Survival of homogeneous fragmentation processes with killing
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 476-491.

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.

Nous considérons un processus de fragmentation homogène tué à une barrière exponentielle. À l'aide de deux familles de martingales nous analysons la décroissance du plus gros fragment pour des valeurs des paramètres permettant la survie du système. Cet article traite aussi de la probabilité d'extinction du processus tué.

DOI: 10.1214/12-AIHP520
Classification: 60J25, 60G09
Keywords: homogeneous fragmentation, scale functions, additive martingales, multiplicative martingales, largest fragment
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Knobloch, Robert; Kyprianou, Andreas E. Survival of homogeneous fragmentation processes with killing. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 476-491. doi : 10.1214/12-AIHP520. http://archive.numdam.org/articles/10.1214/12-AIHP520/

[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. | MR | Zbl

[2] J. Berestycki, N. Berestycki and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Ann. Probab. To appear. | MR

[3] J. Berestycki, S. C. Harris and A. E. Kyprianou. Travelling waves and homogeneous fragmenation. Ann. Appl. Probab. 21 (2011) 1749-1794. | MR | Zbl

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

[5] J. Bertoin. Asymptotic behaviour of fragmentation processes. J. Europ. Math. Soc. 5 (2003) 395-416. | MR | Zbl

[6] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge, 2006. | MR | Zbl

[7] J. Bertoin and A. Rouault. Additive martingales and probability tilting for homogeneous fragmentations. Preprint, 2003.

[8] J. Bertoin and A. Rouault. Discritization methods for homogeneous fragmentations. J. London Math. Soc. 72 (2005) 91-109. | MR | Zbl

[9] L. Breiman. Probability, 2nd edition. SIAM, Philadelphia, PA, 1992. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] R. Durrett. Probability: Theory and Examples. Duxbury Press, N. Scituate, 1991. | MR | Zbl

[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. | Numdam | MR | Zbl

[14] J. W. Harris and S. C. Harris. Survival probabilities for branching Brownian motion with absorption. Elect. Comm. Probab. 12 (2007) 81-92. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

[17] R. Knobloch. Asymptotic properties of fragmentation processes. Ph.D. thesis, Univ. Bath, 2011.

[18] R. Knobloch. One-sided FKPP travelling waves in the context of homogeneous fragmentation processes. Preprint, 2012. Available at arXiv:1204.0758.

[19] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006.

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