Finite time extinction of superprocesses with catalysts

Consider a catalytic super-Brownian motion X = X-Gamma 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 cata lyst is a stable random measure Gamma on R of index 0 < <gamma> < 1. Conseq uently, here the catalyst is located in a countable dense subset of R. Star ting with a finite reactant mass Xo supported by a compact set, X is shown to die in finite time. We also deal with two other cases, with a power low catalyst and with a super-random walk on Z(d) with an i.i.d. catalyst. Our probabilistic argument uses the idea of good and bad historical paths o f reactant "particles" during time periods [T-n, Tn+1). Good paths have a s ignificant collision local time with the catalyst, and extinction can be sh own by individual time change according to the collision local time and a c omparison with Feller's branching diffusion. On the other hand, the remaini ng 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 collis ion local time spent on them.