The celebrated result by Biane-Bougerol-O'Connell relates Duistermaat-Heckman (DH) measures for coadjoint orbits of a compact Lie group G with the multi-dimensional Pitman transform of the Wiener process on its Cartan subalgebra. The DH theory admits several non-trivial generalizations. In this paper, we consider the case of G=SU(2), and we give an interpretation of DH measures for SU(2) \cong S^3 valued moment maps in terms of an interesting stochastic process on the unit disc, and an interpretation of the DH measures for Poisson \mathbb{H}^3 valued moment maps (in the sense of Lu) in terms of a stochastic process on the interior of a hyperbola.
