Non co-adapted couplings of Brownian motions on free, step 2 Carnot groups

On the free, step $2$ Carnot groups of rank n Gₙ, the subRiemannian Brownian motion consists in a Rⁿ-Brownian motion together with its n(n-1)/2 L{\'e}vy areas. In this article we construct an explicit successful non co-adapted coupling of Brownian motions on Gₙ. We use this construction to obtain gradient inequalities for the heat semi-group on all the homogeneous step $2$ Carnot groups. Comparing the first coupling time and the first exit time from a domain, we also obtain gradient inequalities for harmonic functions on Gₙ. These results generalize the coupling strategy by Banerjee, Gordina and Mariano on the Heisenberg group.