Overview
- Includes a new chapter on one-dimensional stochastic equations and their links to stochastic flows
- The chapter on state-dependent immigration has been completely rewritten
- New results and examples have been added all throughout the book
Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 103)
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About this book
This book provides a compact introduction to the theory of measure-valued branching processes, immigration processes and Ornstein–Uhlenbeck type processes.
Measure-valued branching processes arise as high density limits of branching particle systems. The first part of the book gives an analytic construction of a special class of such processes, the Dawson–Watanabe superprocesses, which includes the finite-dimensional continuous-state branching process as an example. Under natural assumptions, it is shown that the superprocesses have Borel right realizations. Transformations are then used to derive the existence and regularity of several different forms of the superprocesses. This technique simplifies the constructions and gives useful new perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The second part investigates immigration structures associated with the measure-valued branching processes. The structures are formulated by skewconvolution semigroups, which are characterized in terms of infinitely divisible probability entrance laws. A theory of stochastic equations for one-dimensional continuous-state branching processes with or without immigration is developed, which plays a key role in the construction of measure flows of those processes. The third part of the book studies a class of Ornstein-Uhlenbeck type processes in Hilbert spaces defined by generalized Mehler semigroups, which arise naturally in fluctuation limit theorems of the immigration superprocesses.
This volume is aimed at researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.
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Keywords
- Measure-valued branching processes
- Immigration processes
- Dawson-Watanabe superprocess
- Skew convolution semigroup
- Generalized Ornstein-Uhlenbeck process
- Laplace functionals
- Entrance laws for Dawson-Watanabe superprocess
- Structures of state-dependent immigration
- Poisson random measures
- Fluctuation limit theorems
Table of contents (14 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
About the author
Zenghu Li is a Changjiang Scholar Distinguished Professor at Beijing Normal University and a Fellow of the Institute of Mathematical Statistics. He serves on the editorial board for several mathematical journals and book series, including Acta Mathematica Sinica and Stochastic Processes and Their Applications. His research interests include branching Markov processes, measure-valued Markov processes, stochastic differential equations and applications.
Bibliographic Information
Book Title: Measure-Valued Branching Markov Processes
Authors: Zenghu Li
Series Title: Probability Theory and Stochastic Modelling
DOI: https://doi.org/10.1007/978-3-662-66910-5
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag GmbH Germany, part of Springer Nature 2022
Hardcover ISBN: 978-3-662-66909-9Published: 14 March 2023
Softcover ISBN: 978-3-662-66912-9Published: 15 March 2024
eBook ISBN: 978-3-662-66910-5Published: 13 March 2023
Series ISSN: 2199-3130
Series E-ISSN: 2199-3149
Edition Number: 2
Number of Pages: XV, 475
Number of Illustrations: 1 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Statistics and Computing/Statistics Programs, Statistics and Computing/Statistics Programs