Science in China Series A: Mathematics

, Volume 44, Issue 11, pp 1373–1380 | Cite as

Shift ergodicity for stationary Markov processes

  • Jinwen Chen


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.


ergodicity extremality stationary process interacting particle system 


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Copyright information

© Science in China Press 2001

Authors and Affiliations

  • Jinwen Chen
    • 1
  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina

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