Finite time extinction of superprocesses with catalysts

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

Ann. Probab., Tome 28 (2000) no. 1, p. 603-642 / Harvested from Project Euclid

Résumé

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 catalyst is a stable random measure $\Gamma$ on $\mathsf{R}$ of index $0 <\gamma<1$. Consequently, here the catalyst is located in a countable dense subset of $\mathsf{R}$. Starting with a finite reactant mass $X_0$ 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 $\mathsf{Z^d}$ withan i.i.d.catalyst. ¶ Our probabilistic argument uses the idea of good and bad historical paths of reactant “particles ”during time periods $[T_n, T_{n +1}$. Good paths have a signi .cant 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 $T_{n +1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.

Détail

Publié le : 2000-04-14
Classification:  Catalytic super-Brownian motion,  historical superprocess,  critical branching,  finite time extinction,  finite time survival,  measure-valued branching,  random medium,  good and bad paths,  stopped measures,  collision local time,  comparison,  stopped historical superprocess,  branching rate functional,  super-random walk,  interacting Feller’s branching diffusion,  stable catalyst.,  60J80,  60J55,  60G57