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 skew convolution 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.
Author(s): Zenghu Li
Series: Probability Theory and Stochastic Modelling 103
Edition: 2
Publisher: Springer-Verlag GmbH
Year: 2023
Language: English
Pages: 475
City: Berlin, Germany
Tags: branching Processes, Random Measures, Particle Systems, Kuznetsov Measures, Generalized Ornstein-Uhlenbeck Processes
Preface to the Second Edition
Preface to the First Edition
Contents
Conventions and Notations
Chapter 1 Random Measures on Metric Spaces
1.1 Borel Measures
1.2 Laplace Functionals
1.3 Poisson Random Measures
1.4 Infinitely Divisible Random Measures
1.5 Lévy–Khintchine Type Representations
1.6 Notes and Comments
Chapter 2 Measure-Valued Branching Processes
2.1 Definitions and Basic Properties
2.2 Integral Evolution Equations
2.3 Dawson–Watanabe Superprocesses
2.4 Examples of Superprocesses
2.5 Some Moment Formulas
2.6 Variations of Transition Probabilities
2.7 Notes and Comments
Chapter 3 One-Dimensional Branching Processes
3.1 Continuous-State Branching Processes
3.2 Long-Time Evolution Rates
3.3 Immigration and Conditioned Processes
3.4 More Conditional Limit Theorems
3.5 Scaling Limits of Discrete Processes
3.6 Notes and Comments
Chapter 4 Branching Particle Systems
4.1 Particle Systems with Local Branching
4.2 Scaling Limits of Local Branching Systems
4.3 General Branching Particle Systems
4.4 Scaling Limits of General Branching Systems
4.5 Notes and Comments
Chapter 5 Basic Regularities of Superprocesses
5.1 Right Continuous Realizations
5.2 The Strong Markov Property
5.3 Borel Right Superprocesses
5.4 Weighted Occupation Times
5.5 A Counterexample
5.6 Bounds for the Cumulant Semigroup
5.7 Notes and Comments
Chapter 6 Constructions by Transformations
6.1 Spaces of Tempered Measures
6.2 Multitype Superprocesses
6.3 Two-Type Superprocesses
6.4 A Change of the Probability Measure
6.5 Time-Inhomogeneous Superprocesses
6.6 Notes and Comments
Chapter 7 Martingale Problems of Superprocesses
7.1 The Differential Evolution Equation
7.2 Generators and Martingale Problems
7.3 Worthy Martingale Measures
7.4 A Stochastic Convolution Formula
7.5 Transforms by Martingales
7.6 Notes and Comments
Chapter 8 Entrance Laws and Kuznetsov Measures
8.1 Some Simple Properties
8.2 Minimal Probability Entrance Laws
8.3 Infinitely Divisible Probability Entrance Laws
8.4 Kuznetsov Measures and Excursion Laws
8.5 Cluster Representations of the Process
8.6 Super-Absorbing-Barrier Brownian Motions
8.7 Notes and Comments
Chapter 9 Structures of Independent Immigration
9.1 Skew Convolution Semigroups
9.2 Properties of Transition Probabilities
9.3 Regular Immigration Superprocesses
9.4 Characterizations by Martingale Problems
9.5 Constructions of the Trajectories
9.6 Stationary Distributions and Ergodicities
9.7 Notes and Comments
Chapter 10 One-Dimensional Stochastic Equations
10.1 Existence and Uniqueness of Solutions
10.2 The Lamperti Transformations
10.3 Distributional Properties of Jumps
10.4 Local and Global Maximal Jumps
10.5 A Generalized CBI-process
10.6 Notes and Comments
Chapter 11 Path-Valued Processes and Stochastic Flows
11.1 Path-Valued Growing Processes
11.2 The Total Population Process
11.3 Construction by Stochastic Equations
11.4 A Stochastic Flow of Measures
11.5 The Excursion Law
11.6 Notes and Comments
Chapter 12 State-Dependent Immigration Structures
12.1 Inhomogeneous Immigration Rates
12.2 Predictable Immigration Rates
12.3 State-Dependent Immigration Rates
12.4 Changes of the Branching Mechanism
12.5 Notes and Comments
Chapter 13 Generalized Ornstein–Uhlenbeck Processes
13.1 Generalized Mehler Semigroups
13.2 Gaussian Type Semigroups
13.3 Non-Gaussian Type Semigroups
13.4 Extensions of Centered Semigroups
13.5 Construction of the Processes
13.6 Notes and Comments
Chapter 14 Small-Branching Fluctuation Limits
14.1 The Brownian Immigration Superprocess
14.2 Stochastic Processes in Nuclear Spaces
14.3 Fluctuation Limits in the Schwartz Space
14.4 Fluctuation Limits in Sobolev Spaces
14.5 Notes and Comments
Appendix A Markov Processes
A.1 Measurable Spaces
A.2 Stochastic Processes
A.3 Right Markov Processes
A.4 Ray–Knight Completion
A.5 Entrance Space and Entrance Laws
A.6 Concatenations andWeak Generators
A.7 Time–Space Processes
References
Subject Index
Symbol Index
