Ordered Additive Coalescent and Fragmentations Associatedto Lévy Processes with No Positive Jumps

Grégory Miermont (Université Pierre et Marie Curie)

Abstract


We study here the fragmentation processes that can be derived from Lévy processes with no positive jumps in the same manner as in the case of a Brownian motion (cf. Bertoin [4]). One of our motivations is that such a representation of fragmentation processes by excursion-type functions induces a particular order on the fragments which is closely related to the additivity of the coalescent kernel. We identify the fragmentation processes obtained this way as a mixing of time-reversed extremal additive coalescents by analogy with the work of Aldous and Pitman [2], and we make its semigroup explicit.

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Pages: 1-33

Publication Date: June 30, 2001

DOI: 10.1214/EJP.v6-87

References

  1. Aldous, D. J. and Pitman, J. (1998) The standard additive coalescent, Ann. Probab. 26, No. 4, 1703-1726. Math. Review 2000d:60121
  2. Aldous, D. J. and Pitman, J. (2000) Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent, Prob. Theory Relat. Fields 118, No. 4, 455-482. Math. Review 1 808 372
  3. Bertoin, J. (1996) Lévy Processes, Cambridge University Press, Cambridge. Math. Review 98e:60117
  4. Bertoin, J. (2000) A fragmentation process connected to Brownian motion , Prob. Theory Relat. Fields 117, No. 2, 289-301. Math. Review CMP 1 771 665 (2000:15)
  5. Bertoin, J. (2001) Eternal additive coalescents and certain bridges with exchangeable increments, Ann. Probab. 29, No. 1, 344-360. Math. Review CMP 1 825 153 (2001:11)
  6. Bertoin, J. (2000) Clustering statistics for sticky particles with Brownian initial velocity, J. Math. Pures Appl. (9) 79, No. 2, 173-194. Math. Review 2001g:60191
  7. Biane, P. (1986) Relations entre pont et excursion du mouvement brownien réel, Ann. Inst. Henri Poincaré 22, No. 1, 1-7. Math. Review 87k:60186
  8. Chassaing, P. and Louchard, G. (2001) Phase transition for parking blocks, brownian excursion and coalescence, Random Structures and Algorithms (to appear). Math. Review number not available. Available at http://www.iecn.u-nancy.fr/~chassain/.
  9. Chaumont, L. (1994) Sur certains processus de Lévy conditionnés à rester positifs, Stochastics and Stochastic Reports 47, No. 1-2, 1-20. Math. Review 2001e:60093
  10. Chaumont, L. (1997) Excursion normalisée, méandre et pont pour les processus de Lévy stables, Bull. Sci. Math. 121, No. 5, 377-403. Math. Review 99a:60077
  11. Chaumont, L. (2000) An extension of Vervaat's transformation and its consequences, J. Theor. Probab. 13, No. 1, 259-277. Math. Review 2001a:60093
  12. Evans, S. N. and Pitman, J. (1998) Construction of Markovian coalescents, Ann. Inst. Henri Poincaré 34, No. 3, 339-383. Math. Review 99k:60184
  13. Fitzsimmons, P. J., Pitman, J. and Yor, M. (1992) Markovian bridges: construction, Palm interpretation, and splicing, Seminar on Stochastic processes, 1992, pp. 101-134, Birkhäuser, Boston, Mass. Math. Review 95i:60070
  14. Kallenberg, O. (1973) Canonical representations and convergence criteria for processes with interchangeable increments, Z. Wahrscheinlichkeitstheorie verw. Geb. 27, 23-36. Math. Review 52 #15641
  15. Kallenberg, O. (1974) Path properties of processes with independent and interchangeable increments, Z. Wahrscheinlichkeitstheorie verw. Geb. 28, 257-271. Math. Review 53 #6715
  16. Knight, F. B. (1996) The uniform law for exchangeable and Lévy process bridges, Hommage à P. A. Meyer et J. Neveu. Astérisque No. 236, 171-188. Math. Review 97j:60137
  17. Millar, P. W. (1981) Comparison theorems for sample function growth, Ann. Probab. 9, No. 2, 330-334. Math. Review 82d:60077
  18. Norris J. R. (2000) Cluster coagulation, Comm. Math. Phys. 209, No. 2, 407-435. Math. Review 1 737 990
  19. Perman, M. (1993) Order statistics for jumps of normalised subordinators, Stochastic Process. Appl. 46, No. 2, 267-281. Math. Review 94g:60141
  20. Perman, M., Pitman, J. and Yor, M. (1992) Size-biased sampling of Poisson point processes and excursions, Probab. Theory Related Fields 92, No. 1, 21-39. Math. Review 93d:60088
  21. Pitman, J. and Yor, M. (1992) Arcsine laws and interval partitions derived from a stable subordinator, Proc. London Math. Soc. (3) 65, No. 2, 326-356. Math. Review 93e:60152
  22. Sato, K.-I. (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge. Math. Review CMP 1 739 520 (2000:08)
  23. Schweinsberg, J. (2001) Applications of the continuous-time ballot theorem to Brownian motion and related processes, Stochastic Process. Appl. (to appear). Math. Review number not available. Available at http://www.stat.berkeley.edu/users/jason/index.html.
  24. Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes, John Wiley & Sons, Inc., New York-London-Sydney. Math. Review 36 #947
  25. Winkel, M. (2000) Right-inverses of non-symmetric Lévy processes, Preprint. Math. Review number not available. Available at http://www.proba.jussieu.fr/mathdoc/preprints/winkel.Fri_Oct_20_15_57_38_CEST_2000.html.
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