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, (t)T/t+1, with 1<<2. For this system it is shown that asymptotically p(0,t)1/t1/. The form of the profile is shown to be Gaussian near the peak and to fall off like tx-(1+) near the tails, and the survival probability is asymptotically proportional to exp(-Bt/L). These results are confirmed by numerical calculations based on the method of exact enumeration.