Long-term behavior for superprocesses over a stochastic flow

Abstract

We study the limit of a superprocess controlled by a stochastic flow as $t\to\infty$. It is proved that when $d \le 2$, this process suffers long-time local extinction; when $d\ge 3$, it has a limit which is persistent. The stochastic log-Laplace equation conjectured by Skoulakis and Adler (2001) and studied by this author (2004) plays a key role in the proofs like the one played by the log-Laplace equation in deriving long-term behavior for usual super-Brownian motion.