Symmetric Markov processes, time change, and boundary theory / Zhen-Qing Chen, Masatoshi Fukushima.

Format:

Book

Author/Creator:

Chen, Zhen-Qing.

Contributor:

Fukushima, Masatoshi, 1935-

Series:

London Mathematical Society monographs ; vol. 35.

London Mathematical Society monographs ; vol. 35

Language:

English

Subjects (All):

Markov processes.

Physical Description:

xv, 479 pages ; 25 cm.

Place of Publication:

Princeton : Princeton University Press, [2012]

Summary:

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. Book jacket.

Contents:

Chapter 1 Symmetric Markovian Semigroups and Dirichlet Forms 1

1.1 Dirichlet Forms and Extended Dirichlet Spaces 1

1.2 Excessive Functions and Capacities 15

1.3 Quasi-Regular Dirichlet Forms 24

1.4 Quasi-Homeomorphism of Dirichlet Spaces 30

1.5 Symmetric Right Processes and Quasi-Regular Dirichlet Forms 33

Chapter 2 Basic Properties and Examples of Dirichlet Forms 37

2.1 Transience, Recurrence, and Irreducibility 37

2.2 Basic Examples 50

2.3 Analytic Potential Theory for Regular Dirichlet Forms 77

2.4 Local Properties 88

Chapter 3 Symmeteic Hunt Processes and Rrgular Dirichlet Forms 92

3.1 Relations between Probabilistic and Analytic Concepts 92

3.2 Hitting Distributions and Projections I 103

3.3 Quasi Properties, Fine Properties, and Part Processes 106

3.4 Hitting Distributions and Projections II 113

3.5 Transience, Recurrence, and Path Behavior 118

Chapter 4 Additive Functional of Symmetric Merkov Processes 130

4.1 Positive Continuous Additive Functionals and Smooth Measures 130

4.2 Decompositions of Additive Functionals of Finite Energy 143

4.3 Probabilistic Derivation of Beurling-Deny Formula 151

Chapter 5 Time Changes of Symmetrix Markov Processes 166

5.1 Subprocesses and Perturbed Dirichlet Forms 168

5.2 Time Changes and Trace Dirichlet Forms 174

5.3 Examples 190

5.4 Energy Functional for Transient Processes 202

5.5 Trace Dirichlet Forms and Feller Measures 206

5.6 Characterization of Time-Changed Processes 221

5.7 Excursions, Exit System, and Feller Measures 225

5.8 More Examples 232

Chapter 6 Rrflected Dirichlet Spaces 240

6.1 Terminal Random Variables and Harmonic Functions 241

6.2 Reflected Dirichlet Spaces: Transient Case 246

6.3 Recurrent Case 260

6.4 Toward Quasi-Regular Cases 262

6.5 Examples 269

6.6 Silverstein Extensions 275

6.7 Equivalent Notions of Harmonicity 263

Chapter 7 Boundary Theory for Symmetric Markov Processes 300

7.1 Reflected Dirichlet Space for Part Processes 302

7.2 Douglas Integrals and Reflecting Extensions 310

7.3 Lateral Condition for L2-Generator 324

7.4 Countable Boundary 334

7.5 One-Point Extensions 340

7.6 Examples of One-Point Extensions 352

7.7 Many-Point Extensions 369

7.8 Examples of Many-Point Extensions 377.

Notes:

Includes bibliographical references (pages [457]-466) and index.

ISBN:

9780691136059

069113605X

OCLC:

724663149