TRAPPING A MEASURE-VALUED MARKOV BRANCHING-PROCESS CONDITIONED ON NON-EXTINCTION

We consider a class of measure-valued Markov branching processes that are constructed as "superprocesses" over some underlying Markov process. We condition such a process to stay away from the zero measure forever. We show that if there is a set of traps that will eventually catch the underlying process then the mass of the conditioned superprocess will eventually be concentrated on a single one of these traps. Moreover, the distribution of the point of eventual concentration is the same as that of the final destination for the underlying process.