A construction of the measurable Poisson boundary: from discrete to continuous groups.
Résumé
Let Γ be a dense countable subgroup of a locally compact continuous group G, and μ a probability measure on Γ. Two spaces of harmonic functions are naturally associated with μ: the space of μ-harmonic functions on the countable group Γ and the space of μ-harmonic functions seen as functions on G defined a.s. with respect to its Haar measure λ. Correspondingly we have two natural Poisson boundaries: the Γ-Poisson boundary and the G-Poisson bounary. Since boundaries on the countable group are quite well understood, a natural question is to ask how G-boundary is related to the Γ-boundary.
``In this chapter we introduce a general technique that allows us to build the G-Poisson boundary from the Γ-boundary. As an application, we determine the Poisson boundary of the closure of the Baumslag-Solitar group in the group of real matrices. In particular we show that, under suitable moment conditions and assuming that the action on R is not contracting, this boundary is the p-solenoid