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

On the free, step $2$ Carnot groups of rank $n$ ${\mathbb{G}}_n$, the subRiemannian Brownian motion consists in a ${\mathbb{R}}^n$-Brownian motion together with its $\frac{n(n-1)}{2}$ L{é}vy areas. In this article we construct an explicit successful non co-adapted coupling of Brownian motions on ${\mathbb{G}}_n$. 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 ${\mathbb{G}}_n$. These results generalize the coupling strategy by Banerjee, Gordina and Mariano on the Heisenberg group.