On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure

Abstract

Let S be a countable set and let Q = (q ij , 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 = (m j , j ∈ S) such that ∑ i∈S m i q ij = −μ m j , j ≠ b. We prove that there exists a Q-process P(t) = (p ij (t), i, j ∈ S) for which m is a μ-invariant measure, that is ∑ i∈S m i p ij (t) = e−μ t m j , j∈S. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.