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**Brownian excursions, trees and measure-valued branching processes.**
*(English)*
Zbl 0753.60078

Measure-valued branching processes, also called superprocesses, arise as limits of branching particle systems. An important example is the continuous super-Brownian motion and this process has been intensively studied in recent years. In the present paper an ingeneous new construction of this process is given. This construction is based on a description of the underlying branching mechanism of the superprocess in terms of the tree structure of a Poisson process of Brownian excursions. In addition this also allows the construction of the historical process also studied by D. A. Dawson and E. A. Perkins [Mem. Am. Math. Soc. 454 (1991)] and E. B. Dynkin [Probab. Theory Relat. Fields 90, No. 1, 1-36 (1991; Zbl 0727.60095)]. Several applications of this new construction are also given. One application is a derivation of the Palm measure of the fixed time random measure which is also obtained by Dawson and Perkins (loc. cit). In addition a probabilistic description of the historical process at a fixed time is obtained in terms of an infinite binary tree which describes the “family structure”. Another application is to the study of the discontinuities of the support of super-Brownian motion including a new proof of a result of E. Perkins [Ann. Probab. 18, No. 2, 453-491 (1990; Zbl 0721.60046)]. To conclude this new construction promises to provide an important tool for the study of superprocesses.

Reviewer: D.A.Dawson (Ottawa)

### MSC:

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60J65 | Brownian motion |

60J60 | Diffusion processes |

60G57 | Random measures |

60G17 | Sample path properties |

60J55 | Local time and additive functionals |