Longtime behavior of a branching process controlled by branching catalysts

The model under consideration is a catalytic branching model constructed in Dawson and Fleischmann (1997), where the catalysts themselves undergo a spatial branching mechanism. The key result is a convergence theorem in dimension d = 3 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 catalytic occupation measures are shown to be absolutely continuous with respect to Lebesgue measure