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Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Book Description

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes.

This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Notation
  6. Preface
  7. Chapter 1. SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS
    1. 1.1 Dirichlet Forms and Extended Dirichlet Spaces
    2. 1.2 Excessive Functions and Capacities
    3. 1.3 Quasi-Regular Dirichlet Forms
    4. 1.4 Quasi-Homeomorphism of Dirichlet Spaces
    5. 1.5 Symmetric Right Processes and Quasi-Regular Dirichlet Forms
  8. Chapter 2. BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS
    1. 2.1 Transience, Recurrence, and Irreducibility
    2. 2.2 Basic Examples
    3. 2.3 Analytic Potential Theory for Regular Dirichlet Forms
    4. 2.4 Local Properties
  9. Chapter 3. SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS
    1. 3.1 Relations between Probabilistic and Analytic Concepts
    2. 3.2 Hitting Distributions and Projections I
    3. 3.3 Quasi Properties, Fine Properties, and Part Processes
    4. 3.4 Hitting Distributions and Projections II
    5. 3.5 Transience, Recurrence, and Path Behavior
  10. Chapter 4. ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES
    1. 4.1 Positive Continuous Additive Functionals and Smooth Measures
    2. 4.2 Decompositions of Additive Functionals of Finite Energy
    3. 4.3 Probabilistic Derivation of Beurling-Deny Formula
  11. Chapter 5. TIME CHANGES OF SYMMETRIC MARKOV PROCESSES
    1. 5.1 Subprocesses and Perturbed Dirichlet Forms
    2. 5.2 Time Changes and Trace Dirichlet Forms
    3. 5.3 Examples
    4. 5.4 Energy Functionals for Transient Processes
    5. 5.5 Trace Dirichlet Forms and Feller Measures
    6. 5.6 Characterization of Time-Changed Processes
    7. 5.7 Excursions, Exit System, and Feller Measures
    8. 5.8 More Examples
  12. Chapter 6. REFLECTED DIRICHLET SPACES
    1. 6.1 Terminal Random Variables and Harmonic Functions
    2. 6.2 Reflected Dirichlet Spaces: Transient Case
    3. 6.3 Recurrent Case
    4. 6.4 Toward Quasi-Regular Cases
    5. 6.5 Examples
    6. 6.6 Silverstein Extensions
    7. 6.7 Equivalent Notions of Harmonicity
  13. Chapter 7. BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES
    1. 7.1 Reflected Dirichlet Space for Part Processes
    2. 7.2 Douglas Integrals and Reflecting Extensions
    3. 7.3 Lateral Condition for L2-Generator
    4. 7.4 Countable Boundary
    5. 7.5 One-Point Extensions
    6. 7.6 Examples of One-Point Extensions
    7. 7.7 Many-Point Extensions
    8. 7.8 Examples of Many-Point Extensions
  14. Appendix A. ESSENTIALS OF MARKOV PROCESSES
    1. A.1 Markov Processes
    2. A.2 Basic Properties of Borel Right Processes
    3. A.3 Additive Functionals of Right Processes
    4. A.4 Review of Symmetric Forms
  15. Appendix B. SOLUTIONS TO EXERCISES
  16. Notes
  17. Bibliography
  18. Catalogue of Some Useful Theorems
  19. Index