We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.

中文翻译：

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随机泊松系统的保漂移数值积分器

我们对一类随机扰动的哈密顿系统和泊松系统进行了数值分析。对于此类系统所考虑的加性噪声​​扰动，我们展示了能量和二次卡西米尔的长期行为以获得精确解。然后，我们针对具有以下特性的此类问题提出并分析了一种保持漂移的分裂方案：能量的精确漂移保持和二次卡西米尔、收敛的均方阶 1、收敛的弱阶 2。这些特性通过数值实验来说明。