In https://mathoverflow.net/a/52279/24840, the answer claims that the Liouville property, i.e. the property that every bounded harmonic function is constant, on a (presumably locally finite connected) graph $G$ is equivalent to the invariant $\sigma$ field $\mathcal{I} := \{F : \forall n, F \circ \theta_n = F \}$ of a simple random walk $X$ on $G$ being $P^x$ trivial for all $x$, where $X$ is a SRW started at $x$ under $P^x$. Here $\theta_n$ is the $n$-fold composition of the shift operator $\theta ((x_i)_{i\geq 0}) := (x_{i+1})_{i \geq 0}$.
Is there a standard reference for this fact? Or does it follow from some simple argument?