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The Λ-Fleming-Viot Process and a Connection with Wright-Fisher Diffusion

Part of: Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Robert C. Griffiths
Robert C. Griffiths*
Affiliation:
Oxford University
*
∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK. Email address: griff@stats.ox.ac.uk
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Abstract

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The d-dimensional Λ-Fleming-Viot generator acting on functions g(x ), with x being a vector of d allele frequencies, can be written as a Wright-Fisher generator acting on functions g with a modified random linear argument of x induced by partitioning occurring in the Λ-Fleming-Viot process. The eigenvalues and right polynomial eigenvectors are easy to see from this representation. The two-dimensional process, which has a one-dimensional generator, is considered in detail. A nonlinear equation is found for the Green's function. In a model with genic selection a proof is given that there is a critical selection value such that if the selection coefficient is greater than or equal to the critical value then fixation, when the boundary 1 is hit, has probability 1 beginning from any nonzero frequency. This is an analytic proof different from the proofs of Der, Epstein and Plotkin (2011) and Foucart (2013). An application in the infinitely-many-alleles Λ-Fleming-Viot process is finding an interesting identity for the frequency spectrum of alleles that is based on size biasing. The moment dual process in the Fleming-Viot process is the usual Λ-coalescent tree back in time. The Wright-Fisher representation using a different set of polynomials g n (x) as test functions produces a dual death process which has a similarity to the Kingman coalescent and decreases by units of one. The eigenvalues of the process are analogous to the Jacobi polynomials when expressed in terms of g n (x), playing the role of x n . Under the stationary distribution when there is mutation, is analogous to the nth moment in a beta distribution. There is a d-dimensional version g n (X ), and even an intriguing Ewens' sampling formula analogy when d → ∞.

Keywords

Λ-coalescentFleming-Viot processWright-Fisher diffusion process

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 92D15: Problems related to evolution

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1009 - 1035
Copyright
© Applied Probability Trust 

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