Bernstein Functions: Theory and Applications

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. Preface to the second edition Preface Index of notation 1 Completely monotone functions 2 Stieltjes functions 3 Bernstein functions 4 Positive and negative definite functions 5 A probabilistic intermezzo 6 Complete Bernstein functions 6.1 Representation of complete Bernstein functions 6.2 Extended complete Bernstein functions 7 Properties of complete Bernstein functions 8 Thorin-Bernstein functions 9 A second probabilistic intermezzo 10 Transformations of Bernstein functions 11 Special Bernstein functions and potentials 11.1 Special Bernstein functions 11.2 Hirsch’s class 12 The spectral theorem and operator monotonicity 12.1 The spectral theorem 12.2 Operator monotone functions 13 Subordination and Bochner’s functional calculus 13.1 Semigroups and subordination in the sense of Bochner 13.2 A functional calculus for generators of semigroups 13.3 Subordination and functional inequalities 13.4 Eigenvalue estimates for subordinate processes 14 Potential theory of subordinate killed Brownian motion 15 Applications to generalized diffusions 15.1 Inverse local time at zero 15.2 First passage times 16 Examples of complete Bernstein functions 16.1 Special functions used in the tables 16.2 Algebraic functions 16.3 Exponential functions 16.4 Logarithmic functions 16.5 Inverse trigonometric functions 16.6 Hyperbolic functions 16.7 Inverse hyperbolic functions 16.8 Gamma and related special functions 16.9 Bessel functions 16.10 Miscellaneous functions 16.11 CBFs given by exponential representations 16.12 Additional comments Appendix A.1 Vague and weak convergence of measures A.2 Hunt processes and Dirichlet forms Bibliography Index