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

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 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 xⁿ. Under the stationary distribution when there is mutation, E[g_n(X)] 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 → ∞.