Convergence of path measures arising from a mean field or polaron type interaction
type: Item
instance of: scholarly article
title: Convergence of path measures arising from a mean field or polaron type interaction
zbMATH Open document ID: 0791.60093
DOI: 10.1007/BF01192166
author: Erwin Bolthausen
author: Jean-Dominique Deuschel
author: Uwe Schmock
publication date: 1994-05-18T00:00:00Z
review text: 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. \textit{M. D. Donsker} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 36, 505-528 (1983; Zbl 0538.60081)] 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.
reviewed by: Erwin Bolthausen
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 82C44
zbMATH DE Number: 569044
zbMATH Keywords: limiting path measures of Markov processes
zbMATH Keywords: polaron type interaction
zbMATH Keywords: transition from the polaron type to the mean field interaction
zbMATH Keywords: differentiable interaction
zbMATH Keywords: mean field interaction
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569044