Quantum $SL_2$, Infinite curvature and Pitman's 2M-X theorem

The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman's theorem is intimately related to the representation theory of the quantum group $\mathcal{U}_q\left( \mathfrak{sl}_2 \right)$, in the so-called crystal regime $q \rightarrow 0$. On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit $r \rightarrow \infty$ of a Brownian motion on the hyperbolic space $\mathbb{H}^3 = SL_2(\mathbb{C})/SU_2$. This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation $\mathcal{U}_q^\hbar\left( \mathfrak{sl}_2 \right)$ of the Jimbo-Drinfeld quantum group which isolates the role of curvature $r$ and that of the Planck constant $\hbar$. The simple relationship between parameters is $q=e^{-r}$. The semi-classical limits $\hbar \rightarrow 0$ are the Poisson-Lie groups dual to $SL_2(\mathbb{C})$ with varying curvatures $r \in \mathbb{R}_+$. We also construct classical and quantum random walks, drawing a full picture which includes Biane's quantum walks and the construction of Bougerol-Jeulin. Taking the curvature parameter $r$ to infinity leads indeed to the crystal regime at the level of representation theory ($\hbar>0$) and to the Bougerol-Jeulin construction in the classical world ($\hbar=0$). All these results are neatly in accordance with the philosophy of Kirillov's orbit method.