Dislocation measure of the fragmentation of a general Lévy tree

Guillaume Voisin

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 372-389
  • ISSN: 1292-8100

Abstract

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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].

How to cite

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Voisin, Guillaume. "Dislocation measure of the fragmentation of a general Lévy tree." ESAIM: Probability and Statistics 15 (2012): 372-389. <http://eudml.org/doc/222455>.

@article{Voisin2012,
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]. },
author = {Voisin, Guillaume},
journal = {ESAIM: Probability and Statistics},
keywords = {Fragmentation; Lévy CRT; fragmentation},
language = {eng},
month = {1},
pages = {372-389},
publisher = {EDP Sciences},
title = {Dislocation measure of the fragmentation of a general Lévy tree},
url = {http://eudml.org/doc/222455},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Voisin, Guillaume
TI - Dislocation measure of the fragmentation of a general Lévy tree
JO - ESAIM: Probability and Statistics
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 372
EP - 389
AB - 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].
LA - eng
KW - Fragmentation; Lévy CRT; fragmentation
UR - http://eudml.org/doc/222455
ER -

References

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  1. R. Abraham and J.-F. Delmas, Fragmentation associated with Lévy processes using snake. Probab. Th. Rel. Fiel141 (2008) 113–154.  
  2. R. Abraham, J.-F. Delmas and G. Voisin, Pruning a Lévy random continuum tree. preprint  
  3. R. Abraham and L. Serlet, Poisson snake and fragmentation. Elect. J. Probab.7 (2002) 1–15.  
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  15. T. Duquesne and M. Winkel, Growth of Lévy trees. Probab. Th. Rel. Fields139 (2007) 313–371.  
  16. M. Jirina, Stochastic branching processes with continuous state space. Czech. Math. J.83 (1958) 292–312.  
  17. J. Lamperti, The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete7 (1967) 271–288.  
  18. J.-F. Le Gall, Spatial branching processes, random snakes and partial differential equations. Birkhäuser Verlag, Basel (1999).  
  19. J.-F. Le Gall and Y. Le Jan, Branching processes in Lévy processes: the exploration process. Ann. Probab.26 (1998) 213–252.  
  20. K.R. Parthasarathy, Probability measures on metric spaces. Probability and Mathematical Statistics3, Academic, New York (1967).  

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