Measure-valued Branching Markov Processes

by Li, Zenghu

• ISBN13: 9783642150036
• ISBN10: 3642150039
• Format: Hardcover
• Copyright: 2010-11-29
• Publisher: Springer Nature

Summary

Measure-valued branching processes arise as high density limits of branching particle systems. The Dawson-Watanabe superprocess is a special class of those. The author constructs superprocesses with Borel right underlying motions and general branching mechanisms and shows the existence of their Borel right realizations. He then uses transformations to derive the existence and regularity of several different forms of the superprocesses. This treatment simplifies the constructions and gives useful perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The most important feature of the book is the systematic treatment of immigration superprocesses and generalized Ornstein--Uhlenbeck processes based on skew convolution semigroups. The volume addresses researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.

Table of Contents

Preface	p. v
Random Measures on Metric Spaces	p. 1
Borel Measures	p. 1
Laplace Functional	p. 8
Poisson Random Measures	p. 13
Infinitely Divisible Random Measures	p. 15
Lévy-Khintchine Type Representations	p. 21
Notes and Comments	p. 26
Measure-Valued Branching Processes	p. 29
Definitions and Basic Properties	p. 29
Integral Evolution Equations	p. 33
Dawson-Watanabe Superprocesses	p. 38
Examples of Superprocesses	p. 44
Some Moment Formulas	p. 47
Notes and Comments	p. 54
One-Dimensional Branching Processes	p. 57
Continuous-State Branching Processes	p. 57
Long-Time Evolution Rates	p. 63
Immigration and Conditioned Processes	p. 66
More Conditional Limit Theorems	p. 70
Scaling Limits of Discrete Processes	p. 75
Notes and Comments	p. 83
Branching Particle Systems	p. 87
Particle Systems with Local Branching	p. 87
Scaling Limits of Local Branching Systems	p. 92
General Branching Particle Systems	p. 97
Scaling Limits of General Branching Systems	p. 100
Notes and Comments	p. 104
Basic Regularities of Superprocesses	p. 107
Right Continuous Realizations	p. 107
The Strong Markov Property	p. 109
Borel Right Superprocesses	p. 114
Weighted Occupation Times	p. 118
A Counterexample	p. 122
Notes and Comments	p. 125
Constructions by Transformations	p. 127
Spaces of Tempered Measures	p. 127
Multitype Superprocesses	p. 132
A Two-Type Superprocess	p. 135
Change of the Probability Measure	p. 136
Time-In homogeneous Superprocesses	p. 138
Notes and Comments	p. 142
Martingale Problems of Superprocesses	p. 145
The Differential Evolution Equation	p. 145
Generators and Martingale Problems	p. 152
Worthy Martingale Measures	p. 160
A Representation for Superprocesses	p. 168
Transforms by Martingales	p. 172
Notes and Comments	p. 176
Entrance Laws and Excursion Laws	p. 179
Some Simple Properties	p. 179
Minimal Probability Entrance Laws	p. 183
Infinitely Divisible Probability Entrance Laws	p. 187
Kuznetsov Measures and Excursion Laws	p. 193
Super-Absorbing-Barrier Brownian Motions	p. 201
Notes and Comments	p. 205
Structures of Independent Immigration	p. 207
Skew Convolution Semigroups	p. 207
The Canonical Entrance Rule	p. 212
Regular Immigration Superprocesses	p. 215
Constructions of the Trajectories	p. 221
One-Dimensional Stochastic Equations	p. 228
Notes and Comments	p. 231
State-Dependent Immigration Structures	p. 233
Deterministic Immigration Rates	p. 233
Stochastic Immigration Rates	p. 235
Interactive Immigration Rates	p. 241
General Interactive Immigration	p. 247
Notes and Comments	p. 252
Generalized Ornstein-Uhlenbeck Processes	p. 255
Generalized Mehler Semigroups	p. 255
Gaussian Type Semigroups	p. 259
Non-Gaussian Type Semigroups	p. 263
Extensions of Centered Semigroups	p. 266
Construction of the Processes	p. 273
Notes and Comments	p. 279
Small-Branching Fluctuation Limits	p. 281
The Brownian Immigration Superprocess	p. 281
Stochastic Processes in Nuclear Spaces	p. 283
Fluctuation Limits in the Schwartz Space	p. 290
Fluctuation Limits in Sobolev Spaces	p. 296
Notes and Comments	p. 298
Markov Processes	p. 301
Measurable Spaces	p. 301
Stochastic Processes	p. 304
Right Markov Processes	p. 306
Ray-Knight Completion	p. 315
Entrance Space and Entrance Laws	p. 319
Concatenations and Weak Generators	p. 321
Time-Space Processes	p. 329
References	p. 333
Subject Index	p. 345
Symbol Index	p. 349
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