We introduce the concepts of lumpability and commutativity of a continuous time discrete state space Markov process, and provide a necessary and sufficient condition for a lumpable Markov process to be commutative. Under suitable conditions we recover some of the basic quantities of the original Markov process from the jump chain of the lumped Markov process.

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and suucient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete-and continuous-time, we show that any aggregated Markov process which satisses mild regularity conditions can be directly converted to a canonical representation which is unique for ea...

We study a family of Markov processes on P , 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+ ×∏ki=1 P (k) with intensity dt ⊗ (k) ν , where ν is the distribution of the paintbox based on the probability measure ν on Pm, the set of ranked-mass partitions of 1, and (k) ν is the product measure on ∏k i=1 P . We...