Finite time extinction of super-Brownian motions with catalysts

Consider a catalytic super-Brownian motion X = XΓ with finite variance branching. Here "catalytic" means that branching of the reactant X is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure Γ on R of index 0 < γ < 1. Consequently, here the catalyst is located in a countable dense subset of R. Starting with a finite reactant mass X0 supported by a compact set, X is shown to die in finite time. Our probabilistic argument uses the idea of good and bad historical paths of reactant "particles" during time periods [Tn,Tn+1). Good paths have a significant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller's branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time Tn+1 which can be controlled by the hitting probability of point catalysts and the collision local time spent on them