Recurrence on Affine Grassmannians

We study the action of the affine group $G$ of ${\mathbb{R}}^d$ on the space $X_{k,d}$ of $k$-dimensional affine subspaces. Given a compactly-supported Zariski dense probability measure $\mu$ on $G$, we show that $X_{k,d}$ supports a $\mu$-stationary measure $\nu$ if and only if the $(k!+!1)^{\rm th}$-Lyapunovexponentof$\mu$isstrictlynegative\mathrm{.}Inparticular,when$\mu$issymmetric,$\nu$existsifandonlyif$2k\geq d$.