Lower functions for the support of super-Brownian motion

The aim of this paper is to describe the minimum speed at which a super-Brownian motion starting at the Dirac mass at 0 moves away from its initial point. More precisely, we consider the class of functions {ϕκ(t)=2t(log(1/t)+κ log log(1/t)), κ∈R} and then determine the values of κ such that the support of super-Brownian motion exits the ball of radius ϕκ(t) before time t, for every t small enough.