On the equivalence of \(\mu\)-invariant measures for the minimal process and its q-matrix (Q1111245)

We obtain necessary and sufficient conditions for a measure or vector that is \(\mu\)-invariant for a q-matrix, Q, to be \(\mu\)-invariant for the family of transition matrices, \(\{\) P(t)\(\}\), of the minimal process it generates. Sufficient conditions are provided in the case when Q is regular and these are shown not to be necessary. When \(\mu\)-invariant measures and vectors can be identified, they may be used, in certain cases, to determine quasistationary distributions for the process.