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. (C) 2003 Elsevier B.V. All rights reserved.