One-Sided FKPP Travelling Waves for Homogeneous Fragmentation Processes

Abstract

In this paper, we introduce an analogue of the classical one-sided FKPP equation in the context of homogeneous fragmentation processes. The main result of the present paper is concerned with the existence and uniqueness of one-sided FKPP travelling waves in this setting. In addition, we prove some analytic properties of such travelling waves. Our techniques make use of fragmentation processes with killing, an associated product martingale as well as various properties of Levy processes. This paper is mainly concerned with general fragmentation processes, but we also devote a section to related considerations regarding fragmentations with a finite dislocation measure, where we obtain a stronger result than for fragmentation processes with an infinite jump activity over finite time horizons. Furthermore, we discuss the relation of our problem to similar questions in the setting of branching Brownian motions, which provides a motivation for our approach.