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

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.