Maximal displacement in a branching random walk through interfaces

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.