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

Abstract

Thed-dimensional Λ-Fleming-Viot generator acting on functionsg(x), withxbeing a vector ofdallele frequencies, can be written as a Wright-Fisher generator acting on functionsgwith a modified random linear argument ofxinduced 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 polynomialsgn(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 ofgn(x), playing the role ofxn. Under the stationary distribution when there is mutation,is analogous to thenth moment in a beta distribution. There is ad-dimensional versiongn(X), and even an intriguing Ewens' sampling formula analogy whend→ ∞.