Invariant measures for Q-processes when Q is not regular

The problem of determining invariant measures for continuous-time Markov chains directly from their transition rates is considered. The major result provides necessary and sufficient conditions for the existence of a unique .single-exit' chain with a specified invariant measure. This generalizes a result of Hou Chen-Ting and Chen Mufa that deals with symmetrically reversible chains. A simple sufficient condition for the existence of a unique honest chain for which the specified measure is invariant is also presented.