The Choquet-Deny Property for Groupoids

TL;DR

This paper extends the concept of the Choquet-Deny property from groups to discrete measured groupoids, providing a full characterization based on isotropy groups and equivalence relations, and classifies this property for transformation groupoids.

Contribution

It offers the first complete characterization of the Choquet-Deny property for discrete measured groupoids, generalizing previous group-based results.

Findings

Characterization of the Choquet-Deny property in terms of isotropy groups.

Classification of the property for transformation groupoids.

Extension of previous group results to the groupoid setting.

Abstract

A countable discrete group is called Choquet-Deny if for any non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Building on the previous work of Jaworski, a complete characterization of Choquet-Deny groups was recently achieved by Frisch, Hartman, Tamuz, and Ferdowski. In this article, we extend the study of the Choquet-Deny property to the framework of discrete measured groupoids. Our primary result offers a complete characterization of this property in terms of the isotropy groups and the equivalence relation associated with the given groupoid. Additionally, we use the implications derived from our main theorem to classify the Choquet-Deny property of transformation groupoids.