Reversibility, invariance and μ-invariance

Abstract

In this paper we consider a number of questions relating to the problem of determining quasi-stationary distributions for transient Markov processes. First we find conditions under which a measure or vector that is µ-invariant for a matrix of transition rates is also μ-invariant for the family of transition matrices of the minimal process it generates. These provide a means for determining whether or not the so-called stationary conditional quasi-stationary distribution exists in the λ-transient case. The process is not assumed to be regular, nor is it assumed to be uniform or irreducible. In deriving the invariance conditions we reveal a relationship between μ-invariance and the invariance of measures for related processes called the μ-reverse and the μ-dual processes. They play a role analogous to the time-reverse process which arises in the discussion of stationary distributions. Secondly we bring the related notions of detail-balance and reversibility into the realm of quasi-stationary processes. For example, if a process can be identified as being μ-reversible, the problem of determining quasi-stationary distributions is made much simpler. Finally, we consider some practical problems that emerge when calculating quasi-stationary distributions directly from the transition rates of the process. Our results are illustrated with reference to a variety of processes including examples of birth and death processes and the birth, death and catastrophe process.