Stochastic integral representation and regularity of the density for the exit measure of super-Brownian motion

Abstract

This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+β)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu=u^1+β.