A finite characterization of weak lumpable Markov processes. Part II: The continuous time case

We analysed in the companion paper (Stochastic Process. Appl. 38, 1991), the conditions under which the aggregated process constructed from an irreducible and homogeneous discrete time Markov chain over a given partition of its state space is another homogeneous Markov chain. The obtained result is a characterization of this situation by means of a finite algorithm which computes the set of all the initial probability distributions of the starting chain such that the aggregated one is also Markov homogeneous. In this paper, we consider the same problem in continuous time. Our main result is that it is always possible to come back to the discrete time case using uniformization.