From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes , which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing and . We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the i.i.d. case , and when . This is due to the localization of extremal particles at the time of speed change, which depends on α and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterize the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion. © 2020 the Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC