LAUSR

We consider a catalytic branching Brownian motion with general branching which takes place only when particles are at the origin at a rate β > 0 on the local time scale. We first establish a spine decomposition for the case wherein the particles have a positive probability of having no children. Then using this tool, we obtain results regarding the asymptotic behavior of the number of particles above λ t at time t for λ > 0. Under an L log L condition, we prove a strong law of large numbers for this catalytic branching Brownian motion.