The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

The ( Ξ , 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 Kingman’s coalescent is replaced by the Ξ -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 the solution to the associated stochastic partial differential equation.