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

The talk will be concerned with a property of lattices in the geometry of numbers which I refer to as "almost sure grid-DV with respect to a function F". The goal of the talk will be three-fold:

(1) Discuss a theorem relating this property to the dynamics of the lattice under the invariance group of the function.

(2) Apply the theorem to various classical discussions and see what it gives (it relates nicely to Minkowski's conjecture and to diophantine exponents).

(3) Discuss a mistake I have in the published paper and some open problems