The Annals of Probability

Branching Particle Systems and Superprocesses

E. B. Dynkin

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We start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location. Then we consider a limit case when the mass of particles and their life times are small and their density is high. This way, we construct a measure-valued process $X_t$ which we call a superprocess. Replacing the underlying Markov process $\xi_t$ by the corresponding "historical process" $\xi_{\leq t}$, we construct a measure-valued process $M_t$ in functional spaces which we call a historical superprocess. The moment functions for superprocesses are evaluated. Linear positive additive functionals are studied. They are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path (say, at the first exit time from a domain). A related special Markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations. The concluding section is devoted to a survey of the literature, and the terminology on Markov processes used in the paper is explained in the Appendix.

Article information

Ann. Probab., Volume 19, Number 3 (1991), 1157-1194.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Branching particle systems immigration measure-valued processes superprocesses historical processes historical superprocesses moment functions linear additive functionals special Markov property


Dynkin, E. B. Branching Particle Systems and Superprocesses. Ann. Probab. 19 (1991), no. 3, 1157--1194. doi:10.1214/aop/1176990339.

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