Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- Part II Results and Applications
- 5 Growth, Dimension, and Heat Kernel
- 6 Bounded Harmonic functions
- 7 Choquet–Deny Groups
- 8 The Milnor–Wolf Theorem
- 9 Gromov’s Theorem
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
6 - Bounded Harmonic functions
from Part II - Results and Applications
Published online by Cambridge University Press: 16 May 2024
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- Part II Results and Applications
- 5 Growth, Dimension, and Heat Kernel
- 6 Bounded Harmonic functions
- 7 Choquet–Deny Groups
- 8 The Milnor–Wolf Theorem
- 9 Gromov’s Theorem
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
Summary
This chapter is devoted to the study of the space of bounded harmonic functions and the Liouville property. We start with the entropic criterion for the Liouville property. We then investigate the relationship of the Liouville property with amenability, speed of the random walk, and coupling of exit measures.The central example of lamplighter groups is studied.
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- Information
- Harmonic Functions and Random Walks on Groups , pp. 196 - 245Publisher: Cambridge University PressPrint publication year: 2024