Quantum SL2, infinite curvature and Pitman's 2M-X theorem

• 成果类型: Article
• 署名作者: Chapon, Francois; Chhaibi, Reda
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01002-8
• 发表日期: 2021
• 页码: 835-888
• 关键词: weyl chambers; random-walks; paths

摘要

The classical theorem by Pitman states that a Brownianmotionminus 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 U-q (sl(2)), in the so-called crystal regime q -> 0. On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit r. 8 of a Brownian motion on the hyperbolic space H-3 = SL2(C)/SU2. This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation U-q(h) (sl(2)) of the Jimbo-Drinfeld quantum group which isolates the role of curvature r and that of the Planck constant (h) over bar. The simple relationship between parameters is q = e-r. The semi-classical limits (h) over bar. 0 are the Poisson-Lie groups dual to SL2( C) with varying curvatures r. 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 ((h) over bar > 0) and to the Bougerol-Jeulin construction in the classical world ( (h) over bar = 0). All these results are neatly in accordance with the philosophy of Kirillov's orbit method.