A characterization of Liouville property
Poisson boundary provides an integral representation of all bounded harmonic functions. We say that a Markov chain satisfies the Liouville property if all bounded harmonic functions are constant, that is the Poisson boundary is trivial.
The first part of the talk is a gentle introduction to Poisson boundary. Then I will state a new condition that is equivalent to the Liouville property and provide a proof of this equivalence. This talk is based on an ongoing work and will be self-contained.