Liouville extensions by an isometric integer action

Definitions and mathematical statement

Let $G$ be a connected locally finite graph, and let an automorphism $T$ define an action of $\mathbb Z$ by $n\cdot v=T^nv$. Let $\pi:V(G)\to V(G)/\mathbb Z$ be the orbit map and choose a set $F$ containing exactly one representative of each orbit. Its translates are $T^kF$, $k\in\mathbb Z$.

The quotient random walk is the projection of simple random walk on $G$. Its transition kernel is \[ \overline P([v],[w])= \frac{|\{u\sim_Gv:\pi(u)=[w]\}|}{\deg_G(v)}. \] This does not depend on the representative $v$, because $T$ preserves adjacency. Retaining these multiplicities and possible holding probabilities specifies the quotient convention. Suppose the quotient is Liouville: every bounded real $g$ satisfying $\overline Pg=g$ is constant.

Assume that for the chosen $F$, simple random walk $(X_n)$ on $G$, from every starting vertex, satisfies \[ \mathbb P\left(X_n\in T^kF\text{ for infinitely many }n\right)=1 \qquad\text{for every }k\in\mathbb Z. \] Must every bounded harmonic function on $G$ itself be constant?

The recurrence assumption is recurrence to each translated set, which may be infinite; it does not assert recurrence to individual vertices. Countability allows the visits to all translated fundamental sets to hold simultaneously almost surely. The action need not be transitive on $G$. The conclusion asks whether bounded harmonic information can remain in the direction of the integer action when the quotient carries none and the walk repeatedly returns to every translate of a fundamental domain.

Short English statement

Does a Liouville quotient and repeated returns to all translated fundamental domains force an integer extension of a graph to be Liouville?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000049 (AMR-099-0049), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Coarse Geometry and Randomness (Saint-Flour notes), Open problem 13.3, PDF page 107. Original source indexed by AMR-099-0049. https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=107.
• S3 I. Benjamini, Coarse Geometry and Randomness, primary Saint-Flour notes; the numbered question and its surrounding definitions. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.