A one-dimensional Fisher–Wright diffusion process on the interval 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.
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).
 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/j.spa.2017.10.014Search in Google Scholar
 I. I. Gikhman, A short remark on Feller’s square root condition, preprint (2011), https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1756450. 10.2139/ssrn.1756450Search in Google Scholar
 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
 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
 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
 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
© 2021 Walter de Gruyter GmbH, Berlin/Boston