On the identification of continuous-time Markov chains with a given invariant measure

Phil Pollett

Abstract: In [1] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit q-matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

[1] Pollett, P.K. (1991) Invariant measures for Q-processes when Q is not regular, Adv. Appl. Probab. 23 , 277--292.

AMS 1991 Subject Classification: 60J27; 60J35; 60J80.

Keywords: Q-processes; stationary distributions.

Acknowledgement: This worked was funded by the Australian Research Council and the University of Queensland.

The author:

• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au

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Last modified: 26th December 1995