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Chaos hypothesis for a system interacting through shared resources
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  • Published: June 1994

Chaos hypothesis for a system interacting through shared resources

  • Carl Graham1 &
  • Sylvie Méléard2 

Probability Theory and Related Fields volume 100, pages 157–174 (1994)Cite this article

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  • 40 Citations

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Summary

We study the chaos hypothesis for a wide class of pure-jump multitype interacting systems. The interaction may be strong, there is no symmetry assumption, and the system is not necessarily Markovian. We use interaction graphs and coupling and study in a precise way how a chain reaction is constituted by a series of direct interactions. We obtain the chaos hypothesis in variation norm with speed of convergence and deduce from it convergence of general empirical measures. We couple the interaction graph to a Boltzmann tree and show that the variation norm between the processes constructed on each goes to zero. This proves propagation of chaos in total variation with speed of convergence when the Boltzmann trees converge. Under light symmetry assumptions, we characterize the limit law by a nonlinear martingale problem.

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Author information

Authors and Affiliations

  1. Centre de Mathématiques Appliquées, Ecole Polytechnique, F-91128, Palaiseau, France

    Carl Graham

  2. Laboratoire de Probabilités, Université Paris VI, F-75231, Paris, France

    Sylvie Méléard

Authors
  1. Carl Graham
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  2. Sylvie Méléard
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Additional information

We dedicate this paper to Professor Claude Kipnis, who brought these network problems to our attention.

URA CNRS 756

URA CNRS 224

This article was processed by the author using the Springer-Verlag TEX QPMZGHB macro package 1991.

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Graham, C., Méléard, S. Chaos hypothesis for a system interacting through shared resources. Probab. Th. Rel. Fields 100, 157–174 (1994). https://doi.org/10.1007/BF01199263

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  • Received: 07 June 1993

  • Revised: 26 April 1994

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199263

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Mathematics Subject Classifications (1991)

  • 60K35
  • 60J85
  • 68M10
  • 90B12
  • 90B15
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