Anomalous spreading in reducible multitype branching Brownian motion

Abstract

We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motion and create offspring at constant rate. Particles of type 1 can give birth to particles of types 1 and 2, but particles of type 2 only give birth to descendants of type 2. Under some specific conditions, Biggins (2012) shows that this process exhibit an anomalous spreading behaviour: the rightmost particle at time t is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type 1 or of type 2. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer (2014,2016). The aim of this article is to study the asymptotic behaviour of the position of the furthest particles in the two-type reducible branching Brownian motion, obtaining in particular tight estimates for the median of the maximal displacement.