Maxima of branching random walks with piecewise constant variance

Abstract: This article extends the results of Fang & Zeitouni (2012a) on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0,1] under a slight restriction. We find the asymptotics of the maximum up to an OP(1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein (2015b) when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang & Zeitouni (2012a) instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.