Math Seminars.

Emmanuel Breuillard (Lille 1)

Random walks on nilpotent Lie groups

We generalize some well-known limit theorems of classical probability theory in the context of nilpotent Lie groups. In particular, using a harmonic analysis approach, we prove the local limit theorem for products of centered i.i.d. random variables on the Heisenberg group $H$. We also get a precise estimate of the behavior of the walk on $H$ by comparing it to the associated gaussian walk. Using a different approach, weaker estimates for a smaller class of measures are also obtained on arbitrary nilpotent Lie groups. As an application, we show that symmetric unipotent random walks on homogeneous spaces &G/\Gamma$ are equidistributed, thus yielding a kind of probabilistic version of Ratner's equidistribution theorem.