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arXiv · 1904.06758

Stochastic differential equations for Lie group valued moment maps

Anton Alekseev·Elizaveta Arzhakova·Daria Smirnova
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Abstract

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.

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Anton Alekseev, Elizaveta Arzhakova, Daria Smirnova. 2019-04-14. Stochastic differential equations for Lie group valued moment maps. https://arxiv.org/abs/1904.06758

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