State-dependent coupling in general networks (Q5959267)

The aim of the present paper is to extend a result which holds for the class of networks of quasireversible nodes to a class of networks constructed by coupling Markov chains in which the quasireversibility assumption does not necessarily hold, therefore the invariant measure for the primary network does not necessarily have product form. It is shown that if the right conditions hold, the relationship between the invariant measures for the primary and secondary network is so simple that it can be derived from the other, suggesting, in particular, that, if the invariant measure of the primary network can be calculated, either using analytical methods, approximation, or simulation, then the invariant measure of the secondary network follows immediately. For clarification, results for networks with a single class of transitions are considered.