MEASURE-VALUED BRANCHING MARKOV PROCESSES
Zenghu Li
A compact and
rigorous treatment of Dawson--Watanabe superprocesses and immigration
superprocesses is given in the book at the level readable for graduate
students. For the convenience of reference, special attention has been paid to
the generality of the framework. The basic regularities of the models are
established to develop a reasonably rich theory.
In the first
part of the book, the author not only constructs the transition semigroups of
the superprocesses with Borel right spatial motions and general 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 classes of
superprocesses, including those in spaces of tempered measures, multitype
models, age-structured models and time-inhomogeneous models. This unified
treatment of the models simplifies their constructions and gives useful
perspectives for their properties. The first part also 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 third parts of the book, the author 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
integral equations. Those materials are not available in any other books on
superprocesses.
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.
This book was
published by Springer in 2011. Click here
to go to the publisher’s page.
Click here for a review in MathSciNet (AMS), here for a remark on Chapter Ten, and here for a list of corrections.