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2 results

  • 6 - Bounded Harmonic functions

  • from Part II - Results and Applications
  • Ariel Yadin, Ben-Gurion University of the Negev, Israel
  • Book:
    Harmonic Functions and Random Walks on Groups
    Published online:
    16 May 2024
    Print publication:
    23 May 2024, pp 196-245

  • 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.

Harmonic Functions and Random Walks on Groups
  • Ariel Yadin
  • Published online:
    16 May 2024
    Print publication:
    23 May 2024
  • Research in recent years has highlighted the deep connections between the algebraic, geometric, and analytic structures of a discrete group. New methods and ideas have resulted in an exciting field, with many opportunities for new researchers. This book is an introduction to the area from a modern vantage point. It incorporates the main basics, such as Kesten's amenability criterion, Coulhon and Saloff-Coste inequality, random walk entropy and bounded harmonic functions, the Choquet–Deny Theorem, the Milnor–Wolf Theorem, and a complete proof of Gromov's Theorem on polynomial growth groups. The book is especially appropriate for young researchers, and those new to the field, accessible even to graduate students. An abundance of examples, exercises, and solutions encourage self-reflection and the internalization of the concepts introduced. The author also points to open problems and possibilities for further research.