Measure-Valued Branching Markov Processes

Description

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. 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