Equidistribution of random walks on nilpotent Lie groups and homogeneous spaces

We consider random walks on Lie groups, mostly nilpotent Lie groups, and study their equidistribution properties. Chapter 1 is a general introduction where a proof of the Central Limit Theorem on Lie groups can be found and where several special cases of the forthcoming results are proven. Chapter 2 is devoted to the study of equidistribution of finitely supported symmetric walks on nilpotent Lie groups and homogeneous spaces. We derive the following probabilistic analog of Ratner's equidistribution theorem for unipotent random walks on homogeneous spaces: let G be a connected Lie group and Gamma a lattice in G, and let m be a finitely supported symmetric and aperiodic probability measure on a simply connected unipotent subgroup U of G, then for all x ∈ G/Gamma limn→+infinity m*n*dx =mx where mx is the unique U-ergodic probability measure whose support in the closure of the orbit Ux. In Chapter 3, making use of both harmonic analysis and probabilistic methods, we prove a local limit theorem for product of random matrices on the Heisenberg group and a refined comparison theorem with the associated heat kernel. No assumptions of absolute continuity are made on the probability measure.