The star-shaped -coalescent and Fleming-Viot process

The star-shaped -coalescent and corresponding -Fleming-Viot process, where the measure has a single atom at unity, are studied in this article. The transition functions and stationary distribution of the -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 -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 -Fleming-Viot process with mutation and selection. In a general -coalescent explicit formulae for the transition functions and stationary distribution, when there is mutation, are unknown. However, in this article, explicit formulae are derived in the star-shaped coalescent.