Nonlocal branching superprocesses and some related models
The authors propose a new formulation of nonlocal branching superprocesses (or measure-valued branching processes), which enables us to treat in a unified manner distinct types of superprocesses, such as the rebirth, the multitype, the mass-structured, the multilevel and the age-reproduction-structured superprocesses and the superprocess-controlled immigration process. Those models arise in different circumstances of application and have been investigated independently with their own theoretical interests. Actually, such a unified treatment is realized by obtaining the nonlocal branching superprocesses as the so-called high density limit of some specific branching particle systems. Then the above-mentioned distinct sorts of superprocesses are derived in comparatively easy ways from the limit superprocess. It is quite certain that this newly developed treatment simplifies the proof of existence of the old classes of superprocesses. The secret of this unification consists in consideration of an enriched underlying state space \(E \times I\) instead of \(E\) itself (where \(E\) is a Lusin topological space generally). Then the authors show that different types of superprocess models can be derived respectively by proper choices of specific space \(I\): for example, to take \(I = (0, \infty)\) for the mass-structured superprocess, \(I = [0, \infty) \times {\mathbb N}\) for the age-reproduction-structured superprocess, and \(I = M(S)\) (the space of finite measures on \(S\)) for a two-level superprocess, etc.
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