A coupling strategy for Brownian motions at fixed time on Carnot groups using Legendre expansion

We propose a new simple construction of a coupling at a fixed time of two sub-Riemannian Brownian motions on the Heisenberg group and on the free step 2 Carnot groups. The construction is based on a Legendre expansion of the standard Brownian motion and of the L{\'e}vy area. We deduce sharp estimates for the decay in total variation distance between the laws of the Brownian motions. Using a change of probability method, we also obtain the log-Harnack inequality, a Bismut type integration by part formula and reverse Poincar\'e inequalities for the associated semi-group.