# The Liouville property and random walks on topological groups

## TL;DR

This paper characterizes the amenability of second-countable topological groups through the Liouville property and Poisson boundaries, extending classical results to non-locally compact groups and linking harmonic analysis with group properties.

## Contribution

It generalizes existing theories by establishing a topological version of Furstenberg's conjecture and characterizes amenability via Poisson boundaries for broader classes of groups.

## Key findings

- Amenability characterized by trivial Poisson boundary with fully supported measures
- Extension of Furstenberg's conjecture to non-locally compact groups
- Implications for Liouville actions of discrete groups

## Abstract

We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich--Vershik and Rosenblatt, confirms a general topological version of Furstenberg's conjecture, and entails a characterization of the amenability of isometry groups in terms of the Liouville property for induced actions. Moreover, our result has non-trivial consequences concerning Liouville actions of discrete groups on countable sets