Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types

Abstract

Let Z be a Borel right process with Lusin state-space E. For any finite set S it is possible to associate with Z a processof coalescing partitions of S with components labelled by elements of E that evolve as copies of Z. It is shown that, subject to a weak duality hypothesis on Z, there is a Feller process X with state-space a certain space of probability measure valued functions on E. The process X has its “moments” defined in terms of expectations forin a manner suggested by various instances of martingale problem duality between coalescing Markov processes and voter model particle systems, systems of interacting Fisher-Wright and Fleming-Viot diffusions, and stochastic partial differential equations with Fisher-Wright noise. Some sample path properties are examined in the special case where Z is a symmetric stable process on ℝ with index 1 < α ≤ 2. In particular, we show that for fixed t > 0 the essential range of the random probability measure valued function X t is almost surely a countable set of point masses.