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We provide equivalent conditions for Liouville property of actions of groups. As an application, we show that there is a Liouville measure for the action of the Thompson group on dyadic rationals. This result should be compared with a recent result of Kaimanovich, where he shows that the action of the Thompson group on dyadic rationals is not Liouville for all finitely supported measures. As another application we show that there is a Liouville measure for lamplighter actions. This gives more examples of non-amenable Liouville actions.
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Kate Juschenko, Tianyi Zheng, Infinitely supported Liouville measures of Schreier graphs. Groups Geom. Dyn. 12 (2018), no. 3, pp. 911–918DOI 10.4171/GGD/464