Diffusion of walkers with persistent velocities

We describe some properties for a phenomenological model of superdiffusion based on a generalization of the persistent random walk in one dimension to continuous time. The time spent moving to either increasing or decreasing x is characterized by a fractal-time pausing time density, {ψ}(t){~}T^α/t^α+1, with 12. For this system it is shown that asymptotically p(0,t){~}1/t^1/α. The form of the profile is shown to be Gaussian near the peak and to fall off like tx^-(1+α) near the tails, and the survival probability is asymptotically proportional to exp(-Bt/L^α). These results are confirmed by numerical calculations based on the method of exact enumeration.