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Reversible Markov processes on general spaces and spatial migration processes

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Richard F. Serfozo
Richard F. Serfozo*
Affiliation:
Georgia Institute of Technology
*
∗ Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email address: rserfozo@isye.gatech.edu
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Abstract

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In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes.

Keywords

Reversible Markov processKolmogorov criterionmulticlass birth-death processmobile phonebatch birthbatch death

MSC classification

Primary: 60J75: Jump processes
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 801 - 818
Copyright
Copyright © Applied Probability Trust 2005 

References

Dobrushin, R. L., Sukhov, Yu. M. and Fritts, Ī. (1988). A. N. Kolmogorov – founder of the theory of reversible Markov processes. Russian Math. Surveys 43, 157–182.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol 2, 2nd edn. John Wiley, New York.Google Scholar
Glötzl, E. (1981). Time reversible and Gibbsian point processes. I. Markovian spatial birth and death processes on a general phase space. Math. Nachr. 102, 217–222.CrossRefGoogle Scholar
Huang, X. and Serfozo, R. F. (1999). Spatial queueing systems. Math. Operat. Res. 24, 865–886.CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, London.Google Scholar
Kendall, W. S. (1997). On some weighted Boolean models. In Advances in Theory and Applications of Random Sets, ed. Jeulin, D., World Scientific, Singapore, pp. 105–120.Google Scholar
Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.CrossRefGoogle Scholar
Kingman, J. F. C. (1969). Markov population processes. J. Appl. Prob. 6, 1–18.CrossRefGoogle Scholar
Kolmogorov, A. N. (1936). Zur theorie der Markoffschen ketten. Math. Ann. 112, 155–160.CrossRefGoogle Scholar
Lopes Garcia, N. (1995). Birth and death processes as projections of higher-dimensional Poisson processes. Adv. Appl. Prob. 27, 911–930.CrossRefGoogle Scholar
Møller, J. (2000). On the rate of convergence of spatial birth-and-death processes. Ann. Inst. Statist. Math. 41, 565–581.CrossRefGoogle Scholar
Ōsawa, H. (1985). Reversibility of Markov chains with applications to storage models. J. Appl. Prob. 22, 123–137.CrossRefGoogle Scholar
Ōsawa, H. (1988). Reversibility of first-order autoregressive processes. Stoch. Process. Appl. 28, 61–69.CrossRefGoogle Scholar
Preston, C. J. (1977). Spatial birth-and-death processes. Bull. Inst. Internat. Statist. 46, 371–391.Google Scholar
Serfozo, R. F. (1999). Introduction to Stochastic Networks. Springer, New York.CrossRefGoogle Scholar
Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Prob. 8, 1–9.CrossRefGoogle Scholar
Whittle, P. (1986). Systems in Stochastic Equilibrium. John Wiley, New York.Google Scholar