The reversibility and an SPDE for the generalized Fleming-Viot Processes with mutation

TL;DR

This paper introduces a generalized Fleming-Viot process with mutation, explores its reversibility, and establishes an SPDE framework, extending classical models to include simultaneous multiple coalescent events.

Contribution

It proves the existence of the (,)-Fleming-Viot process with mutation for general mutation generators and analyzes its reversibility and associated SPDE uniqueness.

Findings

Existence of the process for general mutation generators.

Reversibility properties of the process.

Uniqueness results for the related SPDE.

Abstract

The (\Xi, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the \Xi-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.