Bernstein Functions: Theory and Applications

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This text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided.

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Product Details

Meet the Author

René Schilling, Dresden University of Technology, Germany; Renming Song, University of Illinois, Urbana, USA; Zoran Vondracek, University of Zagreb, Croatia.

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Table of Contents

Preface vii

Index of notation xii

1 Completely monotone functions 1

2 Stieltjes functions 11

3 Bernstein functions 15

4 Positive and negative definite functions 25

5 A probabilistic intermezzo 34

6 Complete Bernstein functions: representation 49

7 Complete Bernstein functions: properties 62

8 Thorin-Bernstein functions 73

9 A second probabilistic intermezzo 80

10 Special Bernstein functions and potentials 92

10.1 Special Bernstein functions 92

10.2 Hirsch's class 105

11 The spectral theorem and operator monotonicity 110

11.1 The spectral theorem 110

11.2 Operator monotone functions 118

12 Subordination and Bochner's functional calculus 130

12.1 Semigroups and subordination in the sense of Bochner 130

12.2 A functional calculus for generators of semigroups 145

12.3 Eigenvalue estimates for subordinate processes 161

13 Potential theory of subordinate killed Brownian motion 174

14 Applications to generalized diffusions 185

14.1 Inverse local time at zero 185

14.2 First passage times 202

15 Examples of complete Bernstein functions 214

15.1 Special functions used in the tables 215

15.2 Algebraic functions 218

15.3 Exponential functions 226

15.4 Logarithmic functions 228

15.5 Inverse trigonometric functions 244

15.6 Hyperbolic functions 244

15.7 Inverse hyperbolic functions 250

15.8 Gamma and other special functions 254

15.9 Bessel functions 264

15.10 Miscellaneous functions 272

15.11 Additional comments 278

A Appendix 281

A.1 Vague and weak convergence of measures 281

A.2 Hunt processes and Dirichlet forms 283

Bibliography 291

Index 309

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