Wandering Random Measures in the Fleming-Viot Model

Abstract

Fleming and Viot have established the existence of a continuous-state-space version of the Ohta-Kimura ladder or stepwise-mutation model of population genetics for decribing allelic frequencies within a selectively neutral population undergoing mutation and random genetic drift. Their model is given by a probability-measure-valued Markov diffusion process. In this paper, we investigate the qualitative behavior of such measure-valued processes. It is demonstrated that the random measure is supported on a bounded gener-alized Cantor set and that this set performs a "wandering" but "coherent" motion that, if appropriately resealed, approaches a Brownian motion. The method used involves the construction of an interacting infinite particle system determined by the moment measures of the process and an analysis of the function-valued process that is "dual" to the measure-valued process of Fleming and Viot.