Abstract: 

I will discuss joint work with Manuel Luethi and Barak Weiss where we provide a classification of stationary measures for certain random walks on homogeneous spaces arising from natural problems in Diophantine approximation. These random walks admit invariant bundles with negative top Lyapunov exponents, causing them to have stationary, non-invariant, measures. I will describe the difficulties this introduces in the proofs, and how we deal with them.