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.  Zbl1142.60048
  2. R. Abraham, J.-F. Delmas and G. Voisin, Pruning a Lévy random continuum tree. preprint  Zbl1231.60073
  3. R. Abraham and L. Serlet, Poisson snake and fragmentation. Elect. J. Probab.7 (2002) 1–15.  Zbl1015.60046
  4. D. Aldous, The continuum random tree II: an overview. Proc. Durham Symp. Stochastic Analysis. Cambridge univ. press edition (1990) 23–70.  Zbl0791.60008
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  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.  
  12. J.-F. Delmas, Height process for super-critical continuous state branching process. Markov Proc. Rel. Fields.14 (2008) 309–326.  Zbl1149.60057
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  15. T. Duquesne and M. Winkel, Growth of Lévy trees. Probab. Th. Rel. Fields139 (2007) 313–371.  Zbl1126.60068
  16. M. Jirina, Stochastic branching processes with continuous state space. Czech. Math. J.83 (1958) 292–312.  Zbl0168.38602
  17. J. Lamperti, The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete7 (1967) 271–288.  Zbl0154.42603
  18. J.-F. Le Gall, Spatial branching processes, random snakes and partial differential equations. Birkhäuser Verlag, Basel (1999).  Zbl0938.60003
  19. J.-F. Le Gall and Y. Le Jan, Branching processes in Lévy processes: the exploration process. Ann. Probab.26 (1998) 213–252.  Zbl0948.60071
  20. K.R. Parthasarathy, Probability measures on metric spaces. Probability and Mathematical Statistics3, Academic, New York (1967).  Zbl0153.19101

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