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

• Main Author: Brofferio, Sara
• Format: Journal Article
• Language: English
• Published: 11-03-2015
• Subjects: Mathematics - Dynamical Systems; Mathematics - Probability
• DOI: 10.48550/arxiv.1503.03333

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.