Poisson boundary

Summary

The Poisson boundary of a random walk on a group is a measure space that records where the walk goes at infinity. Every bounded harmonic function for the walk is the Poisson integral of a unique bounded function on this boundary. This mirrors the Poisson integral formula for bounded harmonic functions on the disc. The boundary is trivial exactly when all bounded harmonic functions are constant, which is the Liouville property. It was introduced by Furstenberg and developed by Kaimanovich and Vershik, who related its triviality to the entropy of the walk.