On the Existence of Uni-instantaneous Q-processes with a given Finite Μ-invariant Measure

Let S be a countable set and let Q = (qij , i, j ∈ S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (mj , j ∈ S) such that ∑i∈S miqij = −μmj , j = b. We prove that there exists a Q-process P(t) = (pij (t), i, j ∈ S) for which m is a μ-invariant measure, that is ∑ i∈S mipij (t) = e−μtmj , j ∈ S. We illustrate our results with reference to the Kolmogorov ‘K1’chain and a birth–death process with catastrophes and instantaneous resurrection.