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.

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.

DOI : 10.1214/08-AIHP304
Classification : 60G18, 60J65
Mots-clés : self-similar fragmentation, normalized brownian excursion 
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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] R. Abraham and J.-F. Delmas. Fragmentation associated with Lévy processes using snake. Probab. Theory Related Fields 141 (2008) 113-154. | MR | Zbl

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

[4] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[5] J. Bertoin. Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001) 301-318. | MR | Zbl

[6] J. Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319-340. | Numdam | MR | Zbl

[7] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. | MR | Zbl

[8] J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994) 147-166. | MR | Zbl

[9] J. Bertoin and M. Yor. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002) 389-400. | MR | Zbl

[10] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212 (electronic). | MR

[11] P. Biane. Relations entre pont et excursion du mouvement brownien réel. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 1-7. | Numdam | MR | Zbl

[12] P. Biane, J. Pitman and M. Yor. 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

[13] M.-E. Caballero and L. Chaumont. Conditioned stable Lévy processes and Lamperti representation. Technical Report PMA-1066, Laboratoire de Probabilités et Modèles Aléatoires, 2006. | MR

[14] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. | MR | Zbl

[15] L. Chaumont. Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 (1997) 377-403. | MR | Zbl

[16] R. Dong, C. Goldschmidt and J. B. Martin. Coagulation-fragmentation duality, Poisson-Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16 (2006) 1733-1750. | MR | Zbl

[17] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) 1-147. | Numdam | MR | Zbl

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

[19] P. Fitzsimmons, J. Pitman and M. Yor. 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

[20] B. Haas. Equilibrium for fragmentation with immigration. Ann. Appl. Probab. 15 (2005) 1958-1996. | MR | Zbl

[21] B. Haas, J. Pitman and M. Winkel. Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. (2009). To appear. Available at arXiv:0705.3602v1. | MR | Zbl

[22] T. Jeulin. Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics 833. Springer, Berlin, 1980. | MR | Zbl

[23] J. Lamperti. Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225. | MR | Zbl

[24] J. F. Le Gall. Random real trees. Probab. Surv. 2 (2005) 245-311. | MR

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

[26] G. Miermont. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003) 423-454. | MR | Zbl

[27] G. Miermont. Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Related Fields 131 (2005) 341-375. | MR | Zbl

[28] M. Perman, J. Pitman and M. Yor. Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 (1992) 21-39. | MR | Zbl

[29] J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997) 855-900. | MR | Zbl

[30] J. Pitman and M. Yor. Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math. 55 (2003) 292-330. | MR | Zbl

[31] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. 293 Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1999. | MR | Zbl

[32] M. L. Silverstein. Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 (1980) 539-575. | MR | Zbl

[33] W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979) 143-149. | MR | Zbl

[34] T. Watanabe. Sample function behavior of increasing processes of class L. Probab. Theory Related Fields 104 (1996) 349-374. | MR | Zbl

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