The extremal process of two-speed branching Brownian motion

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 $s\le bt$ and ${\sigma }_2$ when $bt\leq s\leq t$\mathrm{.}Inthecase$\sigma_1>\sigma_2$, 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.