Large deviations for the growth rate of the support of supercritical super-Brownian motion

We prove a large deviation result for the growth rate of the support of the d-dimensional (strictly dyadic) branching Brownian motion Z and the d-dimensional (supercritical) super-Brownian motion X. We show that the probability that Z (X) remains in a smaller than typical ball up to time t is exponentially small in t and we compute the cost function. The cost function turns out to be the same for Z and X. In the proof we use a decomposition result due to Evans and O'Connell and elementary probabilistic arguments. Our method also provides a short alternative proof for the lower estimate of the large time growth rate of the support of X, first obtained by Pinsky by pde methods.