Advertisement

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

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. 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. 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. 3.
    Bertoin, J., Le Gall, J.-F., Le Jan, Y.: Spatial branching processes and subordination. Can. Math. J. 49, 24–54 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Curien, N., Kortchemski, I.: Random stable looptrees. Electron. J. Probab. 19(108), 1–35 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Curien, N., Le Gall, J.-F.: The hull process of the Brownian plane. Probab. Theory Relat. Fields 166, 187–231 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Duquesne, T.: The coding of compact real trees by real valued functions. Preprint, arXiv:math/0604106
  7. 7.
    Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi+147 (2002)MATHGoogle Scholar
  8. 8.
    Duquesne, T., Le Gall, J.-F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields 131, 553–603 (2005)CrossRefMATHGoogle Scholar
  9. 9.
    Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab. 19, 1157–1195 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 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. 11.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1971)MATHGoogle Scholar
  12. 12.
    Grey, D.R.: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11, 669–677 (1974)MathSciNetCrossRefMATHGoogle Scholar
  13. 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)MathSciNetCrossRefMATHGoogle Scholar
  14. 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. 15.
    Le Gall, J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Le Gall, J.F.: Geodesics in large planar maps and in the Brownian map. Acta Math. 205, 287–360 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Le Gall, J.-F.: Uniqueness and universality of the Brownian map. Ann. Probab. 41, 2880–2960 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Le Gall, J.-F.: Brownian disks and the Brownian snake. Preprint, arXiv:1704.08987
  19. 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)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Miermont, G.: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210, 319–401 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Miller, J., Sheffield, S.: An axiomatic characterization of the Brownian map. Preprint, arXiv:1506.03806
  22. 22.
    Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric. Preprint, arXiv:1507.00719
  23. 23.
    Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. Preprint, arXiv:1605.03563
  24. 24.
    Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map III: the conformal structure is determined Preprint, arXiv:1608.05391
  25. 25.
    Pardo, J.C., Rivero, V.: Self-similar Markov processes. Bol. Soc. Mat. Mexicana 19, 201–235 (2013)MathSciNetMATHGoogle Scholar
  26. 26.
    Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambrigde University Press, Cambridge (1999)MATHGoogle Scholar
  27. 27.
    Weill, M.: Regenerative real trees. Ann. Probab. 35, 2091–2121 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Université Paris-SudOrsayFrance

Personalised recommendations