Conditioned superprocesses and their weighted occupation times

A superprocess which starts at a finite measure will die in finite time. We consider a class of measure-valued Markov processes obtained by conditioning such a process to stay alive forever. More specifically, we study the asymptotic behavior of the weighted occupation times for these "conditioned superprocesses". We give necessary and sufficient conditions, based on the asymptotics of the underlying motion, for the total occupation time to be infinite. Some special cases are investigated. We also prove that, when properly scaled, the occupation times for conditioned super-Brownian motion converge to a function of the local time when d [less-than-or-equals, slant] 3.