Finite time extinction of super-Brownian motions with catalysts

TL;DR

This paper proves that catalytic super-Brownian motions with finite variance branching and a stable random measure catalyst die out in finite time, using probabilistic path analysis and collision local times.

Contribution

It introduces a novel probabilistic approach analyzing good and bad paths to establish finite time extinction for catalytic super-Brownian motions with stable measure catalysts.

Findings

Super-Brownian motion with stable measure catalyst dies out in finite time.

Path analysis distinguishes between significant collision paths and negligible ones.

Extinction is shown via comparison with Feller's branching diffusion.

Abstract

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 $R$ of index $0< gamma <1$. Consequently, here the catalyst is located in a countable dense subset of $R$. Starting with a finite reactant mass $X_0$ 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 $[T_{n},T_{n+1})$. 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 $T_{n+1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.