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)∼${\mathit{T}}^{\mathrm{\alpha }}$/${\mathit{t}}^{\mathrm{\alpha }+1\mathrm{}}$, with 1p(0,t)∼1/${\mathit{t}}^{1/\mathrm{\alpha }}$. The form of the profile is shown to be Gaussian near the peak and to fall off like ${\mathit{tx}}^{\mathrm{-}(1+\mathrm{\alpha })}$ near the tails, and the survival probability is asymptotically proportional to exp(-Bt/${\mathit{L}}^{\mathrm{\alpha }}$). These results are confirmed by numerical calculations based on the method of exact enumeration.