Survival of homogeneous fragmentation processes with killing

Robert Knobloch; Andreas E. Kyprianou

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 476-491
  • ISSN: 0246-0203

Abstract

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We consider a homogeneous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the decay of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.

How to cite

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Knobloch, Robert, and Kyprianou, Andreas E.. "Survival of homogeneous fragmentation processes with killing." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 476-491. <http://eudml.org/doc/272090>.

@article{Knobloch2014,
abstract = {We consider a homogeneous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the decay of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.},
author = {Knobloch, Robert, Kyprianou, Andreas E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {homogeneous fragmentation; scale functions; additive martingales; multiplicative martingales; largest fragment},
language = {eng},
number = {2},
pages = {476-491},
publisher = {Gauthier-Villars},
title = {Survival of homogeneous fragmentation processes with killing},
url = {http://eudml.org/doc/272090},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Knobloch, Robert
AU - Kyprianou, Andreas E.
TI - Survival of homogeneous fragmentation processes with killing
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 476
EP - 491
AB - We consider a homogeneous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the decay of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.
LA - eng
KW - homogeneous fragmentation; scale functions; additive martingales; multiplicative martingales; largest fragment
UR - http://eudml.org/doc/272090
ER -

References

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  1. [1] J. Bérard and J.-B. Gouéré. Brunet–Derrida behavior of branching-selection particle systems on the line. Comm. Math. Phys.298 (2010) 323–342. Zbl1247.60124MR2669438
  2. [2] J. Berestycki, N. Berestycki and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Ann. Probab. To appear. Zbl1304.60088MR3077519
  3. [3] J. Berestycki, S. C. Harris and A. E. Kyprianou. Travelling waves and homogeneous fragmenation. Ann. Appl. Probab.21 (2011) 1749–1794. Zbl1245.60069MR2884050
  4. [4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. Zbl0938.60005MR1406564
  5. [5] J. Bertoin. Asymptotic behaviour of fragmentation processes. J. Europ. Math. Soc.5 (2003) 395–416. Zbl1042.60042MR2017852
  6. [6] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge, 2006. Zbl1107.60002MR2253162
  7. [7] J. Bertoin and A. Rouault. Additive martingales and probability tilting for homogeneous fragmentations. Preprint, 2003. 
  8. [8] J. Bertoin and A. Rouault. Discritization methods for homogeneous fragmentations. J. London Math. Soc.72 (2005) 91–109. Zbl1077.60053MR2145730
  9. [9] L. Breiman. Probability, 2nd edition. SIAM, Philadelphia, PA, 1992. Zbl0753.60001MR1163370
  10. [10] B. Derrida and D. Simon. The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. EPL 78 (2007) Art. 60006. Zbl1244.82071MR2366713
  11. [11] B. Derrida and D. Simon. Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys.131 (2008) 203–233. Zbl1144.82321MR2386578
  12. [12] R. Durrett. Probability: Theory and Examples. Duxbury Press, N. Scituate, 1991. Zbl1202.60002MR2722836
  13. [13] N. Gantert, Y. Hu and Z. Shi. Asymptotics for the survival probability in a supercritical branching random walk. Ann. Inst. H. Poincaré Probab. Statist.47 (2011) 111–129. Zbl1210.60093MR2779399
  14. [14] J. W. Harris and S. C. Harris. Survival probabilities for branching Brownian motion with absorption. Elect. Comm. Probab.12 (2007) 81–92. Zbl1132.60059MR2300218
  15. [15] J. Harris, S. C. Harris and A. E. Kyprianou. Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: One sided travelling waves. Ann. Inst. H. Poincaré Probab. Statist.42 (2006) 125–145. Zbl1093.60059MR2196975
  16. [16] S. C. Harris, R. Knobloch and A. E. Kyprianou. Strong law of large numbers for fragmentation processes. Ann. Inst. H. Poincaré Probab. Statist.46 (2010) 119–134. Zbl1195.60046MR2641773
  17. [17] R. Knobloch. Asymptotic properties of fragmentation processes. Ph.D. thesis, Univ. Bath, 2011. 
  18. [18] R. Knobloch. One-sided FKPP travelling waves in the context of homogeneous fragmentation processes. Preprint, 2012. Available at arXiv:1204.0758. 
  19. [19] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. Zbl06176054

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