Structure-preserving numerical methods for stochastic Poisson systems

TL;DR

This paper introduces structure-preserving numerical methods for stochastic Poisson systems by transforming them into stochastic Hamiltonian systems, applying symplectic discretizations, and then transforming back, ensuring the preservation of key geometric properties.

Contribution

It develops a novel class of numerical integrators for SPSs that preserve Poisson structures and Casimir functions using coordinate transformations and symplectic discretizations.

Findings

Methods successfully preserve Poisson structure and Casimir functions.

Applications demonstrate efficiency on stochastic rigid body and Lotka-Volterra systems.

Transformations enable structure-preserving discretizations for complex SPSs.

Abstract

We propose a class of numerical integration methods for stochastic Poisson systems (SPSs) of arbitrary dimensions. Based on the Darboux-Lie theorem, we transform the SPSs to their canonical form, the generalized stochastic Hamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. An a-generating function approach with \alpha\in [0,1] is then used to create symplectic discretizations of the SHSs, which are then transformed back by the inverse coordinate transformation to numerical integrators for the SPSs. These integrators are proved to preserve both the Poisson structure and the Casimir functions of the SPSs. Applications to a three-dimensional stochastic rigid body system and a three-dimensional stochastic Lotka-Volterra system show efficiency of the proposed methods.