On extreme events for non-spatial and spatial branching Brownian motions

•Branching process with infinite variance.•Branching Bessel motion.•Hitting probability.•Evolutionary genetics and epidemics.

We study the impact of having a non-spatial branching mechanism with infinite variance on some parameters (height, width and first hitting time) of an underlying Bienaymé–Galton–Watson branching process. Aiming at providing a comparative study of the spread of an epidemics whose dynamics is given by the modulus of a branching Brownian motion (BBM) we then consider spatial branching processes in dimension dd, not necessarily integer. The underlying branching mechanism is either a binary branching model or one presenting infinite variance. In particular we evaluate the chance p(x)p(x) of being hit if the epidemics started away at distance xx. We compute the large xx tail probabilities of this event, both when the branching mechanism is regular and when it exhibits very large fluctuations.