Some properties of the exit measure for super brownian motion

We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ^d. We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂ D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4].