Measure-Valued Branching Markov Processes

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.