We show that a quantum particle in R d , for d 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement x 2 ρ(x, x, t) dx grows like t 3 in any dimension. The white noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. This is a known result in one dimension [7, 15]. The energy of the system is also shown to increase linearly in time. We also prove that for the same white-noise potential model on the lattice Z d , for d 1, the mean square displacement is diffusive growing like t 1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models in the lattice Z d with a time-dependent Markovian potential [17].