Branching particle systems and superprocesses. (English) Zbl 0732.60092

Dynkin, E. B.

Ann. Probab. 19, No. 3, 1157-1194 (1991).

A general class of branching particle systems with immigration and general offspring distribution and with branching rate given by an additive functional is formulated. This leads in the high density limit to the construction of a very general class of superprocesses. For example, this formulation is sufficiently general to include the construction of the corresponding “historical process” in which the state of a particle includes its past trajectory. An important accomplishment of this paper is the study of linear positive functionals of superprocesses which are then used to construct a random measure on trajectories stopped at a random time, for example, the exit time from a domain. A related special Markov property is proved. These results have subsequently been applied to characterize the class of “S-polar” sets and to exploit the relations between superprocesses and a class of nonlinear partial differential equations [see author, A probabilistic approach to one class of nonlinear differential equations (Preprint, 1990)].

Reviewer: D.A.Dawson (Ottawa)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G57	Random measures
60J25	Continuous-time Markov processes on general state spaces
60J50	Boundary theory for Markov processes

Keywords:

branching particle systems; additive functional; historical process; Markov property; superprocesses