Poisson boundary for random walks on free (semi)-groups with infinite moment

The conjecture says that the geometric boundary, equipped with the hitting measure, describes all bounded harmonic information for random walks on free groups and free semigroups, without moment assumptions.

Known results

• The 2017 paper proves the identification under finite entropy or finite logarithmic moment.
• It also proves it under finite logarithmic $w$-moment for a nonempty finite word $w$, including examples with infinite entropy and infinite logarithmic moment.
• Earlier results covered finite-support measures; later work cited in 2025 covers all finite-entropy measures.

June 2025 counterexample claim

The paper Non-Realizability of the Poisson Boundary claims that an irreducible probability measure on the free group $F_2$ has a geometric boundary that is not the Poisson boundary, thereby resolving the open problem negatively. It further claims that the method extends to free semigroups, but the displayed theorem is for $F_2$ and the result remains unverified.

Current status (as of September 2026): The finite-moment cases are established, while the June 2025 free-group counterexample claim, and its asserted free-semigroup extension, have not been independently verified.