The propagation of test particles in a turbulent plasma is considered both analytically and numerically. We split the Green's function into a deterministic and a stochastic part and derive several exact relations between them; these relations show that partial summations of the perturbation series do not lead, in general, to a consistent closure for the hierarchy problem. The mean propagator in velocity space is calculated numerically for an acceleration field given by an Ornstein-Uhlenbeck process. If the relative coherent changeε of the particle velocity remains smaller than 0.1% the Fokker-Planck description may be adequate for the time evolution of the propagator. Otherwise the picture of Brownian motion becomes worse, and incorporating memory effects or renormalization of the mass operator have also been found to disagree with the numerical experiment.