Quasi-stationary Distributions: A Bibliography

Quasi-stationary distributions have been used to model the long-term behaviour of stochastic systems which in some sense terminate, but appear to be stationary over any reasonable time scale. The idea can be traced back to the work of the Russian Mathematician A.M. Yaglom, who showed that the limiting conditional distribution of the number in the $n^{\rm th}$ generation of the Galton Watson branching process always exists in the subcritical case (see Yaglom [370]). But, it was not until the early sixties, and largely stimulated by the remarkable work of Vere-Jones [359], and later Kingman [176], Darroch and Seneta [75], Seneta and Vere-Jones [313], and Darroch and Seneta [76], that a general theory was annunciated. Since then, quasi-stationary distributions have appeared in a variety of diverse contexts, including chemical reaction kinetics, reliability theory, genetics, epidemics, ecology and telecommunications, and this work has stimulated further developments in the theory. Recent key papers in the area are Ferrari, Kesten, Martínez and Picco [91] and Kesten [156].

I present here a bibliography of work on quasi-stationary distributions. This includes work on quasi-stationary distributions per se (stationary conditional distributions), limiting conditional distributions (often called quasi-stationary distributions, and also called Yaglom limits and quasi-limiting distributions), the companion topics of geometric and exponential ergodicity, -classification of states and -invariant measures (et cetera), ratio limit theorems, analysis of processes conditioned to stay within a given region (particularly weak convengence of those processes), and papers dealing with diffusion approximations which specifically describe quasi stationarity of evanescent processes.

Published work is cited under various headings. Several works appear under more than one heading. The final section lists the same works in chronological order.

Whilst I do not claim that the bibliography is exhaustive, I do hope that it includes most of the work published on quasi-stationary distributions. I welcome additions and corrections. I would particularly like to hear about Ph.D. theses in the area (they are very difficult to trace). Please e-mail me at pkp@maths.uq.edu.au. This bibliography is maintained at

http://www.maths.uq.edu.au/~pkp/papers/qsds.html

Quasi-stationary Distributions: A Bibliography

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