga, T., An interacting system in population genetics. I and II. J. Math. Kyoto Univ. 20 (1980), no. 2, 213-242; and no. 4, 723-733.Google Scholar
[14] Shiga, T., A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30 (1990), 245-279.Google Scholar
[15] Shiga, T. and, Uchiyama, K., Stationary states and their stability of the stepping stone model involving mutation and selection. Probab. Theory Relat. Fields 73 (1986), no. 1, 87-117. doi:10.1007/BF01845994Google Scholar