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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 gn(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 gn(x), playing the role of xn. Under the stationary distribution when there is mutation, is analogous to the nth moment in a beta distribution. There is a d-dimensional version gn(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
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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