Bernstein Functions (eBook)

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

Preface to the second edition 5
Preface 6
Index of notation 13
1 Completely monotone functions 15
2 Stieltjes functions 30
3 Bernstein functions 35
4 Positive and negative definite functions 49
5 A probabilistic intermezzo 62
6 Complete Bernstein functions 83
6.1 Representation of complete Bernstein functions 83
6.2 Extended complete Bernstein functions 94
7 Properties of complete Bernstein functions 106
8 Thorin-Bernstein functions 123
9 A second probabilistic intermezzo 131
10 Transformations of Bernstein functions 145
11 Special Bernstein functions and potentials 173
11.1 Special Bernstein functions 173
11.2 Hirsch’s class 186
12 The spectral theorem and operator monotonicity 193
12.1 The spectral theorem 193
12.2 Operator monotone functions 201
13 Subordination and Bochner’s functional calculus 214
13.1 Semigroups and subordination in the sense of Bochner 214
13.2 A functional calculus for generators of semigroups 230
13.3 Subordination and functional inequalities 247
13.4 Eigenvalue estimates for subordinate processes 256
14 Potential theory of subordinate killed Brownian motion 271
15 Applications to generalized diffusions 282
15.1 Inverse local time at zero 282
15.2 First passage times 299
16 Examples of complete Bernstein functions 313
16.1 Special functions used in the tables 314
16.2 Algebraic functions 318
16.3 Exponential functions 326
16.4 Logarithmic functions 328
16.5 Inverse trigonometric functions 344
16.6 Hyperbolic functions 344
16.7 Inverse hyperbolic functions 350
16.8 Gamma and related special functions 354
16.9 Bessel functions 364
16.10 Miscellaneous functions 372
16.11 CBFs given by exponential representations 380
16.12 Additional comments 385
Appendix 388
A.1 Vague and weak convergence of measures 388
A.2 Hunt processes and Dirichlet forms 391
Bibliography 397
Index 420

lt;P>"To sum up, the book collects and integrates results from a number of sources scattered throughout the literature. It is clearly and carefully written. It will certainly be useful for both graduate students and researchers in different areas." Mathematical Reviews (review of the first edition)

"This impressive monograph, which is now in an expanded second edition, does a wonderful job of collecting together and synthesising some of the most important results and applications. Consequently it represents a unique perspective on this important function class which will be a valuable resource for many years to come for both experienced researchers and graduate students." Mathematical Reviews