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

TL;DR

This paper analyzes the long-time behavior of decoupled continuous-time random walks with superheavy-tailed waiting times and heavy-tailed jumps, deriving explicit forms for the limiting probability densities and their asymptotic behaviors.

Contribution

It provides explicit forms of the limiting probability densities for such random walks, including their expression via Fox H-functions and detailed asymptotic analysis.

Findings

The scaled position density converges to a non-degenerate limit under specific scaling.

The limiting density is expressed explicitly and exhibits heavy tails.

Asymptotic behaviors at small and large distances are characterized.

Abstract

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