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**An introduction to superprocesses.**
*(English)*
Zbl 0971.60053

University Lecture Series. 20. Providence, RI: American Mathematical Society (AMS). 187 p. (2000).

Superprocesses are measure-valued processes that model various types of evolving populations. Two main classes of such processes, Dawson-Watanabe (DW) superprocesses and Fleming-Viot (FV) superprocesses, have been the subject of intensive research during the last two decades and are still rapidly growing. DW superprocesses arise as scaling limits of spatial branching particle systems and FV superprocesses arise from models for genotype frequencies in population genetics. The two are related by the fact that a FV superprocess can be regarded as a DW superprocess conditioned to have total mass 1. DW superprocesses have recently been shown to arise also from other classes of interacting particle systems. Superprocesses are difficult for beginners because they involve several different sorts of non-elementary mathematical tools.

The present book is an introduction which concentrates mainly on DW and FV superprocesses. It is written with the newcomer in mind, the emphasis being on explaining the subject in an accessible way, dealing mostly with simple cases and sometimes sacrificing rigor for intuition. However, the reader should have a background on stochastic processes to fill gaps. Results for more general settings are mentioned along the way. Although some topics and techniques are omitted, the author has covered a large part of the background and tools one needs to know in order to read advanced research papers.

A brief description of the contents is as follows: Chapter 1 introduces DW and FV superprocesses as diffusion approximations of two models from population biology, and they are characterized as solutions of martingale problems. Chapter 2 deals with basic qualitative properties of DW superprocesses. Chapter 3 revolves around Le Gall’s representations of DW superprocesses. Chapter 4 is concerned with relationships between DW and FV superprocesses. Chapter 5 is about the Donnelly-Kurtz representations of DW and FV superprocesses. Chapter 6 goes further into the qualitative properties of DW superprocesses. Chapter 7 introduces interactions into the models with emphasis on methods of Dawson and Perkins; the basic DW and FV superprocesses discussed in the previous chapters are used as building blocks. Chapter 8 discusses the interplay between DW superprocesses and partial differential equations, important contributions being due to Dynkin and Le Gall. Chapter 9 refers to other types of extensions of DW superprocesses.

The present book is an introduction which concentrates mainly on DW and FV superprocesses. It is written with the newcomer in mind, the emphasis being on explaining the subject in an accessible way, dealing mostly with simple cases and sometimes sacrificing rigor for intuition. However, the reader should have a background on stochastic processes to fill gaps. Results for more general settings are mentioned along the way. Although some topics and techniques are omitted, the author has covered a large part of the background and tools one needs to know in order to read advanced research papers.

A brief description of the contents is as follows: Chapter 1 introduces DW and FV superprocesses as diffusion approximations of two models from population biology, and they are characterized as solutions of martingale problems. Chapter 2 deals with basic qualitative properties of DW superprocesses. Chapter 3 revolves around Le Gall’s representations of DW superprocesses. Chapter 4 is concerned with relationships between DW and FV superprocesses. Chapter 5 is about the Donnelly-Kurtz representations of DW and FV superprocesses. Chapter 6 goes further into the qualitative properties of DW superprocesses. Chapter 7 introduces interactions into the models with emphasis on methods of Dawson and Perkins; the basic DW and FV superprocesses discussed in the previous chapters are used as building blocks. Chapter 8 discusses the interplay between DW superprocesses and partial differential equations, important contributions being due to Dynkin and Le Gall. Chapter 9 refers to other types of extensions of DW superprocesses.

Reviewer: L.G.Gorostiza (Mexico City)