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On the long term behavior of some finite particle systems
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  • Published: June 1990

On the long term behavior of some finite particle systems

  • J. T. Cox1 &
  • A. Greven2 

Probability Theory and Related Fields volume 85, pages 195–237 (1990)Cite this article

Summary

We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ d. Letη t denote this system, and letη N t denote a finite version ofη t defined on the torus [−N,N]d∩Z d. Ford≧3 we prove that for stationary, shift ergodic initial measures with density θ, that ifT(N)→∞ andT(N)/(2N+1)d →s∈[0,∞] asN→∞, then

{v θ}, θ≧0 is the set of extremal invariant measures for the infinite systemη t andQ s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Syracuse University, 13244-1150, Syracuse, NY, USA

    J. T. Cox

  2. Institut für Angewandte Mathematik, Im Neuenheimer Feld 294, D-6900, Heidelberg, Federal Republic of Germany

    A. Greven

Authors
  1. J. T. Cox
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  2. A. Greven
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Additional information

Work supported in part by the National Science Foundation under Grant DMS-8802055, by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by the Deutsche Forschungsgemeinschaft through the SFB 123 at the Universität Heidelberg

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Cox, J.T., Greven, A. On the long term behavior of some finite particle systems. Probab. Th. Rel. Fields 85, 195–237 (1990). https://doi.org/10.1007/BF01277982

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  • Received: 03 November 1988

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01277982

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Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Specific Model
  • Invariant Measure
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