Couplings of Brownian Motions on SU(2, C)

Abstract

We study couplings of Brownian motions on the subRiemannian manifold SU(2, C), that is diffusion processes having the subLaplacian operator as infinitesimal generator. Using some interesting geometric interpretations of the Brownian motion on this space, we present some basic examples of co-adapted couplings. Inspiring ourselves with works on the Heisenberg group, we also construct two successful couplings on SU(2, C) : one co-adapted and one not co-adapted but having a good coupling rate. Finally, using the similar structure of SU(2, C) and SL(2, R) we generalise some of our results for the case of this second subRiemannian manifold.