The star-shaped Lambda-coalescent and Fleming-Viot process

TL;DR

This paper investigates the star-shaped Lambda-coalescent and Fleming-Viot processes, deriving explicit formulas for transition functions and stationary distributions, and extending models to multiple types, mutation, and selection.

Contribution

It provides explicit formulas for transition functions and stationary distributions in the star-shaped Lambda-coalescent and Fleming-Viot processes, including multi-type and selection models.

Findings

Derived transition functions and stationary distributions for the star-shaped Lambda-Fleming-Viot process.

Extended models to multiple types with mutation and selection.

Found polynomial eigenfunctions and hyperfunction eigenfunctions for transition analysis.

Abstract

The star-shaped $\Lambda$-coalescent and corresponding $\Lambda$-Fleming-Viot process where the $\Lambda$ measure has a single atom at unity are studied in this paper. The transition functions and stationary distribution of the $\Lambda$-Fleming-Viot process are derived in a two-type model with mutation. The distribution of the number of non-mutant lines back in time in the star-shaped $\Lambda$-coalescent is found. Extensions are made to a model with $d$ types, either with parent independent mutation or general Markov mutation, and an infinitely-many-types model when $d\to \infty$. An eigenfunction expansion for the transition functions is found which has polynomial right eigenfunctions and left eigenfunctions described by hyperfunctions. A further star-shaped model with general frequency dependent change is considered and the stationary distribution in the Fleming-Viot process derived. This model includes a star-shaped $\Lambda$-Fleming-Viot process with mutation and selection. In a general $\Lambda$-coalescent explicit formulae for the transition functions and stationary distribution when there is mutation are unknown, however in this paper explicit formulae are derived in the star-shaped coalescent.