Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions

摘要

We study the long-time behavior of decoupled continuous-time random walkscharacterized by superheavy-tailed distributions of waiting times and symmetricheavy-tailed distributions of jump lengths. Our main quantity of interest isthe limiting probability density of the position of the walker multiplied by ascaling function of time. We show that the probability density of the scaledwalker position converges in the long-time limit to a non-degenerate one onlyif the scaling function behaves in a certain way. This function as well as thelimiting probability density are determined in explicit form. Also, we expressthe limiting probability density which has heavy tails in terms of the Fox$H$-function and find its behavior for small and large distances.