  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.
