Continuous-time random walk with a superheavy-tailed distribution of waiting times

TL;DR

This paper analyzes the long-time behavior of continuous-time random walks with superheavy-tailed waiting times, showing convergence to exponential densities and characterizing superslow diffusion.

Contribution

It provides a detailed analysis of the asymptotic behavior of such random walks, including convergence results and examples of different superheavy-tailed distributions.

Findings

Convergence to exponential density for unbiased walks with finite second moment.

Moments grow slower than any power of time, indicating superslow diffusion.

Different superheavy-tailed waiting time distributions lead to various superslow diffusion laws.

Abstract

We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that if the random walk is unbiased (biased) and the jump distribution has a finite second moment then the properly scaled probability density converges in the long-time limit to a symmetric two-sided (an asymmetric one-sided) exponential density. The convergence occurs in such a way that all the moments of the probability density grow slower than any power of time. As a consequence, the reference random walk can be viewed as a generic model of superslow diffusion. A few examples of superheavy-tailed distributions of waiting times that give rise to qualitatively different laws of superslow diffusion are considered.