Transport properties of the continuous-time random walk with a long-tailed waiting-time density

We derive asymptotic properties of the propagator p(r, t) of a continuous-time random walk (CTRW) in which the waiting time density has the asymptotic form ψ(t)∼T^α/t^α+1 when t≫T and 0<α<1. Several cases are considered; the main ones are those that assume that the variance of the displacement in a single step of the walk is finite. Under this assumption we consider both random walks with and without a bias. The principal results of our analysis is that one needs two forms to characterize p(r, t), depending on whether r is large or small, and that the small-r expansion cannot be characterized by a scaling form, although it is possible to find such a form for large r. Several results can be demonstrated that contrast with the case in which 〈t〉=∫₀^∞τψ(τ)dτ is finite. One is that the asymptotic behavior of p(0, t) is dominated by the waiting time at the origin rather than by the dimension. The second difference is that in the presence of a field p(r, t) no longer remains symmetric around a moving peak. Rather, it is shown that the peak of this probability always occurs at r=0, and the effect of the field is to break the symmetry that occurs when 〈t〉∞. Finally, we calculate similar properties, although in not such great detail, for the case in which the single-step jump probabilities themselves have an infinite mean.