The Annals of Probability

Branching Particle Systems and Superprocesses

E. B. Dynkin

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990339

Digital Object Identifier
doi:10.1214/aop/1176990339

Mathematical Reviews number (MathSciNet)
MR1112411

Zentralblatt MATH identifier
0732.60092

JSTOR
links.jstor.org

Subjects
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

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

Citation

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


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