Measure-Valued Branching Markov Processes

ISBN-13: 9783642266201 / Angielski / Miękka / 2013 / 350 str.

Zenghu Li

Opis

A compact and rigorous treatment of measure-valued branching processes and immigration processes is given in the book at the level readable for graduate students. For the convenience of references, special attention has been paid to the generality of the framework. To develop a reasonably rich theory, the basic regularities of the models are certainly necessary. In the first part of the book, the author not only constructs the transition semigroups of superprocesses with general spatial motions and branching mechanisms, but also proves the existence of their Borel right realizations. Based on the general existence and regularity results, he uses transformations to derive the existence and regularity of several different forms of the superprocesses, including those in spaces of tempered measures, time-inhomogeneous superprocesses, multitype models and age-structured models. This unified treatment of the models simplifies their constructions and gives useful perspectives for their properties. The first part of the book contains two chapters discussing respectively one-dimensional branching processes and limit theorems of branching particle systems, which give necessary interpretations and intuitions of the superprocesses. Martingale problems of superprocesses are discussed under Feller type assumptions. In the second and the third parts of the book, the authors gives systematic treatments of immigration superprocesses and generalized Ornstein-Uhlenbeck processes based on skew convolution semigroups. The connection of those two classes of processes is established by certain fluctuation limit theorems. A brief account is also given to state-dependent immigration involving a class of stochastic equations. Those materials are not available in any other books in the subject. Some chapters of the book lean on the general theory of Markov processes. A summary of that is given in the Appendix for the convenience of the reader. In the last section of each chapter, comments on the history and recent developments are given. Those guide the reader to the frontiers of the ongoing research.

Recenzje

From the book reviews:

"The present book appears to be the first monograph in textbook format that provides a rigorous treatment of general theory for a wide class of measure-valued processes. ... A selection of good examples throughout the book helps clarify the results. ... This is a well-written, concise and dedicated book. It will serve excellently as a platform and reference for the next phase of development of superprocesses and measure-valued branching processes." (Ingemar Kaj, Mathematical Reviews, Issue 2012 c)

"The present book is written for the purpose of providing a compact and rigorous basic theory of measured-valued branching processes and immigration processes. ... this book is well organized, the presentation is very systematic and sufficiently simple, and the key points are easy to understand. ... It is recommendable for graduate students and researchers working in the field of measure-valued processes and their related topics, including stochastic processes and probability theory." (Isamu Dôku, Zentralblatt MATH, Vol. 1235, 2012)

"The main purpose of this book is to give a compact and rigorous treatment of the basic theory of measure-valued branching processes and immigration processes. ... This book can be used as a reference of the basics of Dawson-Watanabe superprocesses and immigration superprocesses. It can also be used in a course for graduate students specialized in probability and stochastic processes." (Gyula Pap, Acta Scientiarum Mathematicarum (Szeged), Vol. 77 (3-4), 2011)

Spis treści

Preface.- 1. Random Measures on Metric Spaces.- 2. Measure-valued Branching Processes.- 3. One-dimensional Branching Processes.- 4. Branching Particle Systems.- 5. Basic Regularities of Superprocesses.- 6. Constructions by Transformations.- 7. Martingale Problems of Superprocesses.- 8. Entrance Laws and Excursion Laws.- 9. Structures of Independent Immigration.- 10. State-dependent Immigration Structures.- 11. Generalized Ornstein-Uhlenbeck Processes.- 12. Small Branching Fluctuation Limits.- 13. Appendix: Markov Processes.- Bibliography.- Index.

Blurb

Measure-valued branching processes arise as high density limits of branching particle systems. The Dawson-Watanabe superprocess is a special class of those. The author constructs superprocesses with Borel right underlying motions and general branching mechanisms and shows the existence of their Borel right realizations. He then uses transformations to derive the existence and regularity of several different forms of the superprocesses. This treatment simplifies the constructions and gives useful perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The most important feature of the book is the systematic treatment of immigration superprocesses and generalized Ornstein--Uhlenbeck processes based on skew convolution semigroups.

The volume addresses researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.