Splitting integrators for stochastic Lie--Poisson systems

TL;DR

This paper introduces explicit splitting integrators for stochastic Lie--Poisson systems that preserve geometric properties like Casimir functions, providing theoretical analysis and numerical validation for their effectiveness.

Contribution

It proposes new stochastic Poisson integrators based on splitting strategies that maintain key geometric invariants and analyzes their convergence and stability properties.

Findings

Preservation of Casimir functions in stochastic systems

Existence of strong and weak convergence rates

Numerical experiments confirm theoretical properties

Abstract

We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie--Poisson systems, namely: stochastically perturbed Maxwell--Bloch, rigid body and sine--Euler equations.