Anomalous diffusion in a space- and time-dependent energy landscape

We study the influence on diffusion in one dimension of a potential energy perturbation varying as a power in space and time. We concentrate on the case of a parabolic perturbation in space decaying as $t^{-\omega}$ which shows a rich variety of scaling behaviours. When $\omega=1$, the perturbation is truly marginal and leads to anomalous (super)diffusion with a dynamical exponent varying continuously with the perturbation amplitude below some negative threshold value. For slower decay, $\omega<1$, the perturbation becomes relevant and the system is either subdiffusive for an attractive potential or displays a stretched-exponential behaviour for a repulsive one. Exact results are obtained for the mean value and the variance of the position as well as for the surviving probability.