The uniform random tree in a Brownian excursion |
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Authors: | Jean-François Le Gall |
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Institution: | (1) Laboratoire de Probabilités, Université P. & M. Curie, 4, place Jussieu-Tour 56, F-75252 Paris Cedex 05, France |
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Abstract: | Summary To any Brownian excursione with duration (e) and anyt
1, ...,t
p
0, (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, (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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Keywords: | 60J65 60J80 |
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