Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T14:25:59.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Dislocation measure of the fragmentation of a general Lévy tree

Published online by Cambridge University Press:  05 January 2012

Guillaume Voisin
Guillaume Voisin*
Affiliation:
MAPMO CNRS UMR 6628, Fédération Denis Poisson 2964, Université d'Orléans, B.P. 6759, 45067 Orléans Cedex 2 France; guillaume.voisin@uni-due.de
Get access

Abstract

Given a general critical or sub-critical branching mechanism and its associated Lévy continuum random tree, we consider a pruning procedure on this tree using a Poisson snake. It defines a fragmentation process on the tree. We compute the family of dislocation measures associated with this fragmentation. This work generalizes the work made for a Brownian tree [R. Abraham and L. Serlet, Elect. J. Probab.7 (2002) 1–15] and for a tree without Brownian part [R. Abraham and J.-F. Delmas, Probab. Th. Rel. Fiel141 (2008) 113–154].

Keywords

FragmentationLévy CRT
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 372 - 389
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abraham, R. and Delmas, J.-F., Fragmentation associated with Lévy processes using snake. Probab. Th. Rel. Fiel 141 (2008) 113–154. CrossRef
R. Abraham, J.-F. Delmas and G. Voisin, Pruning a Lévy random continuum tree. preprint
Abraham, R. and Serlet, L., Poisson snake and fragmentation. Elect. J. Probab. 7 (2002) 1–15.
D. Aldous, The continuum random tree II: an overview. Proc. Durham Symp. Stochastic Analysis. Cambridge univ. press edition (1990) 23–70.
Aldous, D., The continuum random tree I. Ann. Probab. 19 (1991) 1–28. CrossRef
Aldous, D., The continuum random tree III. Ann. Probab. 21 (1993) 248–289. CrossRef
Aldous, D. and Pitman, J., Inhomogeneous continuum trees and the entrance boundary of the additive coalescent. Probab. Th. Rel. Fields 118 (2000) 455–482. CrossRef
Aldous, D. and Piman, J., The standard additive coalescent. Ann. Probab. 26 (1998) 1703–1726.
J. Bertoin, Lévy processes. Cambridge University Press, Cambridge (1996).
J. Bertoin, Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge 102 (2006).
D.A. Dawson, Measure-valued Markov processes, in École d'été de Probabilités de Saint-Flour 1991, Lect. Notes Math. Springer Verlag, Berlin 1541 (1993) 1–260.
Delmas, J.-F., Height process for super-critical continuous state branching process. Markov Proc. Rel. Fields. 14 (2008) 309–326.
T. Duquesne and J.-F. Le Gall, Random trees, Lévy processes and spatial branching processes 281. Astérisque (2002).
T. Duquesne and J.-F. Le Gall, Probabilistic and fractal aspects of Lévy trees, Probab. Th. Rel. Fields 131 (2005) 553–603. CrossRef
Duquesne, T. and Winkel, M., Growth of Lévy trees. Probab. Th. Rel. Fields 139 (2007) 313–371. CrossRef
Jirina, M., Stochastic branching processes with continuous state space. Czech. Math. J. 83 (1958) 292–312.
Lamperti, J., The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 7 (1967) 271–288. CrossRef
J.-F. Le Gall, Spatial branching processes, random snakes and partial differential equations. Birkhäuser Verlag, Basel (1999).
Le Gall, J.-F. and Le Jan, Y., Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 (1998) 213–252.
K.R. Parthasarathy, Probability measures on metric spaces. Probability and Mathematical Statistics 3, Academic, New York (1967).