Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T00:14:29.285Z Has data issue: false hasContentIssue false
  • English
  • Français

On three classical problems for Markov chains with continuous time parameters

Published online by Cambridge University Press:  14 July 2016

Mu-Fa Chen
Mu-Fa Chen*
Affiliation:
Beijing Normal University
*
∗Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, The People's Republic of China.
Get access Rights & Permissions [Opens in a new window]

Abstract

For a given transition rate, i.e., a Q-matrix Q = (qij) on a countable state space, the uniqueness of the Q-semigroup P(t) = (Pij(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1.11), (1.17) and (1.18). Their proofs are quite straightforward.

Keywords

UNIQUENESSRECURRENCEPOSITIVE RECURRENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 2 , June 1991 , pp. 305 - 320
Copyright
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author was partially supported by the Ying-Tung Fok Education Foundation and the Natural Science Foundation of China.

References

Chen, M. F. (1986a) Coupling of jump processes. Acta Math. Sinica NS 2, 123–136.Google Scholar
Chen, M. F. (1986b) Jump Processes and Particle Systems. Beijing Normal University Press.Google Scholar
Chen, M. F. (1989) Stationary distributions of infinite particle systems with non-compact state spaces. Acta Math. Sci. 9, 7–19.10.1016/S0252-9602(18)30325-4Google Scholar
Chen, M. F. (1990) Ergodic theorems for reaction-diffusion processes. J. Statist. Phys. 58, 939–966.10.1007/BF01026558Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.Google Scholar
Dobrushin, R. L. (1970) Prescribing a system of random variables by conditional distributions. Theory Prob. Appl. 15, 458–486.10.1137/1115049Google Scholar
Feller, W. (1957) On boundaries and related conditions for the Kolmogorov differential equations. Ann. Math. 65, 527–570.10.2307/1970064Google Scholar
Gihman, I. I. and Skorohod, A. V. (1983) The Theory of Stochastic Processes. Springer-Verlag, Berlin.Google Scholar
Haken, H. (1983) Synergetics. Springer-Verlag, Berlin.10.1007/978-3-642-88338-5Google Scholar
Hou, Z. T. and Guo, Q. F. (1978) Homogeneous Markov Process with Countable State Space. Science Press, Beijing.Google Scholar
Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, Berlin.10.1007/978-1-4613-8542-4Google Scholar
Loeve, M. (1977) Probability Theory. Springer-Verlag, Berlin.Google Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes. Acta Math. 97, 1–46.10.1007/BF02392391Google Scholar
Schlögl, F. (1972) Chemical reaction models for phase transition. Z. Phys. 253, 147–161.10.1007/BF01379769Google Scholar
Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.Google Scholar
Yan, S. J. and Chen, M. F. (1986) Multidimensional Q-processes. Chinese Ann. Math. 7, 90–110.Google Scholar