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Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 819–864 | Cite as

Subordination of trees and the Brownian map

  • Jean-François Le Gall
Article
  • 104 Downloads

Abstract

We discuss subordination of random compact \({\mathbb R}\)-trees. We focus on the case of the Brownian tree, where the subordination function is given by the past maximum process of Brownian motion indexed by the tree. In that particular case, the subordinate tree is identified as a stable Lévy tree with index 3/2. As a more precise alternative formulation, we show that the maximum process of the Brownian snake is a time change of the height process coding the Lévy tree. We then apply our results to properties of the Brownian map. In particular, we recover, in a more precise form, a recent result of Miller and Sheffield identifying the metric net associated with the Brownian map.

Mathematics Subject Classification

60J80 60D05 

Notes

Acknowledgements

I thank the referee for a careful reading of the manuscript and for several useful suggestions.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Université Paris-SudOrsayFrance

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