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Energy of Markov chains

Published online by Cambridge University Press:  01 July 2016

R. Syski
R. Syski*
Affiliation:
University of Maryland
*
∗Postal address: Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.
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Abstract

After preliminaries on Markov chains, supermartingales and potential theory (Section 1), the energy of a potential supermartingale generated by an increasing process is defined. The paper examines some properties of the energy of potentials of the form Ut = p(Xt) where p is a purely excessive function (which is also a potential of a charge) for a Markov chain (Xt). Also, the mutual energy of two potentials associated with the same Markov chain is discussed. Finally, several applications and examples are worked out in detail (Section 3).

Keywords

BALAYAGEENERGYEXCESSIVE FUNCTIONSINCREASING PROCESSMARKOV CHAINPOTENTIALSSUPERMARTINGALES
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979 , pp. 542 - 575
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

This paper is an expanded version of one presented at the Fourth Conference on Stochastic processs and their Applications, York University, Ontario, Canada, in August 1974.

References

1. Blake, L. H. (1978) Every Amart is a martingale in the limit. J. London Math. Soc. (2) 18, 381–384.CrossRefGoogle Scholar
2. Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
3. Chung, K. L. (1967) Markov Chains. Springer-Verlag, Berlin.Google Scholar
4. Dellacherie, (1972) Capacités et processus stochastiques. Springer-Verlag, Berlin.Google Scholar
5. Denzel, G. E., Kemeny, J. G. and Snell, J. L. (1967) Excessive functions for a class of continuous time Markov chains. In Markov Processes and Potential Theory, ed. Chover, J. Wiley, New York.Google Scholar
6. Helms, L. L. (1969) Introduction to Potential Theory. Wiley, New York.Google Scholar
7. Keilson, J. and Syski, R. (1974) Compensation measures in the theory of Markov chains. Stoch. Proc. Appl. 2, 59–72.Google Scholar
8. Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1966) Denumerable Markov Chains. Van Nostrand, Princeton, N.J. Google Scholar
9. Meyer, P. A. (1965) Probabilistic interpretation of the notion of energy. In Sem. Théor. Potentiel, Inst. H. Poincaré, Paris, 7e année, No. 5.Google Scholar
10. Meyer, P. A. (1965) Probability and Potentials. Blaisdell, New York.Google Scholar
11. Nelson, E. (1967) Dynamical Theories of Brownian Motion. Princeton University Press.Google Scholar
12. Prabhu, N. U. (1966) Stochastic Processes. Macmillan, London.Google Scholar
13. Silverstein, M. L. (1974) Symmetric Markov chains. Ann. Prob. 2, 681–701.CrossRefGoogle Scholar
14. Syski, R. (1967) Stochastic differential equations. In Modern Nonlinear Equations, ed. Saaty, T. L. McGraw-Hill, New York.Google Scholar
15. Syski, R. (1973) Potential theory for Markov chains. In Probabilistic Methods in Applied Mathematics, Vol. 3, ed. Bharucha-Reid, A. T. Academic Press, New York.Google Scholar
16. Syski, R. (1975) Queues and potentials. In Proc. XX Internal. Meeting TIMS (Tel Aviv, 1973) Vol. 2, ed. Schlifer, E. Jerusalem Academic Press, 547–554.Google Scholar
17. Syski, R. (1974) Reverse Markov chains. TIMS, Puerto Rico.Google Scholar
18. Syski, R. (1977) Perturbation models. Stoch. Proc. Appl. 5, 93–130.CrossRefGoogle Scholar
19. Syski, R. (1978) Ergodic potential. Stoch. Proc. Appl. 7, 311–330.Google Scholar