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

Alexander Alban,Anton Bovier, Annabell Gros,Lisa Hartung


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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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