Non-Realizability of the Poisson Boundary

We show that for any countable group $G$ equipped with a probability measure $\mu$, there exists a randomized stopping time $\tau$ such that $(G,{\mu }_{\tau })$ admits a strictly larger space of bounded harmonic functions than $(G,\mu )$, unless this space is trivial for all measures on $G$. In particular, we exhibit an irreducible probability measure on the free group $F_2$ such that the Poisson boundary is strictly larger than the geometric boundary equipped with the hitting measure, resolving a longstanding open problem. As another consequence, there is never a nontrivial universal topological realization of the Poisson boundary for any countable group.