The Annals of Probability

Compensated fragmentation processes and limits of dilated fragmentations

Jean Bertoin

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Abstract

A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $\nu$ which governs their evolutions has only to fulfill the integral condition $\int_{\mathcal{P}}(1-p_{1})^{2}\nu({\mathrm{d}}{\mathbf{p}})<\infty$, where ${\mathbf{p}}=(p_{1},\ldots)$ denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $\int_{\mathcal{P}}(1-p_{1})\nu({\mathrm{d}}{\mathbf{p}})<\infty$ for $\nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Lévy processes.

Article information

Source
Ann. Probab. Volume 44, Number 2 (2016), 1254-1284.

Dates
Received: April 2014
Revised: December 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960395

Digital Object Identifier
doi:10.1214/14-AOP1000

Mathematical Reviews number (MathSciNet)
MR3474471

Zentralblatt MATH identifier
1344.60033

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; Lévy processes 60G80

Keywords
Homogeneous fragmentation dilation compensation dislocation measure

Citation

Bertoin, Jean. Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab. 44 (2016), no. 2, 1254--1284. doi:10.1214/14-AOP1000. https://projecteuclid.org/euclid.aop/1457960395


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References

  • [1] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362–1426.
  • [2] Aïdékon, E., Berestycki, J., Brunet, É. and Shi, Z. (2013). Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 405–451.
  • [3] Arguin, L.-P., Bovier, A. and Kistler, N. (2011). Genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64 1647–1676.
  • [4] Banasiak, J. (2004). Conservative and shattering solutions for some classes of fragmentation models. Math. Models Methods Appl. Sci. 14 483–501.
  • [5] Berestycki, J. (2002). Ranked fragmentations. ESAIM Probab. Stat. 6 157–175 (electronic).
  • [6] Berestycki, J. (2003). Multifractal spectra of fragmentation processes. J. Stat. Phys. 113 411–430.
  • [7] Berestycki, J., Harris, S. C. and Kyprianou, A. E. (2011). Traveling waves and homogeneous fragmentation. Ann. Appl. Probab. 21 1749–1794.
  • [8] Bertoin, J. (2003). The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. (JEMS) 5 395–416.
  • [9] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [10] Bertoin, J., Le Gall, J.-F. and Le Jan, Y. (1997). Spatial branching processes and subordination. Canad. J. Math. 49 24–54.
  • [11] Bertoin, J. and Rouault, A. (2005). Discretization methods for homogeneous fragmentations. J. Lond. Math. Soc. (2) 72 91–109.
  • [12] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
  • [13] Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 137–151.
  • [14] Cáceres, M. J., Cañizo, J. A. and Mischler, S. (2011). Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations. J. Math. Pures Appl. (9) 96 334–362.
  • [15] Doumic, M., Hoffmann, M., Krell, N. and Robert, L. (2015). Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli 21 1760–1799.
  • [16] Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Probab. Theory Related Fields 139 313–371.
  • [17] Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 245–277.
  • [18] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [19] Kolmogoroff, A. N. (1941). Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 99–101.
  • [20] Lieb, E. H. and Loss, M. (2001). Analysis, 2nd ed. Graduate Studies in Mathematics 14. Amer. Math. Soc., Providence, RI.
  • [21] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [22] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [23] Uchiyama, K. (1982). Spatial growth of a branching process of particles living in ${\mathbf{R}}^{d}$. Ann. Probab. 10 896–918.