we will start by reviewing recent developments in random walks on homogeneous spaces. in a second part, we will discuss the notion of a $h$-expanding probability measure on a connected semisimple lie group $h$. as we shall see, for a $h$-expanding $\mu$ with $h < g$, on the one hand, one can obtain a description of $\mu$-stationary probability measures on the homogeneous space $g / \lambda$ ($g$ lie group, $\lambda$ lattice) using the measure classification results of eskin-lindenstrauss, and on the other hand, the recurrence techniques of benoist-quint and eskin-mirzakhani-mohammadi can be adapted to this setting. with some further work, these allow us to deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains zariski-dense subgroups and further epimorphic subgroups of $h$. if time allows, we will see how, utilizing an idea of simmons-weiss, these also allow us to deduce birkhoff genericity of a class of fractal measures with respect to certain diagonal flows, which, in turn, has applications in diophantine approximation problems. \\ \\ joint work with roland prohaska and ronggang shi.