The Behavior of Superprocesses Near Extinction

Abstract

In this paper we use a martingale problem characterization to study the behavior of finite measure valued superprocesses with a variety of spatial motions. In general the superprocess, when normalized to be a probability, will converge to a point mass at its extinction time. For some spatial motions we prove that there are times near extinction at which the closed support of the process is concentrated near one point. We obtain a Tanaka formula for the measure of a half space under a one dimensional symmetric stable superprocess of index $\alpha$ and we show this process fails to be a semimartingale if $1 < \alpha \leq 2$.