A consistent Markov partition process generated from the paintbox process

Abstract

We study a family of Markov processes on P^(k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R⁺ x ∏_i=1^kP^(k) with intensity dt ⊗ ϱ_ν^(k), where ϱ_ν is the distribution of the paintbox based on the probability measure ν on P_m, the set of ranked-mass partitions of 1, and ϱ_ν^(k) is the product measure on ∏_i=1^kP^(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.