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Recurrence and ergodicity of interacting particle systems
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  • Published: February 2000

Recurrence and ergodicity of interacting particle systems

  • J. Theodore Cox1 &
  • Achim Klenke2 

Probability Theory and Related Fields volume 116, pages 239–255 (2000)Cite this article

  • 115 Accesses

  • 14 Citations

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Abstract.

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense.

We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching.

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Authors and Affiliations

  1. Mathematics Department, Syracuse University, Syracuse, NY 13244, USA, , , , , , US

    J. Theodore Cox

  2. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße 1½, 91054 Erlangen, Germany. e-mail: klenke@mi.uni-erlangen.de, , , , , , DE

    Achim Klenke

Authors
  1. J. Theodore Cox
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  2. Achim Klenke
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Received: 22 January 1999 / Revised version: 24 May 1999

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Cox, J., Klenke, A. Recurrence and ergodicity of interacting particle systems. Probab Theory Relat Fields 116, 239–255 (2000). https://doi.org/10.1007/PL00008728

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  • Issue Date: February 2000

  • DOI: https://doi.org/10.1007/PL00008728

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  • Mathematics Subject Classification (1991): Primary 60K35
  • Key words and phrases: Interacting particle systems – Longtime behavior – Clustering – Recurrence – Ergodicity – Mutually catalytic branching
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