Measure-Valued Branching Markov Processes

This book provides a comprehensive introduction to measure-valued branching processes, immigration processes, and Ornstein-Uhlenbeck type processes. It offers an analytic construction of Dawson-Watanabe superprocesses, discusses Martingale problems, investigates immigration structures, and studies Ornstein-Uhlenbeck type processes in Hilbert spaces. It is suitable for researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.

Format: Hardback
Length: 475 pages
Publication date: 14 March 2023
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG

This comprehensive book offers a concise yet thorough introduction to the theory of measure-valued branching processes, immigration processes, and Ornstein-Uhlenbeck type processes. Measure-valued branching processes emerge as high-density limits of branching particle systems, and the first part of the book focuses on their analytic construction. It introduces a special class of such processes, known as Dawson-Watanabe superprocesses, which encompass a wide range of branching phenomena, including finite-dimensional continuous-state branching processes. Through the use of transformations, the book demonstrates how these superprocesses can be derived and analyzed, simplifying the constructions and providing valuable new perspectives. Martingale problems related to superprocesses are explored under Feller type assumptions, while the second part delves into the investigation of immigration structures associated with measure-valued branching processes. These structures are formulated using skewconvolution semigroups, characterized by infinitely divisible probability entrance laws. A theory of stochastic equations is developed for one-dimensional continuous-state branching processes with or without immigration, which plays a crucial role in constructing measure flows of these processes. The third part of the book explores a class of Ornstein-Uhlenbeck type processes in Hilbert spaces defined by generalized Mehler semigroups. These processes arise naturally in fluctuation limit theorems of the immigration superprocesses and offer insights into complex systems with branching and migration dynamics.

This volume is designed for researchers and graduate students interested in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and probability theory and stochastic processes. It provides a solid foundation for understanding these complex phenomena and their applications in various fields.

Weight: 898g
Dimension: 235 x 155 (mm)
ISBN-13: 9783662669099
Edition number: 2nd ed. 2022