Antoine Derimay

In the Poincaré disk, a bounded harmonic function can always be written as an integral of the Poisson kernel over the boundary, a formula which solves the Dirichlet problem in this context.
For random walks on graphs, there is a notion of harmonic functions and one can define a measure theoretic boundary for which an analog of the Poisson formula remains true. If our graph was moreover a group, this group acts on its Poisson boundary with nice ergodicity properties and more.
If time allows it, I will then explain how these properties can be used to derive rigidity properties of the group.