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

Stochastic differential equations for Lie group valued moment maps