The classical work of Pitman in probability theory establishes a surprizing link between the Brownian motion in dimensions one and three. This relation was interpreted by Biane-Bougerol-O'Connell in terms of the Duistermaat-Heckman measure from symplectic geometry.
We generalize these constructions for the case of Brownian motion on curved three dimensional spaces: the 3-sphere and the hyperbolic space. The case of the hyperbolic space is intimately related to the quantum group U_q(sl(2)). We method is a combination of analytic results and numerical experiements which allowed to rule out some of the scenarios.