Traveling waves and homogeneous fragmentation

Abstract

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.