We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative It\^o and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.

翻译：我们研究一类由Wiener噪声驱动的线性Vlasov偏微分方程。处理了不同类型的随机扰动：加性噪声、乘性Itô和Stratonovich噪声以及输运噪声。我们提出采用裂步积分器对这些随机偏微分方程进行时间离散化。这些积分器的设计旨在根据随机扰动类型保持精确解的定性性质，例如解的范数守恒性或正性。我们通过数值实验说明了所提积分器的性质，并研究了均方收敛阶。