The extremal process of a cascading family of branching Brownian motion

Abstract

We study the asymptotic behaviour of the extremal process of a cascading family of branching Brownian motions. This is a particle system on the real line such that each particle has a type in addition to his position. Particles of type 1 move on the real line according to Brownian motions and branch at rate 1 into two children of type 1. Furthermore, at rate α, they give birth to children too of type 2. Particles of type 2 move according to standard Brownian motion and branch at rate 1, but cannot give birth to descendants of type 1. We obtain the asymptotic behaviour of the extremal process of particles of type 2.

Extremal process for irreducible multitype branching Brownian motion

“…Remark 1.7 The asymptotic behavior above for irreducible multi-type branching Brownian motion is similar to the one obtained in [2,4] for a single-type branching Brownian motion. Belloum and Mallein [6] and Belloum [7] considered a 2-type reducible branching Brownian motion and their results are quite different. In their model, particles of type 1 move as a Brownian motion with diffusion coefficient σ 2 , reproduce with branching rate β + α and offspring distribution {p k (1)} satisfying p (2,0) (1) = β β+α , p (1,1) (1) = α α+β .…”