Pitman's theorem states that a Brownian motion minus twice its current minimum is a Markov process. We will focus on two seemingly distinct approaches to this theorem: (1) Biane's approach, which uses a non-commutative random walk on the quantum group deforming \( SL_2 \) in the crystal regime q = 0, and (2) the Bougerol-Jeulin approach, involving Brownian motion on the hyperbolic space with infinite curvature. A unified version of these two approaches will be presented through a double deformation of the quantum group that isolates a curvature parameter and Planck's constant.
This talk is based on joint work with François Chapon (Toulouse)