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
7 - Markov and Feller Semigroups
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
Markov and Feller semigroups are introduced, together with the corresponding stochastic processes. As all generators of Feller semigroups satisfy the positive maximum principle, we focus on that property and discuss the associated Hille–Yosida–Ray theorem. The main result of the chapter is proof of the Courrege theorem, which gives a Levy–Khinchine representation (but with variable coefficients) for all linear operators satisfying the positive maximum principle. We conclude with a brief discussion of the martingale problem and sub-Feller semigroups.
- Type
- Chapter
- Information
- Semigroups of Linear OperatorsWith Applications to Analysis, Probability and Physics, pp. 129 - 148Publisher: Cambridge University PressPrint publication year: 2019