On the exit measure of super-Brownian motion
We study some properties of the exit measure of super-Brownian motion from a smooth domain \(D\) in \(\mathbb{R}^ d\). In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of \(\Delta u = u^ 2\) in \(D\), and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of \(\Delta u=u^ 2\) in \(D\) that tends to \(\infty\) on an open subset \(O\) of \(\partial D\) and to 0 on the complement in \(\partial D\) of the closure of \(O\). Our proofs use the path-valued process.
characterization of removable singularities at the boundary
scientific article; zbMATH DE number 639359