Open Access
October 2011 Traveling waves and homogeneous fragmentation
J. Berestycki, S. C. Harris, A. E. Kyprianou
Ann. Appl. Probab. 21(5): 1749-1794 (October 2011). DOI: 10.1214/10-AAP733


We formulate the notion of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323–331] and [Comm. Pure Appl. Math. 29 (1976) 553–554], Neveu [In Seminar on Stochastic Processes (1988) 223–242 Birkhäuser] and Chauvin [Ann. Probab. 19 (1991) 1195–1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump–Mode–Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365–395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553–570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 119–134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503–517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609–631]) showing their mathematical robustness even within the context of fragmentation theory.


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J. Berestycki. S. C. Harris. A. E. Kyprianou. "Traveling waves and homogeneous fragmentation." Ann. Appl. Probab. 21 (5) 1749 - 1794, October 2011.


Published: October 2011
First available in Project Euclid: 25 October 2011

zbMATH: 1245.60069
MathSciNet: MR2884050
Digital Object Identifier: 10.1214/10-AAP733

Primary: 60G09 , 60J25

Keywords: Additive martingales , Fisher–Kolmogorov–Petrovskii–Piscounov equation , homogeneous fragmentation processes , product martingales , spine decomposition , stopping lines , Traveling waves

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.21 • No. 5 • October 2011
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