Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

Roland Prohaska

Published 2019-10-25Version 1

The purpose of this note is to discuss three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/\Gamma$ of semisimple real Lie groups. First, we investigate the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish strong uniformity for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.