Book contents
- Frontmatter
- Dedication
- Epigraph
- Contents
- Introduction
- 1 Semigroups and Generators
- 2 The Generation of Semigroups
- 3 Convolution Semigroups of Measures
- 4 Self-Adjoint Semigroups and Unitary Groups
- 5 Compact and Trace Class Semigroups
- 6 Perturbation Theory
- 7 Markov and Feller Semigroups
- 8 Semigroups and Dynamics
- 9 Varopoulos Semigroups
- Notes and Further Reading
- Appendix A The Space C0(Rd)
- Appendix B The Fourier Transform
- Appendix C Sobolev Spaces
- Appendix D Probability Measures and Kolmogorov’s Theorem on Construction of Stochastic Processes
- Appendix E Absolute Continuity, Conditional Expectation and Martingales
- Appendix F Stochastic Integration and Itô’s Formula
- Appendix G Measures on Locally Compact Spaces – Some Brief Remarks
- References
- Index
3 - Convolution Semigroups of Measures
Published online by Cambridge University Press: 27 July 2019
Book contents
- Frontmatter
- Dedication
- Epigraph
- Contents
- Introduction
- 1 Semigroups and Generators
- 2 The Generation of Semigroups
- 3 Convolution Semigroups of Measures
- 4 Self-Adjoint Semigroups and Unitary Groups
- 5 Compact and Trace Class Semigroups
- 6 Perturbation Theory
- 7 Markov and Feller Semigroups
- 8 Semigroups and Dynamics
- 9 Varopoulos Semigroups
- Notes and Further Reading
- Appendix A The Space C0(Rd)
- Appendix B The Fourier Transform
- Appendix C Sobolev Spaces
- Appendix D Probability Measures and Kolmogorov’s Theorem on Construction of Stochastic Processes
- Appendix E Absolute Continuity, Conditional Expectation and Martingales
- Appendix F Stochastic Integration and Itô’s Formula
- Appendix G Measures on Locally Compact Spaces – Some Brief Remarks
- References
- Index
Summary
Chapter 3 studies semigroups on function spaces obtained via convolution semigroups of probability measures. Motivating examples that are studied in detail are the heat kernel (Brownian motion) and the Poisson kernel (Cauchy process). The characteristic functional (Fourier transform) is used to establish the Levy–Khinchine formula, and applications are given to stable laws. The generator and the semigroup are written as pseudo-differential operators.
Keywords
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- Semigroups of Linear OperatorsWith Applications to Analysis, Probability and Physics, pp. 46 - 82Publisher: Cambridge University PressPrint publication year: 2019