  Let $G$ be a real Lie group, $\Lambda<G$ a lattice and $H<G$ a connected
semisimple subgroup without compact factors and with finite center. We define
the notion of $H$-expanding measures $\mu$ on $H$ and, applying recent work of
Eskin-Lindenstrauss, prove that $\mu$-stationary probability measures on
$G/\Lambda$ are homogeneous. Transferring a construction by Benoist-Quint and
drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis
functions to show that $H$-expanding random walks on $G/\Lambda$ satisfy a
recurrence condition and that homogeneous subspaces are repelling. Combined
with a countability result, this allows us to prove equidistribution of
trajectories in $G/\Lambda$ for $H$-expanding random walks and to obtain orbit
closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we
deduce Birkhoff genericity of a class of measures with respect to some diagonal
flows and extend their applications to Diophantine approximation on similarity
fractals to a non-conformal and weighted setting.
