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 22 and 33 as constructed by Fleischmann and Mueller in 2004. Using analytic facts we derive the long time behavior of the mean in dimensions 22 and 33 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.