The extremal process of two-speed branching Brownian motion. (English) Zbl 1288.60108

Bovier, Anton; Hartung, Lisa Bärbel

Electron. J. Probab. 19, Paper No. 18, 28 p. (2014).

Authors’ abstract: We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by M. Fang and O. Zeitouni [J. Stat. Phys. 149, No. 1, 1–9 (2012; Zbl 1259.82141)], 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 L.-P. Arguin et al. [Probab. Theory Relat. Fields 157, No. 3–4, 535–574 (2013; Zbl 1286.60045)].

Reviewer: P. R. Parthasarathy (Chennai)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Keywords:

branching Brownian motion; extremal processes; extreme values; F-KPP equation; cluster \(t\) processes

Citations:

Zbl 1259.82141; Zbl 1286.60045