# Subordination of trees and the Brownian map

- 113 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.

## References

- 1.Abraham, C., Le Gall, J.-F.: Excursion theory for Brownian motion indexed by the Brownian tree. J. Eur. Math. Soc. (
**to appear**). arXiv:1509.06616 - 2.Addario-Berry, L., Albenque, M.: The scaling limit of random simple triangulations and random simple quadrangulations. Ann. Probab. (
**to appear**). arXiv:1306.5227 - 3.Bertoin, J., Le Gall, J.-F., Le Jan, Y.: Spatial branching processes and subordination. Can. Math. J.
**49**, 24–54 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Curien, N., Kortchemski, I.: Random stable looptrees. Electron. J. Probab.
**19**(108), 1–35 (2014)MathSciNetzbMATHGoogle Scholar - 5.Curien, N., Le Gall, J.-F.: The hull process of the Brownian plane. Probab. Theory Relat. Fields
**166**, 187–231 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Duquesne, T.: The coding of compact real trees by real valued functions. Preprint, arXiv:math/0604106
- 7.Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque
**281**, vi+147 (2002)zbMATHGoogle Scholar - 8.Duquesne, T., Le Gall, J.-F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields
**131**, 553–603 (2005)CrossRefzbMATHGoogle Scholar - 9.Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab.
**19**, 1157–1195 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Evans, S.N.: Probability and Real Trees. Lectures from the 35th Saint-Flour Summer School on Probability Theory. Lecture Notes in Mathematics, vol. 1920. Springer, Berlin (2008)Google Scholar
- 11.Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1971)zbMATHGoogle Scholar
- 12.Grey, D.R.: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab.
**11**, 669–677 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Le Gall, J.-F.: The Brownian snake and solutions of \(\Delta u = u^2\) in a domain. Probab. Theory Relat. Fields
**102**, 393–432 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Le Gall, J.-F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (1999)CrossRefGoogle Scholar
- 15.Le Gall, J.-F.: Random trees and applications. Probab. Surv.
**2**, 245–311 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Le Gall, J.F.: Geodesics in large planar maps and in the Brownian map. Acta Math.
**205**, 287–360 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Le Gall, J.-F.: Uniqueness and universality of the Brownian map. Ann. Probab.
**41**, 2880–2960 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Le Gall, J.-F.: Brownian disks and the Brownian snake. Preprint, arXiv:1704.08987
- 19.Le Gall, J.-F., Paulin, F.: Scaling limits of bipartite planar maps are homeomorphic to the \(2\)-sphere. Geom. Funct. Anal.
**18**, 893–918 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Miermont, G.: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math.
**210**, 319–401 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Miller, J., Sheffield, S.: An axiomatic characterization of the Brownian map. Preprint, arXiv:1506.03806
- 22.Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric. Preprint, arXiv:1507.00719
- 23.Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. Preprint, arXiv:1605.03563
- 24.Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map III: the conformal structure is determined Preprint, arXiv:1608.05391
- 25.Pardo, J.C., Rivero, V.: Self-similar Markov processes. Bol. Soc. Mat. Mexicana
**19**, 201–235 (2013)MathSciNetzbMATHGoogle Scholar - 26.Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambrigde University Press, Cambridge (1999)zbMATHGoogle Scholar
- 27.Weill, M.: Regenerative real trees. Ann. Probab.
**35**, 2091–2121 (2007)MathSciNetCrossRefzbMATHGoogle Scholar