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The uniform random tree in a Brownian excursion
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  • Published: September 1993

The uniform random tree in a Brownian excursion

  • Jean-François Le Gall1 

Probability Theory and Related Fields volume 96, pages 369–383 (1993)Cite this article

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Summary

To any Brownian excursione with duration σ(e) and anyt 1, ...,t p ∈[0,σ(e)], we associate a branching tree withp branches denoted byT p (e, t 1,...,t p ), which is closely related to the structure of the minima ofe. Our main theorem states that, ife is chosen according to the Itô measure and (t 1, ...,t p ) according to Lebesgue measure on [0,σ(e)]p, the treeT p (e, t 1, ...,t p ) is distributed according to the uniform measure on the set of trees withp branches. The proof of this result yields additional information about the “subexcursions” ofe corresponding to the different branches of the tree, thus generalizing a well-known representation theorem of Bismut. If we replace the Itô measure by the law of the normalized excursion, a simple conditioning argument leads to another remarkable result originally proved by Aldous with a very different method.

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References

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Authors and Affiliations

  1. Laboratoire de Probabilités, Université P. & M. Curie, 4, place Jussieu-Tour 56, F-75252, Paris Cedex 05, France

    Jean-François Le Gall

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  1. Jean-François Le Gall
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Le Gall, JF. The uniform random tree in a Brownian excursion. Probab. Th. Rel. Fields 96, 369–383 (1993). https://doi.org/10.1007/BF01292678

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  • Received: 12 October 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01292678

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Mathematics Subject Classification

  • 60J65
  • 60J80
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