The stationary distribution of a sample from the Wright-Fisher diffusion\n model with general small mutation rates

2018/02/21 by Conrad J. Burden, Burden, Conrad J., Robert Griffiths +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Evolution and Genetic Dynamics #Stochastic processes and statistical mechanics #Mathematical and Theoretical Epidemiology and Ecology Models

paper · pdf · doi:10.48550/arxiv.1802.07875

Abstract

The stationary distribution of a sample taken from a Wright-Fisher diffusion\nwith general small mutation rates is found using a coalescent approach. The\napproximation is equivalent to having at most one mutation in the coalescent\ntree to the first order in the rates. The sample probabilities characterize an\napproximation for the stationary distribution from the Wright-Fisher diffusion.\nThe approach is different from Burden and Tang (2016,2017) who use a\nprobability flux argument to obtain the same results from a forward diffusion\ngenerator equation. The solution has interest because the solution is not known\nwhen rates are not small. An analogous solution is found for the configuration\nof alleles in a general exchangeable binary coalescent tree. In particular an\nexplicit solution is found for a pure birth process tree when individuals\nreproduce at rate lambda.\n

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