Une étude additionnelle de la fragmentation de hauteur brownienne est présentée. Plus précisément, une représentation de la masse du fragment marqué en termes d'une transformation de Doob du subordinateur stable d'indice 1/2 est décrite puis utilisée pour étudier les sauts du processus de masse; ceci nous renseigne sur la façon dans laquelle un fragment typique se casse. Ces résultats se généralisent au cadre des fragmentations de hauteur de l'arbre stable. Enfin, nous donnons un théorème limite de la fragmentation de l'excursion Brownienne par les hauteurs, centrée autour du dernier fragment qui se décompose en poussière.
We present a further analysis of the fragmentation at heights of the normalized brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the brownian height fragmentation when it is about to reduce to dust are described in a limit theorem.
Mots clés : self-similar fragmentation, normalized brownian excursion
@article{AIHPB_2009__45_4_1130_0, author = {Uribe Bravo, Ger\'onimo}, title = {The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1130--1149}, publisher = {Gauthier-Villars}, volume = {45}, number = {4}, year = {2009}, doi = {10.1214/08-AIHP304}, mrnumber = {2572168}, zbl = {1208.60036}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP304/} }
TY - JOUR AU - Uribe Bravo, Gerónimo TI - The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 1130 EP - 1149 VL - 45 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP304/ DO - 10.1214/08-AIHP304 LA - en ID - AIHPB_2009__45_4_1130_0 ER -
%0 Journal Article %A Uribe Bravo, Gerónimo %T The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 1130-1149 %V 45 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP304/ %R 10.1214/08-AIHP304 %G en %F AIHPB_2009__45_4_1130_0
Uribe Bravo, Gerónimo. The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1130-1149. doi : 10.1214/08-AIHP304. http://www.numdam.org/articles/10.1214/08-AIHP304/
[1] Fragmentation associated with Lévy processes using snake. Probab. Theory Related Fields 141 (2008) 113-154. | MR | Zbl
and .[2] Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 (1999) 3-48. | MR | Zbl
.[3] The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. | MR | Zbl
and .[4] Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl
.[5] Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001) 301-318. | MR | Zbl
.[6] Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319-340. | Numdam | MR | Zbl
.[7] Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. | MR | Zbl
.[8] Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994) 147-166. | MR | Zbl
and .[9] The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002) 389-400. | MR | Zbl
and .[10] Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212 (electronic). | MR
and .[11] Relations entre pont et excursion du mouvement brownien réel. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 1-7. | Numdam | MR | Zbl
.[12] Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38 (2001) 435-465 (electronic). | MR | Zbl
, and .[13] Conditioned stable Lévy processes and Lamperti representation. Technical Report PMA-1066, Laboratoire de Probabilités et Modèles Aléatoires, 2006. | MR
and .[14] Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. | MR | Zbl
.[15] Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 (1997) 377-403. | MR | Zbl
.[16] Coagulation-fragmentation duality, Poisson-Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16 (2006) 1733-1750. | MR | Zbl
, and .[17] Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) 1-147. | Numdam | MR | Zbl
and .[18] Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553-603. | MR | Zbl
and .[19] Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992) 101-134. Progr. Probab. 33. Birkhäuser Boston, Boston, MA, 1993. | MR | Zbl
, and .[20] Equilibrium for fragmentation with immigration. Ann. Appl. Probab. 15 (2005) 1958-1996. | MR | Zbl
.[21] Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. (2009). To appear. Available at arXiv:0705.3602v1. | MR | Zbl
, and .[22] Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics 833. Springer, Berlin, 1980. | MR | Zbl
.[23] Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225. | MR | Zbl
.[24] Random real trees. Probab. Surv. 2 (2005) 245-311. | MR
.[25] Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. | MR | Zbl
and .[26] Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003) 423-454. | MR | Zbl
.[27] Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Related Fields 131 (2005) 341-375. | MR | Zbl
.[28] Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 (1992) 21-39. | MR | Zbl
, and .[29] The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997) 855-900. | MR | Zbl
and .[30] Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math. 55 (2003) 292-330. | MR | Zbl
and .[31] Continuous Martingales and Brownian Motion, 3rd edition. 293 Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1999. | MR | Zbl
and .[32] Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 (1980) 539-575. | MR | Zbl
.[33] A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979) 143-149. | MR | Zbl
.[34] Sample function behavior of increasing processes of class L. Probab. Theory Related Fields 104 (1996) 349-374. | MR | Zbl
.Cité par Sources :