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.
Publié le : 1991-07-14
Classification:
Branching particle systems,
immigration,
measure-valued processes,
superprocesses,
historical processes,
historical superprocesses,
moment functions,
linear additive functionals,
special Markov property,
60J80,
60G57,
60J25,
60J50
@article{1176990339,
author = {Dynkin, E. B.},
title = {Branching Particle Systems and Superprocesses},
journal = {Ann. Probab.},
volume = {19},
number = {4},
year = {1991},
pages = { 1157-1194},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990339}
}
Dynkin, E. B. Branching Particle Systems and Superprocesses. Ann. Probab., Tome 19 (1991) no. 4, pp. 1157-1194. http://gdmltest.u-ga.fr/item/1176990339/