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 , and we give an interpretation of DH measures for valued moment maps in terms of an interesting stochastic process on the unit disc, and an interpretation of the DH measures for Poisson valued moment maps (in the sense of Lu) in terms of a stochastic process on the interior of a hyperbola.
Funding Statement
The research of AA and DS was supported in part by the grants 178794, 178828, 182767 and by the NCCR SwissMAP of the Swiss National Science Foundation. The research of AA was supported in part by the project MODFLAT of the European Research Council (ERC).
Acknowledgments
We are indebted to D. Chelkak and L. Parnovksi for interesting suggestions. We are grateful to the referee of this paper for useful comments. This work is partly based on the Master thesis of EA at the University of Geneva.
Citation
Anton Alekseev. Elizaveta Arzhakova. Daria Smirnova. "Stochastic differential equations for Lie group valued moment maps." Electron. Commun. Probab. 26 1 - 9, 2021. https://doi.org/10.1214/21-ECP427
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