Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 8, 2021

On recurrent properties of Fisher--Wright's diffusion on (0,1) with mutation

Roman Sineokiy and Alexander Veretennikov


A one-dimensional Fisher–Wright diffusion process on the interval ( 0 , 1 ) with mutations is considered. This is a widely known model in population genetics. The goal of this paper is an exponential recurrence of the process, which also implies an exponential rate of convergence towards the invariant measure.

MSC 2010: 60J60; 37A25

Communicated by Nikolai Leonenko

Funding source: Russian Science Foundation

Award Identifier / Grant number: 17-11-0198

Funding statement: For the second author of this study, part of Proposition 2.3 was prepared within the framework of the HSE University Basic Research Program, and part of Corollary 2.2 it was funded by the Russian Science Foundation grant 17-11-0198 (extended).


[1] L. Chen and D. W. Stroock, The fundamental solution to the Wright–Fisher equation, SIAM J. Math. Anal. 42 (2010), no. 2, 539–567. 10.1137/090764207Search in Google Scholar

[2] L. H. Duc, T. D. Tran and J. Jost, Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients, Stochastic Process. Appl. 128 (2018), no. 10, 3253–3272. 10.1016/ in Google Scholar

[3] C. L. Epstein and R. Mazzeo, Wright–Fisher diffusion in one dimension, SIAM J. Math. Anal. 42 (2010), no. 2, 568–608. 10.1137/090766152Search in Google Scholar

[4] W. Feller, Two singular diffusion problems, Ann. of Math. (2) 54 (1951), 173–182. 10.1007/978-3-319-16856-2_9Search in Google Scholar

[5] I. I. Gikhman, A short remark on Feller’s square root condition, preprint (2011), 10.2139/ssrn.1756450Search in Google Scholar

[6] R. Z. Khas’minskii, Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations, Theory Probab. Appl. 5 (1960), no. 2, 179–196. 10.1137/1105016Search in Google Scholar

[7] N. V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 691–708. Search in Google Scholar

[8] M. Steinrücken, R. Wang and Y. S. Song, An explicit transition density expansion for a multi-allelic Wright–Fisher diffusion with general diploid selection, Theor. Popul. Biol. 83 (2013), 1–14. 10.1016/j.tpb.2012.10.006Search in Google Scholar

[9] A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stochastic Process. Appl. 70 (1997), no. 1, 115–127. 10.1016/S0304-4149(97)00056-2Search in Google Scholar

[10] A. Yu. Veretennikov, On polynomial mixing and the rate of convergence for stochastic differential and difference equations, Theory Probab. Appl. 44 (2000), no. 2, 361–374. 10.1137/S0040585X97977550Search in Google Scholar

[11] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. 10.1215/kjm/1250523691Search in Google Scholar

Received: 2021-05-05
Accepted: 2021-05-10
Published Online: 2021-08-08
Published in Print: 2021-09-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow