Subcritical superprocesses conditioned on non-extinction

Abstract

We consider a class of subcritical superprocesses (Xt)t≥0 with general spatial motions and general branching mechanisms. We study the asymptotic behaviors of Qt,r, the distribution of Xt conditioned on Xt+r not being a null measure. We first give the existence of limt→∞Qt,r and limr→∞Qt,r, and then show that an LlogL-type condition is equivalent to the existence of the iterated limits: limr→∞limt→∞Qt,r and limt→∞limr→∞Qt,r. Finally, when the LlogL-type condition holds, we show that those iterated limits, and the double limit limr,t→∞Qt,r, are the same.