Random walks on Lie groups

The goal of these notes is to give an introduction to random walks and limit theorems on Lie groups, mostly amenable Lie groups, with an emphasis on equidistribution problems. We also state a number of open problems. In Section 1 we define the basic notions regarding probability measures on groups, con-vergence in distribution, recurrence, equidistribution, Brownian motions, limit theorems, etc. Most notably we state and prove the Central Limit Theorem on Lie groups. Oddly enough, the proof of this fundamental theorem cannot be found in the literature in a simple and complete form, although it was established almost 45 years ago by Wehn. The exposition here will I hope make up for that. In Section 2 we discuss the equidistribution properties of random walks in Lie groups and survey the existing ratio limit theorems and local limit theorems in this context. We give several examples including the Itô-Kawada equidistribution theorem for compact groups and the local limit theorem for random walks by isometries on the plane, of which we give a full proof thus generalizing an old theorem of Kazhdan. In Section 3 we study in more detail the case of nilpotent Lie groups, rephrase th