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

arxiv（2024）

摘要

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,
1/2√(2)ln t, and that of standard BBM, 3/2√(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
3/2√(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.

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