MEASURE-VALUED MARKOV BRANCHING-PROCESSES CONDITIONED ON NON-EXTINCTION

We consider a particular class of measure-valued Markov branching processes that are constructed as "superprocesses" over some underlying Markov process. Such a process X dies out almost surely, so we introduce various conditioning schemes which keep X alive at large times. Under suitable hypotheses, which include the convergence of the semigroup for the underlying process to some limiting probability measure ν, we show that the conditional distribution of t -1 X t converges to that of Zν as t → ∞, where Z is some strictly positive, real random variable. © 1990 Hebrew University.