Longtime behavior of a branching process controlled by branching catalysts

The model under consideration is a catalytic branching model constructed in [DF96], where the catalysts themselves suffer a spatial branching mechanism. Main attention is paid to dimension d=3. The key result is a convergence theorem towards a limit with full intensity (persistence), which in a sense is comparable with the situation for the "classical" continuous super-Brownian motion. As by-products, strong laws of large numbers are derived for the Brownian collision local time controlling the branching of reactants, and for the catalytic occupation time process. Also, the occupation measures are shown to be absolutely continuous