Markov processes with product-form stationary distribution

This research has been inspired by several papers on processes with inert drift [5, 6, 4, 3, 1]. The model involves a “particle” X and an “inert drift” L, neither of which is a Markov process by itself, but the vector process (X, L) is Markov. It turns out that for a number of diffusions with inert drift, the stationary measure has the product form; see [1]. The purpose of this note is to characterize processes with product form stationary distributions. This seems to be a difficult task for the family of all Markov processes so we limit ourselves to finite state spaces. These processes are simple on the technical side so we can analyze them in a quite explicit way. In particular, we obtain a number of examples that can generate conjectures for diffusions with inert drift.