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**Dawson-Watanabe superprocesses and measure-valued diffusions.**
*(English)*
Zbl 1020.60075

Bolthausen, Erwin et al., Lectures on probability theory and statistics. Ecole d’été de probabilités de Saint-Flour XXIX - 1999, Saint-Flour, France, July 8-24, 1999. Berlin: Springer. Lect. Notes Math. 1781, 125-329 (2002).

These lecture notes on superprocesses and related measure-valued processes, by one of the leading specialists in the field, are a welcome addition to the basic literature on the subject. Dawson-Watanabe superprocesses are measure-valued processes that are obtained as rescaling limits of spatial branching particle systems. They have many facets due to their links with nonlinear partial differential equations, stochastic partial differential equations, martingale problems and other topics, and this also allows to approach superprocesses with different perspectives and methods. The term “superprocess” was introduced by Dynkin to convey the idea that such a measure-valued process is built over the Markov process which governs the individual migration in the underlying branching particle system. Various aspects of superprocesses have been covered in lecture notes and monographs by D. A. Dawson [“Measure-valued Markov processes” (1993; Zbl 0799.60080)], E. B. Dynkin [“An introduction to branching measure-valued processes” (1994; Zbl 0824.60001)], A. M. Etheridge [“An introduction to superprocesses” (2000; Zbl 0971.60053)] and J.-F. Le Gall [“Spatial branching processes, random snakes and partial differential equations” (1999; Zbl 0938.60003)].

The goals of the author in the present lecture notes are to give a largely self-contained graduate level course on superprocesses and to cover some of the topics and methods used in the study of interactive measure-valued models. The basic theory of superprocesses, which is treated in Chapters II and III, is a well developed subject. Here the author relies mostly on the approximating branching particle system approach, not striving for generality but rather with the intent of helping the reader to develop an intuition that the particle system can provide towards a better comprehension of the intricacies of superprocesses. Whereas the methods used in the theory of ordinary superprocesses depend substantially on the strong independence assumptions on the approximating particle system, in the study of interacting measure-valued models, where various forms of dependence are considered (e.g. state-dependent drifts, spatial motions and branching rates), the previous techniques are no longer valid, and they must be replaced by suitable modifications and new ideas. This is an area of active current research, and in Chapters IV and V the author presents some of the methods that have been developed and results that have been obtained during the last decade.

The diversity of ideas and assortment of tools that make the theory of superprocesses and measure-valued diffusions such a rich subject, at the same time make its learning a difficult endeavor, specially for students and nonspecialists. The author has written these lecture notes with this concern in mind, and the product will undoubtedly be well appreciated by anyone interested in this topic, from beginners to experts.

For the entire collection see [Zbl 0996.00039].

The goals of the author in the present lecture notes are to give a largely self-contained graduate level course on superprocesses and to cover some of the topics and methods used in the study of interactive measure-valued models. The basic theory of superprocesses, which is treated in Chapters II and III, is a well developed subject. Here the author relies mostly on the approximating branching particle system approach, not striving for generality but rather with the intent of helping the reader to develop an intuition that the particle system can provide towards a better comprehension of the intricacies of superprocesses. Whereas the methods used in the theory of ordinary superprocesses depend substantially on the strong independence assumptions on the approximating particle system, in the study of interacting measure-valued models, where various forms of dependence are considered (e.g. state-dependent drifts, spatial motions and branching rates), the previous techniques are no longer valid, and they must be replaced by suitable modifications and new ideas. This is an area of active current research, and in Chapters IV and V the author presents some of the methods that have been developed and results that have been obtained during the last decade.

The diversity of ideas and assortment of tools that make the theory of superprocesses and measure-valued diffusions such a rich subject, at the same time make its learning a difficult endeavor, specially for students and nonspecialists. The author has written these lecture notes with this concern in mind, and the product will undoubtedly be well appreciated by anyone interested in this topic, from beginners to experts.

For the entire collection see [Zbl 0996.00039].

Reviewer: L.G.Gorostiza (Mexico City)