The Annals of Probability

Branching Exit Markov Systems and Superprocesses

E.B. Dynkin

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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

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

First available in Project Euclid: 5 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Superprocesses exit measures branching property Markov property transition operators branching particle systems


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

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  • [1] Berg, C., Christiansen, J. P. T. and Ressel, P. (1984). Harmonic Analysis on Semigroups. Springer, New York.
  • [2] Dynkin, E. B. (1991). A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Related Fields 89 89-115.
  • [3] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
  • [4] Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branchingprocesses. Israel J. Math. 64 337-361.
  • [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.