A Consistent Markov Partition Process Generated from the Paintbox Process

We study a family of Markov processes onP(k), the space of partitions of the natural numbers with at mostkblocks. The process can be constructed from a Poisson point process onR+x ∏i=1kP(k)with intensity dt⊗ ϱν(k), where ϱνis the distribution of the paintbox based on the probability measure ν onPm, the set of ranked-mass partitions of 1, and ϱν(k)is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.