Bernstein Functions
Description
Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis - often with different definitions and under different names. Among the synonyms are `Laplace exponent' instead of Bernstein function, and complete Bernstein functions are sometimes called `Pick functions', `Nevanlinna functions' or `operator monotone functions'.
This monograph - now in its second revised and extended edition - offers a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.
Reviews / Votes
"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
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Content
2 - Preface [Seite 6]
3 - Index of notation [Seite 13]
4 - 1 Completely monotone functions [Seite 15]
5 - 2 Stieltjes functions [Seite 30]
6 - 3 Bernstein functions [Seite 35]
7 - 4 Positive and negative definite functions [Seite 49]
8 - 5 A probabilistic intermezzo [Seite 62]
9 - 6 Complete Bernstein functions [Seite 83]
9.1 - 6.1 Representation of complete Bernstein functions [Seite 83]
9.2 - 6.2 Extended complete Bernstein functions [Seite 94]
10 - 7 Properties of complete Bernstein functions [Seite 106]
11 - 8 Thorin-Bernstein functions [Seite 123]
12 - 9 A second probabilistic intermezzo [Seite 131]
13 - 10 Transformations of Bernstein functions [Seite 145]
14 - 11 Special Bernstein functions and potentials [Seite 173]
14.1 - 11.1 Special Bernstein functions [Seite 173]
14.2 - 11.2 Hirsch's class [Seite 186]
15 - 12 The spectral theorem and operator monotonicity [Seite 193]
15.1 - 12.1 The spectral theorem [Seite 193]
15.2 - 12.2 Operator monotone functions [Seite 201]
16 - 13 Subordination and Bochner's functional calculus [Seite 214]
16.1 - 13.1 Semigroups and subordination in the sense of Bochner [Seite 214]
16.2 - 13.2 A functional calculus for generators of semigroups [Seite 230]
16.3 - 13.3 Subordination and functional inequalities [Seite 247]
16.4 - 13.4 Eigenvalue estimates for subordinate processes [Seite 256]
17 - 14 Potential theory of subordinate killed Brownian motion [Seite 271]
18 - 15 Applications to generalized diffusions [Seite 282]
18.1 - 15.1 Inverse local time at zero [Seite 282]
18.2 - 15.2 First passage times [Seite 299]
19 - 16 Examples of complete Bernstein functions [Seite 313]
19.1 - 16.1 Special functions used in the tables [Seite 314]
19.2 - 16.2 Algebraic functions [Seite 318]
19.3 - 16.3 Exponential functions [Seite 326]
19.4 - 16.4 Logarithmic functions [Seite 328]
19.5 - 16.5 Inverse trigonometric functions [Seite 344]
19.6 - 16.6 Hyperbolic functions [Seite 344]
19.7 - 16.7 Inverse hyperbolic functions [Seite 350]
19.8 - 16.8 Gamma and related special functions [Seite 354]
19.9 - 16.9 Bessel functions [Seite 364]
19.10 - 16.10 Miscellaneous functions [Seite 372]
19.11 - 16.11 CBFs given by exponential representations [Seite 380]
19.12 - 16.12 Additional comments [Seite 385]
20 - Appendix [Seite 388]
20.1 - A.1 Vague and weak convergence of measures [Seite 388]
20.2 - A.2 Hunt processes and Dirichlet forms [Seite 391]
21 - Bibliography [Seite 397]
22 - Index [Seite 420]
