Pré-Publication, Document De Travail Année : 2023

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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Dates et versions

hal-04099288 , version 1 (16-05-2023)
hal-04099288 , version 2 (22-12-2023)
hal-04099288 , version 3 (30-03-2024)

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Magalie Bénéfice. Couplings of Brownian motions on $SU(2,\mathbb{C})$ and $SL(2,\mathbb{R})$. 2023. ⟨hal-04099288v2⟩
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