The extremal process of two-speed branching Brownian motion

TL;DR

This paper analyzes the extremal process of two-speed branching Brownian motion with piecewise constant speeds, revealing different behaviors depending on the speed parameters and introducing new cluster point processes.

Contribution

It constructs and describes the extremal process for variable speed BBM with piecewise constant speeds, including the novel case when the faster speed occurs later.

Findings

Extremal process is a concatenation of two BBM extremal processes when the first speed is higher.

A new family of cluster point processes emerges when the second speed is higher.

The approach follows the strategy of Arguin, Bovier, and Kistler.

Abstract

We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $\sigma_1$ for $s\leq bt$ and $\sigma_2$ when $bt\leq s\leq t$. In the case $\sigma_1>\sigma_2$, the process is the concatenation of two BBM extremal processes, as expected. In the case $\sigma_1<\sigma_2$, a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.