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

Grummt, R. and Kolb, M. (2013) Law of large numbers for super-brownian motions with a single point source. Stochastic Processes and their Applications, 123 (4). pp. 1183-1212. ISSN 0304-4149 doi: 10.1016/j.spa.2012.12.002

Abstract/Summary

We investigate the super-Brownian motion with a single point source in dimensions 2 and 3 as constructed by Fleischmann and Mueller in 2004. Using analytic facts we derive the long time behavior of the mean in dimension 2 and 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.