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Superprocess over a stochastic flow with superprocess catalyst. (English) Zbl 1063.60123

Summary: We study the catalytic superprocesses under a stochastic flow where the catalyst itself is a superprocess under the same flow. Comparing with the study of the superprocess under a stochastic flow with deterministic catalyst, here we encounter a serious adaptability problem caused by this common stochastic flow in proving the uniqueness by using the usual conditional log-Laplace transform approach. To overcome this difficulty, we find a limiting moment dual and show that the moments increase not too fast so that the moments determine the distribution. We also prove the state property by the moment method.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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