The falling apart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 45, Issue 4, 2009, Pages 1130-1149

  Bravo, Gerónimo Uribe ^a  

^a UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO   (Mexico)

Author keywords

Normalized brownian excursion; Self similar fragmentation

Indexed keywords

EID: 77952350510     PISSN: 02460203     EISSN: 02460203     Source Type: Journal    
DOI: 10.1214/08-AIHP304     Document Type: Article

References (34)

1. R. Abraham and J.-F. Delmas. Fragmentation associated with Lévy processes using snake. Probab. Theory Related Fields 141 (2008) 113-154. MR2372967.
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. MR1673235.
3. D. Aldous and J. Pitman. The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. MR1675063.
4. J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. MR1406564.
5. J. Bertoin. Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001) 301-318. MR1867425.
6. J. Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319-340. MR1899456.
7. J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. MR2253162.
8. J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994) 147-166. MR1268525.
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. MR1918243.
10. J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212 (electronic). MR2178044.
11. P. Biane. Relations entre pont et excursion du mouvement brownien réel. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 1-7. MR0838369.
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). MR1848256. (Pubitemid 33720543)
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.
14. L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. MR1419491.
15. L. Chaumont. Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 (1997) 377-403. MR1465814.
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. MR2288702.
17. T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) 1-147. MR1954248.
18. T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553-603. MR2147221.
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. MR1278079.
20. B. Haas. Equilibrium for fragmentation with immigration. Ann. Appl. Probab. 15 (2005) 1958-1996. MR2152250.
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.
22. T. Jeulin. Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics 833. Springer, Berlin, 1980. MR0604176.
23. J. Lamperti. Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225. MR0307358.
24. J. F. Le Gall. Random real trees. Probab. Surv. 2 (2005) 245-311. MR2203728.
25. J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. MR1617047.
26. G. Miermont. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003) 423-454. MR2018924.
27. G. Miermont. Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Related Fields 131 (2005) 341-375. MR2123249.
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. MR1156448.
29. J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997) 855-900. MR1434129.
30. J. Pitman and M. Yor. Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math. 55 (2003) 292-330. MR1969794.
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.
32. M. L. Silverstein. Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 (1980) 539-575. MR0573292.
33. W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979) 143-149. MR0515820.
34. T. Watanabe. Sample function behavior of increasing processes of class L. Probab. Theory Related Fields 104 (1996) 349-374. MR1376342.