Random walks on projective spaces

Abstract

Let G be a connected real semisimple Lie group, V be a finite-dimensional representation of G and μ be a probability measure on G whose support spans a Zariski-dense subgroup. We prove that the set of ergodic μ-stationary probability measures on the projective space P(V ) is in one-to-one correspondence with the set of compact G-orbits in P(V ). When V is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic μ-stationary measures on the ag variety of G.