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. It features a self-contained and unified approach to the topic. It comes with applications to various fields of mathematics, such as probability theory, potential theory, operator theory, integral equations, functional calculi and complex analysis. It also comes with an extensive list of complete Bernstein functions.
Author(s): Rene Schilling
Series: De Gruyter Studies in Mathematics
Publisher: Gruyter
Year: 2010
Language: English
Pages: 328
Contents ......Page 6
Preface ......Page 8
Index of notation ......Page 13
1 Completely monotone functions ......Page 14
2 Stieltjes functions ......Page 24
3 Bernstein functions ......Page 28
4 Positive and negative definite functions ......Page 38
5 A probabilistic intermezzo ......Page 47
6 Complete Bernstein functions: representation ......Page 62
7 Complete Bernstein functions: properties ......Page 75
8 Thorin¨CBernstein functions ......Page 86
9 A second probabilistic intermezzo ......Page 93
10.1 Special Bernstein functions ......Page 105
10.2 Hirsch?ˉs class ......Page 118
11.1 The spectral theorem ......Page 123
11.2 Operator monotone functions ......Page 131
12.1 Semigroups and subordination in the sense of Bochner ......Page 143
12.2 A functional calculus for generators of semigroups ......Page 158
12.3 Eigenvalue estimates for subordinate processes ......Page 174
13 Potential theory of subordinate killed Brownian motion ......Page 187
14.1 Inverse local time at zero ......Page 198
14.2 First passage times ......Page 215
15 Examples of complete Bernstein functions ......Page 227
15.1 Special functions used in the tables ......Page 228
15.2 Algebraic functions ......Page 231
15.3 Exponential functions ......Page 239
15.4 Logarithmic functions ......Page 241
15.6 Hyperbolic functions ......Page 257
15.7 Inverse hyperbolic functions ......Page 263
15.8 Gamma and other special functions ......Page 267
15.9 Bessel functions ......Page 277
15.10 Miscellaneous functions ......Page 285
15.11 Additional comments ......Page 291
A.1 Vague and weak convergence of measures ......Page 294
A.2 Hunt processes and Dirichlet forms ......Page 296
Bibliography ......Page 304
Index ......Page 322
