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

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.

7 pages

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Cited by in corpus (10)

• Subdiffusion equation with Caputo fractional derivative with respect to another function
• Localization and universal fluctuations in ultraslow diffusion processes
• Slow Lévy flights
• Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions
• Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion
• Model of anomalous diffusion--absorption process in a system consisting of two different media separated by a thin membrane
• Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion
• Subdiffusion equation with Caputo fractional derivative with respect to another function in modelling diffusion in a complex system consisting of matrix and channels
• Limiting distributions of continuous-time random walks with superheavy-tailed waiting times
• Statistics of bounded processes driven by Poisson white noise