Stationary measures and invariant closed subset of homogeneous spaces

In a recent joint work with Yves Benoist, we obtained a classification of all $\mu$-stationary Borel probability measures on $G/\Lambda$, where $G$ is a simple Lie group, $\Lambda$ a lattice in $G$ and $\mu$ a compactly supported Borel probability measure on $G$, which support spans a Zariski dense subgroup of $G$. This allows to get a description of the closed
$S$-invariant subsets of $G/\Lambda$, where $S$ is a Zariski dense sub-semigroup of $G$. In this talk, I will discuss extensions of these results to the non simple case, and their relations with fine limit theorems for random walk on semisimple groups.