2015/06/24 by Robert Griffiths, Griffiths, Robert, Shuhei Mano +1
Chemistry · #Chemistry and Stereochemistry Studies #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1506.07298
openalex publication_date 2015/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The star-shaped Λ-coalescent and corresponding Λ-Fleming-Viot process where the Λ measure has a single atom at unity are studied in this paper. The transition functions and stationary distribution of the Λ-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 Λ-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→ ∞. 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 Λ-Fleming-Viot process with mutation and selection. In a general Λ-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.