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Large deviation results for a class of Markov chains arising from population genetics. (English) Zbl 0684.60018

Consider stationary Markov chains \((X^ N_ n)_{n\in {\mathbb{N}}}\) on a bounded subset of \({\mathbb{R}}^ d\) depending on a parameter \(N\in {\mathbb{N}}\) with \(E_ x(X^ N_ 1)=:f_ N(x)\). Let \(x_ 0\) be a common stable fixed point of the functions \(f_ N\) and D an open neighborhood of \(x_ 0\). Assume that N \(cov_ x(X^ N_ 1)\) is bounded away from 0 and \(\infty\) uniformly in \(N\in {\mathbb{N}}\) and \(x\in D.\)
Under various regularity assumptions it is proved that the asymptotics (N\(\to \infty)\) of the mean exit times from D and the invariant measures are governed by a quasi-potential as in § 4.4 of M. I. Frejdlin and A. D. Venttsel’, Random perturbations of dynamical systems (1984; Zbl 0522.60055). The results apply to the Wright-Fisher model in populations genetics.
Reviewer: K.U.Schaumlöffel

MSC:

60F10 Large deviations
60J60 Diffusion processes
92D10 Genetics and epigenetics
60G40 Stopping times; optimal stopping problems; gambling theory

Citations:

Zbl 0522.60055
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