Travelling waves and homogeneous fragmentation
Résumé
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.