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 LรฉvyโKhintchine 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 DawsonโWatanabe 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 OrnsteinโUhlenbeck 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 RayโKnight Completion
A.5 Entrance Space and Entrance Laws
A.6 Concatenations andWeak Generators
A.7 TimeโSpace Processes
References
Subject Index
Symbol Index