Some properties of superprocesses conditioned on non-extinction

Abstract

We simply call a superprocess conditioned on non-extinction a conditioned superprocess. In this study, we investigate some properties of the conditioned superprocesses (subcritical or critical). Firstly, we give an equivalent description of the probability of the event that the total occupation time measure on a compact set is finite and some applications of this equivalent description. Our results are extensions of those of Krone (1995) from particular branching mechanisms to general branching mechanisms. We also prove a claim of Krone for the cases of d = 3, 4. Secondly, we study the local extinction property of the conditioned binary super-Brownian motion {X _t , P ^∞_µ }. When d = 1, as t goes to infinity, X _t /$\sqrt t$ converges to ηλ in weak sense under P ^∞_µ , where η is a nonnegative random variable and λ is the Lebesgue measure on ℝ. When d ⩾ 2, the conditioned binary super-Brownian motion is locally extinct under P ^∞_µ .