From $1$ to $6$: a finer analysis of perturbed branching Brownian motion

TL;DR

This paper investigates how the maximum of two-speed branching Brownian motion transitions smoothly between different regimes as the slopes approach the standard case, revealing new localization phenomena and characterizing extremal processes.

Contribution

It introduces a detailed analysis of the transition in the maximum's logarithmic correction for two-speed branching Brownian motion near the standard case, with new localization insights.

Findings

Logarithmic correction interpolates smoothly between iid and standard BBM cases.

Extremal particles localize at the speed change time depending on alpha.

Extremal process essentially matches that of standard branching Brownian motion.

Abstract

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $\sigma_1^2=\sigma_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $\sigma_1^2=1\pm t^{-\alpha}$ and $\sigma_2^2=1\pm t^{-\alpha}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<\alpha<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depends on $\alpha$ and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion.