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 motions and create offspring at a 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  shows that this process exhibits 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 [26, 27]. 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.
The authors are partially funded by ANR-16-CE93-0003 (ANR MALIN). Additionally, M.A.B. is partially supported by Cofund MathInParis project from FSMP. This program has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 754362.
We are grateful to Nina Gantert for several discussions that helped improve the quality of this paper, as well as to the anonymous referee for helpful comments on the previous version of the manuscript.
"Anomalous spreading in reducible multitype branching Brownian motion." Electron. J. Probab. 26 1 - 39, 2021. https://doi.org/10.1214/21-EJP629