Couplings of Brownian motions on {$SU(2,\mathbb{C})$} and {$SL(2,\mathbb{R})$}

Magalie Bénéfice (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 UB - Université de Bordeaux

In the subRiemannian manifolds $SU(2,\mathbb{C})$ (resp. $SL(2,\mathbb{R})$), the choice of cylindrical coordinates gives an interesting interpretation of the Brownian motion as a Brownian motion on the sphere (resp. the hyperbolic plane) together with its swept area. After a detailed account of this geometrical interpretation, we use Itô depiction of processes in a moving frame to present models of co-adapted couplings of these Brownian motions. In particular we propose a successful co-adapted coupling on $SU(2,\mathbb{C})$ inspiring ourselves with previous works on the Heisenberg group.

Mots clés: SubRiemannian manifolds; Brownian motion; Coupling; Co-adapted coupling; Successful coupling

Domaines: Probabilités [math.PR]

Magalie Bénéfice. Couplings of Brownian motions on {$SU(2,\mathbb{C})$} and {$SL(2,\mathbb{R})$}. 2023. ⟨hal-04099288v1⟩