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In this article we continue the study of couplings of subelliptic Brownianmotions on the subRiemannian manifolds SU (2) and SL(2, R). Similar to the caseof the Heisenberg group, this subelliptic Brownian motion can be considered asa Brownian motion on the sphere (resp. the hyperbolic plane) together with itsswept area modulo 4$\pi$. Using this structure, we construct an explicit nonco-adapted successful coupling on SU (2) and, under strong conditions on thestarting points, on SL(2, R) too. This strategy uses couplings of Brownianbridges, taking inspiration into the work from Banerjee, Gordina and Mariano[3] on the Heisenberg group. We prove that the coupling rate associated tothese constructions is exponentially decreasing in time and proportionally tothe subRiemannian distance between the starting points. We also give somegradient inequalities that can be deduced from the estimation of this couplingrate.

Non co-adapted couplings of Brownian motions on subRiemannian manifolds