Measure-Valued Branching Markov Processes

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 6 Preface to the First Edition 8 Contents 11 Conventions and Notations 15 Chapter 1 Random Measures on Metric Spaces 16 1.1 Borel Measures 16 1.2 Laplace Functionals 24 1.3 Poisson Random Measures 28 1.4 Infinitely Divisible Random Measures 31 1.5 Lévy–Khintchine Type Representations 36 1.6 Notes and Comments 43 Chapter 2 Measure-Valued Branching Processes 45 2.1 Definitions and Basic Properties 45 2.2 Integral Evolution Equations 50 2.3 Dawson–Watanabe Superprocesses 55 2.4 Examples of Superprocesses 62 2.5 Some Moment Formulas 64 2.6 Variations of Transition Probabilities 72 2.7 Notes and Comments 76 Chapter 3 One-Dimensional Branching Processes 79 3.1 Continuous-State Branching Processes 79 3.2 Long-Time Evolution Rates 86 3.3 Immigration and Conditioned Processes 89 3.4 More Conditional Limit Theorems 94 3.5 Scaling Limits of Discrete Processes 100 3.6 Notes and Comments 108 Chapter 4 Branching Particle Systems 113 4.1 Particle Systems with Local Branching 113 4.2 Scaling Limits of Local Branching Systems 118 4.3 General Branching Particle Systems 123 4.4 Scaling Limits of General Branching Systems 126 4.5 Notes and Comments 130 Chapter 5 Basic Regularities of Superprocesses 132 5.1 Right Continuous Realizations 132 5.2 The Strong Markov Property 134 5.3 Borel Right Superprocesses 139 5.4 Weighted Occupation Times 143 5.5 A Counterexample 149 5.6 Bounds for the Cumulant Semigroup 152 5.7 Notes and Comments 155 Chapter 6 Constructions by Transformations 156 6.1 Spaces of Tempered Measures 156 6.2 Multitype Superprocesses 162 6.3 Two-Type Superprocesses 165 6.4 A Change of the Probability Measure 166 6.5 Time-Inhomogeneous Superprocesses 168 6.6 Notes and Comments 173 Chapter 7 Martingale Problems of Superprocesses 177 7.1 The Differential Evolution Equation 177 7.2 Generators and Martingale Problems 185 7.3 Worthy Martingale Measures 195 7.4 A Stochastic Convolution Formula 204 7.5 Transforms by Martingales 208 7.6 Notes and Comments 212 Chapter 8 Entrance Laws and Kuznetsov Measures 216 8.1 Some Simple Properties 216 8.2 Minimal Probability Entrance Laws 220 8.3 Infinitely Divisible Probability Entrance Laws 226 8.4 Kuznetsov Measures and Excursion Laws 232 8.5 Cluster Representations of the Process 240 8.6 Super-Absorbing-Barrier Brownian Motions 245 8.7 Notes and Comments 250 Chapter 9 Structures of Independent Immigration 252 9.1 Skew Convolution Semigroups 252 9.2 Properties of Transition Probabilities 258 9.3 Regular Immigration Superprocesses 262 9.4 Characterizations by Martingale Problems 267 9.5 Constructions of the Trajectories 272 9.6 Stationary Distributions and Ergodicities 280 9.7 Notes and Comments 287 Chapter 10 One-Dimensional Stochastic Equations 289 10.1 Existence and Uniqueness of Solutions 289 10.2 The Lamperti Transformations 300 10.3 Distributional Properties of Jumps 303 10.4 Local and Global Maximal Jumps 306 10.5 A Generalized CBI-process 310 10.6 Notes and Comments 315 Chapter 11 Path-Valued Processes and Stochastic Flows 318 11.1 Path-Valued Growing Processes 318 11.2 The Total Population Process 324 11.3 Construction by Stochastic Equations 330 11.4 A Stochastic Flow of Measures 333 11.5 The Excursion Law 339 11.6 Notes and Comments 343 Chapter 12 State-Dependent Immigration Structures 346 12.1 Inhomogeneous Immigration Rates 346 12.2 Predictable Immigration Rates 351 12.3 State-Dependent Immigration Rates 360 12.4 Changes of the Branching Mechanism 368 12.5 Notes and Comments 372 Chapter 13 Generalized Ornstein–Uhlenbeck Processes 374 13.1 Generalized Mehler Semigroups 374 13.2 Gaussian Type Semigroups 378 13.3 Non-Gaussian Type Semigroups 382 13.4 Extensions of Centered Semigroups 385 13.5 Construction of the Processes 392 13.6 Notes and Comments 397 Chapter 14 Small-Branching Fluctuation Limits 400 14.1 The Brownian Immigration Superprocess 400 14.2 Stochastic Processes in Nuclear Spaces 402 14.3 Fluctuation Limits in the Schwartz Space 409 14.4 Fluctuation Limits in Sobolev Spaces 416 14.5 Notes and Comments 417 Appendix A Markov Processes 419 A.1 Measurable Spaces 419 A.2 Stochastic Processes 422 A.3 Right Markov Processes 424 A.4 Ray–Knight Completion 433 A.5 Entrance Space and Entrance Laws 437 A.6 Concatenations andWeak Generators 441 A.7 Time–Space Processes 451 References 456 Subject Index 473 Symbol Index 479