This paper is concerned with stable, conservative \(q\)-matrices \(Q = (q_{ij})\) and Markov chain transition functions \(P = (p_{ij}(t))\) which satisfy \(p'_{ij}(0) = q_{ij} (i,j \in S)\). Such a transition function is called a \(Q\)-function, and under these conditions, will satisfy the Kolmogorov backward equations. Of special importance is the minimal solution \(F\); this is the unique solution iff \(Q\) is regular (non-explosive).
A collection \((m_{j}, j \in S)\) of strictly positive numbers is a measure on \(S\); it is called subinvariant (respectively, invariant) for \(Q\) if \(\sum_{i} m_{i}q_{ij}(t) \leq m_{j}\) (\(\text{respectively,} = m_{j}\)), and subinvariant (respectively, invariant) for \(P\) provided \(\sum_{i}m_{i}p_{ij}(t) \leq m_{j}\) (respectively, \( = m_{j})\). The principal results relating (sub)invariance for \(Q\) and (sub)invariance for \(F\) are due to D. G. Kendall [Proc. Lond. Math. Soc. (3) 9, 417–431 (1959; Zbl 0117.35802)] and F. P. Kelly [Probability, statistics and analysis, Lond. Math. Soc. Lect. Notes Ser. 79, 143–160 (1983; Zbl 0498.60077)]:
(i) A measure \(m\) is subinvariant for \(F\) iff it is subinvariant for \(Q\);
(ii) If \(m\) is invariant for \(F\), it will be invariant for \(Q\);
(iii) If \(m\) is invariant for \(Q\), it will be invariant for \(F\) iff a related \(q\)-matrix (the time-reverse of \(Q\) with respect to \(m\)) is regular.
The purpose of this paper is to examine how such results extend to \(Q\)-functions other than \(F\) (when \(Q\) is non-regular). The author produces an example which shows that, in contrast to (ii), invariant measures for arbitrary \(Q\)-functions \(P\) need not be invariant for \(Q\). He proves that a subinvariant measure for \(P\) will always be subinvariant for \(Q\), but an invariant measure for \(P\) is invariant for \(Q\) iff \(P\) satisfies the Kolmogorov forward equations.
Starting with a specific (sub)invariant measure \(m\) for \(Q\), the author asks for which \(Q\)-functions will \(m\) be invariant or subinvariant. He answers this question when \(Q\) is what he calls “single exit.” (“Single exit” means the space of bounded vectors \(x\) on \(S\) satisfying \((\alpha - Q)x = 0\) for any \(\alpha > 0\) has dimension 1; the “exit boundary” of any associated \(Q\)-process consists of a single point.) In this case, he gives a necessary and sufficient condition for \(m\) to be subinvariant for a given non-minimal \(Q\)-function, and obtains necessary and sufficient conditions for the existence of a unique honest \(Q\)- function for which a specified subinvariant measure for \(Q\) is invariant. He also gives a sufficient condition for the existence of a unique honest \(Q\)-function for which the specified measure is invariant.