Couplings of Brownian motions on $SU(2,\mathbb{C})$ and $SL(2,\mathbb{R})$
Résumé
The Lie groups $SU(2)$ and $SL(2,\mathbb{R})$ are some model spaces in subelliptic geometry. Similar to the Heisenberg group, the subelliptic Brownian motion on $SU(2)$ (resp. $SL(2,\mathbb{R})$) may be seen as a Brownian motion on the sphere (resp. the hyperbolic plane) together with its swept area. We propose an explicit construction of a co-adapted successful coupling on $SU(2)$. The strategy is to alternate between reflection and synchronous (with noise) coupling on the sphere. We also evaluate the subRiemannian distance between general co-adapted couplings of subelliptic Brownian motions in $SU(2)$ and $SL(2,\mathbb{R})$.
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