Abstract
We consider a two-type reducible branching Brownian motion, defined as a two type branching particle system on the real line, in which particles of type $1$ can give birth to particles of type $2$, but not reciprocally. This process has been shown by Biggins to exhibit an anomalous spreading behaviour under specific conditions: in that situation, the rightmost particle at type $t$ 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. The aim of this article is to refine the previous results and study the asymptotic behaviour of the extremal process of the two-type reducible branching Brownian motion. If the branching Brownian motion exhibits an anomalous spreading behaviour, its asymptotic differs from what it typically expected in branching Brownian motions.
Highlights
The standard branching Brownian motion is a particle system on the real line that can be constructed as follows
We compare our main results for the asymptotic behaviour of the two-type reducible branching Brownian motion to the pre-existing literature
We begin by introducing the optimization problem associated to the computation of the speed of the rightmost particle in this process
Summary
The standard branching Brownian motion is a particle system on the real line that can be constructed as follows. The behaviour of the particles at the tip of branching Brownian motions was later investigated by Aidékon, Berestycki, Brunet and Shi [1] as well as Arguin, Bovier and Kistler [2, 3, 4] They proved that the centred extremal process of the standard BBM, defined by. Maillard [36] obtained a characterization of this type of point processes as satisfying a stability by superposition property This characterization was used in [35] to prove a similar convergence in distribution to a DPPP for the shifted extremal process of the branching random walk. We begin with the asymptotic behaviour of extremal particles when (β, σ2) ∈ CI , in which case the extremal point measure is similar to the one observed in a branching Brownian motion of particles of type 1.
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