Non-Realizability of the Poisson Boundary

TL;DR

This paper demonstrates that for any countable group with a probability measure, one can find a randomized stopping time that enlarges the space of bounded harmonic functions, resolving a longstanding open problem about the Poisson boundary.

Contribution

It proves the non-realizability of the Poisson boundary as a universal topological space for any countable group, providing explicit examples and resolving open questions.

Findings

Existence of a measure on free group F_2 with a larger Poisson boundary than the geometric boundary.

No universal topological realization of the Poisson boundary exists for any countable group.

Construction of a randomized stopping time that enlarges harmonic function spaces.

Abstract

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.