Bernstein Functions

by Schilling, Rene L.

• ISBN13: 9783110215304
• ISBN10: 3110215306
• Format: Hardcover
• Copyright: 2010-01-31
• Publisher: De Gruyter

Summary

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. A self-contained and unified approach to the topic With applications to various fields of mathematics, such as probability theory, potential theory, operator theory, integral equations, functional calculi and complex analysis With an extensive list of complete Bernstein functions.

Table of Contents

Preface	p. vii
Index of notation	p. xii
Completely monotone functions	p. 1
Stieltjes functions	p. 11
Bernstein functions	p. 15
Positive and negative definite functions	p. 25
A probabilistic intermezzo	p. 34
Complete Bernstein functions: representation	p. 49
Complete Bernstein functions: properties	p. 62
Thorin-Bernstein functions	p. 73
A second probabilistic intermezzo	p. 80
Special Bernstein functions and potentials	p. 92
Special Bernstein functions	p. 92
Hirsch's class	p. 105
The spectral theorem and operator monotonicity	p. 110
The spectral theorem	p. 110
Operator monotone functions	p. 118
Subordination and Bochner's functional calculus	p. 130
Semigroups and subordination in the sense of Bochner	p. 130
A functional calculus for generators of semigroups	p. 145
Eigenvalue estimates for subordinate processes	p. 161
Potential theory of subordinate killed Brownian motion	p. 174
Applications to generalized diffusions	p. 185
Inverse local time at zero	p. 185
First passage times	p. 202
Examples of complete Bernstein functions	p. 214
Special functions used in the tables	p. 215
Algebraic functions	p. 218
Exponential functions	p. 226
Logarithmic functions	p. 228
Inverse trigonometric functions	p. 244
Hyperbolic functions	p. 244
Inverse hyperbolic functions	p. 250
Gamma and other special functions	p. 254
Bessel functions	p. 264
Miscellaneous functions	p. 272
Additional comments	p. 278
Appendix	p. 281
Vague and weak convergence of measures	p. 281
Hunt processes and Dirichlet forms	p. 283
Bibliography	p. 291
Index	p. 309
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