Timothée Bénard

Local limit theorem on nilpotent Lie groups

We prove the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of a walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures.
This is joint work with Emmanuel Breuillard.