WIAS Preprint No. 431, (1998)

Finite time extinction of super-Brownian motions with catalysts



Authors

  • Dawson, Donald A.
  • Fleischmann, Klaus
  • Mueller, Carl

2010 Mathematics Subject Classification

  • 60J80 60J55 60G57

Keywords

  • catalytic super-Brownian motion, historical superprocess, critical branching, finite time extinction, measure-valued branching, random medium, good and bad paths, stopped measures, collision local time, comparison, coupling, stopped historical superprocess, branching rate functional, super-random walk, interacting Feller's branching diffusion

Abstract

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.

Appeared in

  • Ann. Probab., 28(2) (2000), pp. 603-642

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