From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion

TL;DR

This paper investigates the maximum of variable-speed branching Brownian motion, revealing how the log-correction depends on the speed function's convergence and form, and showing the extremal process resembles that of standard BBM.

Contribution

It extends the understanding of maxima in variable-speed BBM, providing explicit log-corrections and demonstrating the extremal process's similarity to standard BBM.

Findings

Log-correction depends on convergence rate of speed functions near 0 and 1.

Any log-correction larger than 3/2√2 ln t can be achieved with certain speed functions.

Limiting law of maximum and extremal process matches standard BBM.

Abstract

We study the extremes of variable speed branching Brownian motion (BBM) where the time-dependent "speed functions", which describe the time-inhomogeneous variance, converge to the identity function. We consider general speed functions lying strictly below their concave hull and piecewise linear, concave speed functions. In the first case, the log-correction for the order of the maximum depends only on the rate of convergence of the speed function near 0 and 1 and exhibits a smooth interpolation between the correction in the i.i.d. case, $\frac{1}{2\sqrt{2}} \ln t$, and that of standard BBM, $\frac{3}{2\sqrt{2}} \ln t$. In the second case, we describe the order of the maximum in dependence of the form of speed function and show that any log-correction larger than $\frac{3}{2\sqrt{2}} \ln t$ can be obtained. In both cases, we prove that the limiting law of the maximum and the extremal process essentially coincide with those of standard BBM, using a first and second moment method which relies on the localisation of extremal particles. This extends the results of Bovier and Hartung for two-speed BBM.