One-Sided FKPP Travelling Waves for Homogeneous Fragmentation Processes

Abstract

In this paper we introduce the 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. Moreover, 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 Lévy processes.2010 Mathematics Subject Classification: 60G09, 60J25.In the context of homogeneous fragmentation processes we prove the existence and uniqueness of one-sided travelling waves within a certain range of wave speeds. More precisely, the problem we are concerned with in this paper can be roughly described as follows. Consider the integro-differential equationfor certain c ∈ R + := (0, ∞) and all x ∈ R + 0 := [0, ∞), where the product is taken over all n ∈ N with |π n | ∈ R + . Here the space P is the space of partitions (π n ) n∈N of N and µ is the so-called dislocation measure on P. This notation is introduced in more detail in the next section. We are interested in solutions f : R → [0, 1] of the above equation that satisfy

Survival of homogeneous fragmentation processes with killing

“…The motivation for the killing procedure that we introduce in the present paper, partly stems from its relation to the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) equation. In the context of fragmentation processes this connection is studied in [18]. The role an analogous killing plays with regard to solutions of the FKPP equation in the setting of branching Brownian motions (BBM) was investigated in [15].…”