A construction of the measurable Poisson boundary: from discrete to continuous groups

Let $\Gamma$ be a dense countable subgroup of a locally compact continuous group $G$. Take a probability measure $\mu$ on $\Gamma$. There are two natural spaces of harmonic functions: the space of $\mu$-harmonic functions on the countable group $\Gamma$ and the space of $\mu$-harmonic functions seen as functions on $G$ defined a.s. with respect to its Haar measure $\lambda$. This leads to two natural Poisson boundaries: the $\Gamma$-Poisson boundary and the $G$-Poisson boundary. Since boundaries on the countable group are quite well understood, a natural question is to ask how $G$-boundary is related to the $\Gamma$-boundary. In this paper we present a theoretical setting to build the $G$-Poisson boundary from the $\Gamma$-boundary. We apply this technics to build the Poisson boundary of the closure of the Baumslag-Solitar group in the group of real matrices. In particular we show that, under moment condition and in the case that the action on $\mathbf{R}$ is not contracting, this boundary is the $p$-solenoid.