# Stochastic differential equations for Lie group valued moment maps

## TL;DR

This paper explores stochastic processes related to Duistermaat-Heckman measures for Lie group valued moment maps, focusing on SU(2) and hyperbolic space, providing new probabilistic interpretations.

## Contribution

It introduces stochastic process interpretations for DH measures of SU(2) and hyperbolic space valued moment maps, extending classical results to new geometric contexts.

## Key findings

- DH measures for SU(2) relate to processes on the unit disc
- DH measures for Poisson H^3 relate to processes on hyperbolas
- Provides probabilistic interpretations of geometric measures

## 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.