Let G be a semisimple Lie group, and \Gamma a lattice. in G. Let \mu be a measure on G. We show that under a certain condition on \mu, any \mu-stationary measure on G/\Gamma is in fact invariant under the group generated by the support of \mu. We give an alternative argument (which bypasses the Local Limit Theorem) for some of the breakthrough results of Benoist and Quint in this area. This is joint work with Elon Lindenstrauss.