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Convergence of path measures arising from a mean field or polaron type interaction
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  • Published: September 1993

Convergence of path measures arising from a mean field or polaron type interaction

  • Erwin Bolthausen1,
  • Jean-Dominique Deuschel2 &
  • Uwe Schmock1 

Probability Theory and Related Fields volume 95, pages 283–310 (1993)Cite this article

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Summary

We discuss the limiting path measures of Markov processes with either a mean field or a polaron type interaction of the paths. In the polaron type situation the strength is decaying at large distances on the time axis, and so the interaction is of short range in time. In contrast, in the mean field model, the interaction is weak, but of long range in time. Donsker and Varadhan proved that for the partition functions, there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range. The discussion of the path measures is more subtle. We treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples.

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Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik der Universität Zürich, Rämistrasse 74, CH-8001, Zürich, Switzerland

    Erwin Bolthausen & Uwe Schmock

  2. Mathematik, ETH Zentrum, CH-8092, Zürich, Switzerland

    Jean-Dominique Deuschel

Authors
  1. Erwin Bolthausen
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  2. Jean-Dominique Deuschel
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  3. Uwe Schmock
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Additional information

Research supported by the Swiss National Foundation (21-29833.90)

This article was processed by the authors using the Springer-Verlag TEX ProbTh macro package 1991.

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Bolthausen, E., Deuschel, JD. & Schmock, U. Convergence of path measures arising from a mean field or polaron type interaction. Probab. Th. Rel. Fields 95, 283–310 (1993). https://doi.org/10.1007/BF01192166

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  • Received: 10 February 1992

  • Revised: 13 October 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01192166

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Keywords

  • Stochastic Process
  • Partition Function
  • Probability Theory
  • Large Distance
  • Long Range
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