Abstract
In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.
Similar content being viewed by others
References
Liggett, T. M., Interacting Particle Systems, New York: Springer-Verlag, 1985.
Andjel, E. D., Ergodic and mixing properties of equilibrium measures for Markov processes, Trans. of the AMS, 1990, 318: 601–614.
Franchi, J., Asymptotic windings of ergodic diffusion, Stoch. Processes Appl., 1996, 62: 277–298.
Golden, K., Goldstein, S., Lebowitz, J. L., Nash estimates and the asymptotic behavior of diffusion, Ann. Prob., 1988, 16: 1127–1146.
Gordin. M. I., Lifsic, B. A., The central limit theorem for stationary ergodic Markov process, Dokl, Akad. Nauk SSSR, 1978, 19: 392–393.
Orey, S., Large deviations in ergodic theory, Seminar on Stochastic Processes, 1985, 12: 195–249.
Veretenikov, A. Y., On large deviations for ergodic process empirical measures, Adv. Sov. Math., 1992, 12: 125–133.
Deuschel, J. D., Stmock, D. W., Large Deviations, San Diego, CA: Academic Press, 1989.
Rosenblatt, M., Markov Processes, Structure and Asymptotic Behavior, Berlin: Springer-Verlag, 1971.
Liggett, T. M., Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes, Berlin: Springer-Verlag, 1999.
Albevrio, S., Kondratiev, Y. G., Rökner, M., Ergodicity of L2-semigroups and extremality of Gibbs states, J. Funct. Anal., 1997, 144: 293–423.
Liverani, C., Olla, S., Ergodicity in infinite Hamiltonian systems with conservative noise, Probab. Th. Re1. Fields, 1996, 106: 401–445.
Varadhan, S. R. S., Large Deviations and Applications, Philadelphia: SIAM, 1984.
Chen, J. W., A variational principle for Markov processes, J. Stat. Phys., 1999, 96: 1359–1364.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, J. Shift ergodicity for stationary Markov processes. Sci. China Ser. A-Math. 44, 1373–1380 (2001). https://doi.org/10.1007/BF02877065
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02877065