pp. 2, 9, bundle reading — for a non-degenerate μ, the Poisson boundary is non-trivial iff some bounded μ-harmonic function on the group is not constant

Let $G$ be a group and $\mu : G \to \mathbb R$ non-degenerate (IsNondegenerate): the subsemigroup generated by $\{g : \mu(g) \ne 0\}$ is all of $G$. Then the Poisson boundary of $(G, \mu)$ is non-trivial (HasNontrivialPoissonBoundary μ) if and only if there is a function $f : G \to \mathbb R$ that is bounded, $|f(x)| \le C$ for all $x$, that is $\mu$-harmonic (IsHarmonic μ f), and that is not constant: $f(x) \ne f(y)$ for some $x, y \in G$. The statement does not assume that $\mu$ is a probability.

This is not a result of the paper. It backs the sentence of the Walks bundle note ErschlerZheng_Walks saying that for a non-degenerate $\mu$ the definition of HasNontrivialPoissonBoundary is exactly the quoted criterion of p. 2, and the sentence of the note on KaimanovichVershik.not_hasNontrivialPoissonBoundary_iff_asymptoticEntropy_eq_zero saying that for a non-degenerate $\mu$ its condition is the quoted one word for word. With ErschlerZheng.isNondegenerate_muBeta it also backs the note on ErschlerZheng.hasNontrivialPoissonBoundary_muBeta, where for $\mu_\beta$ the definition is the criterion of p. 2.