Measure-Valued Branching Markov Processes

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.

• 60J80; 60G57; 60J70; 60J85; 60J35; 60J40
• Dawson-Watanabe superprocess
• entrance law
• generalized Ornstein-Uhlenbeck process
• immigration superprocess
• skew convolution semigroup

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)

• School of Mathematical Sciences, Beijing Normal University, Beijing, China, People's Republic

  Zenghu Li

Policies and ethics