On the boundary of the support of super-Brownian motion

We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x).a) as a.0 as well as the Hausdorff dimension of the boundary of the support of X(t,.). The answers are in terms of the leading eigenvalue of the Ornstein.Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.