Advances in Applied Probability

The Λ-Fleming-Viot process and a connection with Wright-Fisher diffusion

Robert C. Griffiths

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Abstract

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, E[gn(X)] 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 → ∞.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 4 (2014), 1009-1035.

Dates
First available in Project Euclid: 12 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1418396241

Digital Object Identifier
doi:10.1239/aap/1418396241

Mathematical Reviews number (MathSciNet)
MR3290427

Zentralblatt MATH identifier
1305.60038

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

Keywords
Λ-coalescent Fleming-Viot process Wright-Fisher diffusion process

Citation

Griffiths, Robert C. The Λ-Fleming-Viot process and a connection with Wright-Fisher diffusion. Adv. in Appl. Probab. 46 (2014), no. 4, 1009--1035. doi:10.1239/aap/1418396241. https://projecteuclid.org/euclid.aap/1418396241


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