Electronic Communications in Probability

On the re-rooting invariance property of Lévy trees

Thomas Duquesne and Jean-Francois Le Gall

Full-text: Open access


We prove a strong form of the invariance under re-rooting of the distribution of the continuous random trees called Lévy trees. This expends previous results due to several authors.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 31, 317-326.

Accepted: 12 August 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G51: Processes with independent increments; Lévy processes

continuous tree stable tree re-rooting Lévy process

This work is licensed under aCreative Commons Attribution 3.0 License.


Duquesne, Thomas; Le Gall, Jean-Francois. On the re-rooting invariance property of Lévy trees. Electron. Commun. Probab. 14 (2009), paper no. 31, 317--326. doi:10.1214/ECP.v14-1484. https://projecteuclid.org/euclid.ecp/1465234740

Export citation


  • R. Abraham, J.F. Delmas, Williams' decomposition of the Levy continuous random tree and simultaneous extinction probability for populations with neutral mutations. Stoch. Process. Appl. 119 (2009), 1124-1143.
  • D. Aldous, The continuum random tree. I, Ann. Probab. 19 (1991), 1–28.
  • D. Aldous, The continuum random tree. II. An overview, in: Stochastic analysis (Durham, 1990), 23–70, Cambridge Univ. Press, Cambridge.
  • D. Aldous, The continuum random tree. III, Ann. Probab. 21 (1993), 248–289.
  • D. Aldous, Tree-based models for random distribution of mass, J. Statist. Phys. 73 (1993), 625–641.
  • J. Bertoin, Levy Processes. Cambridge University Press, Cambridge, 1996.
  • B. Chen, D. Ford, M. Winkel, A new family of Markov branching trees: the alpha-gamma model. Electr. J. Probab. 14 (2009), 400-430.
  • T. Duquesne, A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 (2003), 996-1027.
  • T. Duquesne, J.F. Le Gall, Random Trees, Levy Processes and Spatial Branching Processes. Asterisque 281 (2002)
  • T. Duquesne, J.F. Le Gall, Probabilistic and fractal aspects of Levy trees. Probab. Th. Rel. Fields 131 (2005), 553-603.
  • S.N. Evans, J.W. Pitman, A. Winter, Rayleigh processes, real trees and root growth with re-grafting. Probab. Th. Rel. Fields 134 (2006), 81-126.
  • M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics. Birkh"auser, Boston, 1999.
  • B. Haas, G. Miermont, The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electr. J. Probab. 9 (2004), 57-97.
  • B. Haas, G. Miermont, J.W. Pitman, M. Winkel, Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 (2008), 1790-1837.
  • B. Haas, J.W. Pitman, M. Winkel, Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab., to appear.
  • J.F. Le Gall, The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007), 621-670.
  • J.F. Le Gall, Y. Le Jan, Branching processes in Levy processes: The exploration process. Ann. Probab. 26 (1998), 213-252.
  • J.F. Le Gall, M. Weill, Conditioned Brownian trees. Ann. Inst. H. Poincare, Probab. Stat. 42 (2006), 455-489.
  • P. Marchal, A note on the fragmentation of the stable tree. In: Fifth Colloquium on Mathematics and Computer Science. DMCTS Proceedings AI (2008), 489-500.
  • J.F. Marckert, A. Mokkadem, A., Limit of normalized quadrangulations: the Brownian map. Ann. Probab. 34 (2006), 2144–2202.
  • J. Pitman, Combinatorial Stochastic Processes. Lectures Notes Math. 1875. Springer, Berlin, 2006.
  • M. Weill, Regenerative real trees. Ann. Probab. 35 (2007), 2091-2121.