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 x x), with x x 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 x x induced by partitioning occurring in the Λ-Fleming-Viot process. The eigenvalues and right polynomial eigenvectors are easy to see from the 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 the fixation probability can be computed from an algorithm in the paper.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 sizebiassing.The moment dual process in the Fleming-Viot process is the usual Λ-coalescent tree back in time. In 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 .

The seed bank coalescent with simultaneous switching

“…where W := U Y for a random variable U uniformly distributed on [0, 1] independent of Y ∼ Λ([0, 1]) −1 Λ. The last equality is inspired by a represantation in Theorem 2 of [Gri14]. Therefore P ∞,m 0 (mig(n − 1)) = γ(n) Borel-Cantelli gives that almost surely only finitely many of the events mig(n), for n ∈ N happen, and thus both sums in (22) Otherwise, the process stays infinite, at least provided Λ({1}) = 0, since in that case by (a) infinitely many blocks migrating to the seed bank in an arbitrarily short time implies that the process stays infinite.…”

“…This feature extends the individual switching of the seed bank coalescent and leads to new qualitative behaviour. The switching of multiple lines at the same time is reminiscent of multiple merger events in Lambda-coalescents ( [Sag99], [Pit99], [DK99]), yet leads to different tree structures, which is reflected in a new type of criterion for "coming down from infinity", interestingly involving arguments from rather elegant recent work by Griffiths [Gri14].…”

The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

“…The process X := (X t ) t∈R describing the type-0 frequency in the population then has the generator (cf. [7,10])…”

“…Combining results of [9] and [10], one infers that Assumption 2.1 is equivalent to the positive recurrence of the process K on N. Indeed, it is proved in [10, Theorem 3] (for the case σ * < ∞) and [9, Theorem 1.1] (for the case σ * = ∞) that Assumption 2.1 is equivalent to P[X ∞ = 1 | X 0 = x] < 1 for all x < 1, where X ∞ denotes the a.s. limit of X t as t → ∞.…”

“…These, together with their ancestral processes, the so-called Λ-coalescents, have become objects of intensive research in the past two decades. Although less is known for the case with selection, progress has been made in this direction as well (see for example [1,5,6,7,10]).…”