Couplings of Brownian motions on SU(2) and SL(2,R)

Abstract

The Lie groups $SU\left(2\right)$ and $SL(2,R)$ can be viewed as model spaces in subRiemannian geometry. Coupling two subelliptic Brownian motions on $SU\left(2\right)$ (resp. $SL(2,R)$) consists in simultaneously coupling two Brownian motions on the sphere (resp. the hyperbolic plane) and their swept areas. Using this approach we propose an explicit construction of a co-adapted successful coupling on $SU\left(2\right)$. The strategy is to alternate between reflection and synchronous (with noise) couplings on the sphere. We also describe some more general constructions of co-adapted couplings on $SU\left(2\right)$ and on $SL(2,R)$.