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.

中文翻译：

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量子 $$ SL _2$$、无限曲率和皮特曼的 2M-X 定理

Pitman 的经典定理指出，布朗运动减去其运行下界的两倍具有马尔可夫性质。一方面，Biane 明白 Pitman 定理与量子群 $\mathcal{U}_q\left( \mathfrak{sl}_2 \right)$ 的表示理论密切相关，在所谓的晶体状态 $ q \rightarrow 0$。另一方面，Bougerol 和 Jeulin 在双曲空间 $\mathbb{H}^3 = SL_2(\mathbb {C})/SU_2$。本文旨在通过给出统一的观点来理解这一现象。为了做到这一点，我们展示了 Jimbo-Drinfeld 量子群的演示 $\mathcal{U}_q^\hbar\left( \mathfrak{sl}_2 \right)$，它隔离了曲率 $r$ 和普朗克常数 $ 的作用\hbar$。参数之间的简单关系是$q=e^{-r}$。半经典极限 $\hbar \rightarrow 0$ 是对 $SL_2(\mathbb{C})$ 具有不同曲率 $r \in \mathbb{R}_+$ 对偶的泊松-李群。我们还构建了经典和量子随机游走，绘制了一张完整的图片，其中包括 Biane 的量子游走和 Bougerol-Jeulin 的构建。将曲率参数 $r$ 取到无穷大确实会导致表示理论水平的晶体状态（$\hbar>0$）和经典世界中的 Bougerol-Jeulin 构造（$\hbar=0$）。所有这些结果都完全符合基里洛夫的哲学