The extremal process of a cascading family of branching Brownian motion

TL;DR

This paper investigates the long-term behavior of the extremal positions in a multi-type branching Brownian motion system, focusing on particles of type 2, revealing their asymptotic distribution and extremal process characteristics.

Contribution

It introduces a cascading multi-type branching Brownian motion model and characterizes the asymptotic extremal process for particles of type 2, extending classical results to a new multi-type setting.

Findings

Asymptotic distribution of extremal particles of type 2 established

Extremal process of type 2 particles characterized

New multi-type branching Brownian motion model analyzed

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 $\alpha$, 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 $2$.