Probability Theory and Related Fields

, Volume 96, Issue 3, pp 369–383

The uniform random tree in a Brownian excursion

  • Jean-François Le Gall

DOI: 10.1007/BF01292678

Cite this article as:
Le Gall, JF. Probab. Th. Rel. Fields (1993) 96: 369. doi:10.1007/BF01292678


To any Brownian excursione with duration σ(e) and anyt1, ...,tp∈[0,σ(e)], we associate a branching tree withp branches denoted byTp(e, t1,...,tp), 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 (t1, ...,tp) according to Lebesgue measure on [0,σ(e)]p, the treeTp (e, t1, ...,tp) 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.

Mathematics Subject Classification

60J65 60J80 

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Jean-François Le Gall
    • 1
  1. 1.Laboratoire de ProbabilitésUniversité P. & M. CurieParis Cedex 05France

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