Law of large numbers for super-Brownian motions with a single point source

We investigate the super-Brownian motion with a single point source in dimensions 2$2$ and 3$3$ as constructed by Fleischmann and Mueller in 2004. Using analytic facts we derive the long time behavior of the mean in dimensions 2$2$ and 3$3$ thereby complementing previous work of Fleischmann, Mueller and Vogt. Using spectral theory and martingale arguments we prove a version of the strong law of large numbers for the two dimensional superprocess with a single point source and finite variance.