Travelling waves and homogeneous fragmentation

J. Berestycki (1) , S. C. Harris (2) , A. E. Kyprianou (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Department of Mathematical Sciences
3 University of Bath [Bath]

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 travelling waves. Specifically we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [31, 32], Neveu [34] and Chauvin [12] our analysis exposes the relation between travelling waves 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 [33] and in the context of fragmentation processes by Bertoin and Martinez [9] and Harris et al. [17]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion showing their mathematical robustness even within the context of fragmentation theory.

Domains: Probability [math.PR]; Algebraic Geometry [math.AG]

J. Berestycki, S. C. Harris, A. E. Kyprianou. Travelling waves and homogeneous fragmentation. 2009. ⟨hal-00440215⟩