The Annals of Probability

Branching Exit Markov Systems and Superprocesses

E.B. Dynkin

Abstract

Superprocesses (under the name continuous state branchingprocesses) appeared, first, in a pioneering work of S.Watanabe [J. Math. Kyoto Univ. 8 (1968)141 –167 ]. Deep results on paths of the super-Brownian motion were obtained by Dawson, Perkins, Le Gall and others.

In earlier papers, a superprocess was interpreted as a Markov process $X_t$ in the space of measures. This is not sufficient for a probabilistic approach to boundary value problems. A reacher model based on the concept of exit measures was introduced by E.B.Dynkin [Probab. Theory Related Fields 89 (1991) 89 –115 ]. A model of a superprocess as a system of exit measures from time-space open sets was systematically developed in 1993 [E.B. Dynkin, Ann.Probab. 21 (1993)1185 –1262 ]. In particular, branchingand Markov properties of such a system were established and used to investigate partial differential equations. In the present paper, we show that the entire theory of superprocesses can be deduced from these properties.

Article information

Source
Ann. Probab., Volume 29, Number 4 (2001), 1833-1858.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aop/1015345774

Digital Object Identifier
doi:10.1214/aop/1015345774

Mathematical Reviews number (MathSciNet)
MR1880244

Zentralblatt MATH identifier
1014.60079

Citation

Dynkin, E.B. Branching Exit Markov Systems and Superprocesses. Ann. Probab. 29 (2001), no. 4, 1833--1858. doi:10.1214/aop/1015345774. https://projecteuclid.org/euclid.aop/1015345774

References

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• [3] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
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• [5] Sattinger, D. H. (1973). Topics in Stability and Bifurcation Theory. Springer, New York.
• [6] Watanabe, S. (1968). A limit theorem on branchingprocesses and continuous state branchingprocesses. J. Math. Kyoto Univ. 8 141-167.