The behaviour of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyse a time-inhomogeneous model with two level of randomness. In the first step a sequence of branching law is sampled independently according to a distribution on point measures. Conditionally on the realisation of this sequence (called environment) we define a branching random walk and find the asymptotic behaviour of its maximal particle. It is of the form V n − ϕ log n + o P (log n), where V n is a function of the environment that behaves as a random walk and ϕ > 0 is a constant. It turns out that the logarithmic correction ϕ is bigger than the usual one of the homogenous branching case.
In the companion paper REF we show that the randomness of the environment has a slowing effect on the position of the maximal particle.
Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment
Mathematics