Measure-Valued Branching Markov Processes [Paperback]

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.

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 book simplifies the constructions and gives useful perspectives.

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 transformal#Ê