Branching diffusions, superdiffusions and random media. (English) Zbl 1189.60143

Engländer, János

Probab. Surv. 4, 303-364 (2007).

Summary: Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today’s research in probability theory is “random media”, including the now classical problems on “Brownian motion among obstacles” and the more recent “random walks in random environment” and “catalytic branching” models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in deterministic environments (sections 2-6) and in random media (sections 7-11).

MSC:

60J60	Diffusion processes
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60-02	Research exposition (monographs, survey articles) pertaining to probability theory

Keywords:

spatial branching processes; branching diffusions; measure-valued processes; superprocesses; catalytic branching; law of large numbers; spine decomposition; nonlinear h-transform; local extinction; compact support property; mild obstacles; random media; random environment; second order elliptic operators; criticality theory