Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems

We propose a numerical method based on the Lie-Poisson reduction for a class of stochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $\mathfrak{g}^∗$ of the Lie algebra related to the Lie group manifold where the system is located, which is also the reduced form of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentum mapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplectic methods on the cotangent bundle to integrators on $\mathfrak{g}^∗.$ Stochastic generating functions creating stochastic symplectic methods are used to construct the schemes. An application to the stochastic rigid body system illustrates the theory and provides numerical validation of the method.