Stochastic differential equations for Lie group valued moment maps

2019/04/14 by Alekseev, Anton, Arzhakova, Elizaveta, Smirnova, Daria
#53D99 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1904.06758

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) ≅ S³ valued moment maps in terms of an interesting stochastic process on the unit disc, and an interpretation of the DH measures for Poisson ℍ³ valued moment maps (in the sense of Lu) in terms of a stochastic process on the interior of a hyperbola.

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