Maximal displacement of a branching random walk in time-inhomogeneous environment

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length nn of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time nn. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order n1/3n1/3, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement.