Dislocation measure of the fragmentation of a general Lévy tree
ESAIM: Probability and Statistics, Tome 15 (2011), pp. 372-389

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. Fiel 141 (2008) 113-154].

DOI : 10.1051/ps/2010006
Classification : 60J25, 60G57, 60J80
Keywords: fragmentation, Lévy CRT 
@article{PS_2011__15__372_0,
     author = {Voisin, Guillaume},
     title = {Dislocation measure of the fragmentation of a general {L\'evy} tree},
     journal = {ESAIM: Probability and Statistics},
     pages = {372--389},
     year = {2011},
     publisher = {EDP Sciences},
     volume = {15},
     doi = {10.1051/ps/2010006},
     mrnumber = {2870521},
     zbl = {1263.60068},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2010006/}
}
TY  - JOUR
AU  - Voisin, Guillaume
TI  - Dislocation measure of the fragmentation of a general Lévy tree
JO  - ESAIM: Probability and Statistics
PY  - 2011
SP  - 372
EP  - 389
VL  - 15
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps/2010006/
DO  - 10.1051/ps/2010006
LA  - en
ID  - PS_2011__15__372_0
ER  - 
%0 Journal Article
%A Voisin, Guillaume
%T Dislocation measure of the fragmentation of a general Lévy tree
%J ESAIM: Probability and Statistics
%D 2011
%P 372-389
%V 15
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps/2010006/
%R 10.1051/ps/2010006
%G en
%F PS_2011__15__372_0
Voisin, Guillaume. Dislocation measure of the fragmentation of a general Lévy tree. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 372-389. doi: 10.1051/ps/2010006

[1] R. Abraham and J.-F. Delmas, Fragmentation associated with Lévy processes using snake. Probab. Th. Rel. Fiel 141 (2008) 113-154. | Zbl | MR

[2] R. Abraham, J.-F. Delmas and G. Voisin, Pruning a Lévy random continuum tree. preprint | MR

[3] R. Abraham and L. Serlet, Poisson snake and fragmentation. Elect. J. Probab. 7 (2002) 1-15. | Zbl | MR | EuDML

[4] D. Aldous, The continuum random tree II: an overview. Proc. Durham Symp. Stochastic Analysis. Cambridge univ. press edition (1990) 23-70. | Zbl | MR

[5] D. Aldous, The continuum random tree I. Ann. Probab. 19 (1991) 1-28. | Zbl | MR

[6] D. Aldous, The continuum random tree III. Ann. Probab. 21 (1993) 248-289. | Zbl | MR

[7] D. Aldous and J. Pitman, Inhomogeneous continuum trees and the entrance boundary of the additive coalescent. Probab. Th. Rel. Fields 118 (2000) 455-482. | Zbl | MR

[8] D. Aldous and J. Piman, The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. | Zbl | MR

[9] J. Bertoin, Lévy processes. Cambridge University Press, Cambridge (1996). | Zbl | MR

[10] J. Bertoin, Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge 102 (2006). | Zbl | MR

[11] 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. | Zbl | MR

[12] J.-F. Delmas, Height process for super-critical continuous state branching process. Markov Proc. Rel. Fields. 14 (2008) 309-326. | Zbl | MR

[13] T. Duquesne and J.-F. Le Gall, Random trees, Lévy processes and spatial branching processes 281. Astérisque (2002). | Zbl | MR | Numdam

[14] T. Duquesne and J.-F. Le Gall, Probabilistic and fractal aspects of Lévy trees, Probab. Th. Rel. Fields 131 (2005) 553-603. | Zbl | MR

[15] T. Duquesne and M. Winkel, Growth of Lévy trees. Probab. Th. Rel. Fields 139 (2007) 313-371. | Zbl | MR

[16] M. Jirina, Stochastic branching processes with continuous state space. Czech. Math. J. 83 (1958) 292-312. | Zbl | MR | EuDML

[17] J. Lamperti, The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 7 (1967) 271-288. | Zbl | MR

[18] J.-F. Le Gall, Spatial branching processes, random snakes and partial differential equations. Birkhäuser Verlag, Basel (1999). | Zbl | MR

[19] J.-F. Le Gall and Y. Le Jan, Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 (1998) 213-252. | Zbl | MR

[20] K.R. Parthasarathy, Probability measures on metric spaces. Probability and Mathematical Statistics 3, Academic, New York (1967). | Zbl | MR

Cité par Sources :