Existence and Uniqueness of Q-Processes with a Given Finite μ-Invariant Measure

Let Q be a stable and conservative Q‐matrix over a countable state space S consisting of an irreducible class C and a single absorbing state 0 that is accessible from C. Suppose that Q admits a finite μ‐subinvariant measure m on C. We derive necessary and sufficient conditions for there to exist a Q‐process for which m is μ‐invariant on C, as well as a necessary condition for the uniqueness of such a process.