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
8 - Semigroups and Dynamics
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
After a brief discussion of invariant measures and entropy, we introduce semidynamical and dynamical systems. We use Koopmanism to show how to obtain semigroups/groups of linear operators on function spaces, when we have a quasi-invariant measure. Applications are given to solution flows of differential equations. In the last part we discuss “dilations” as a mathematical approach to the origins of irreversibility.
- Type
- Chapter
- Information
- Semigroups of Linear OperatorsWith Applications to Analysis, Probability and Physics, pp. 149 - 165Publisher: Cambridge University PressPrint publication year: 2019