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

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.

16 pages, 1 figure

References in corpus (5)

• Stochastic calculus for uncoupled continuous-time random walks
• Continuous-time random walk theory of superslow diffusion
• Langevin equation with super-heavy-tailed noise
• Continuous-time random walk with a superheavy-tailed distribution of waiting times
• Probability distribution function for systems driven by superheavy-tailed noise

Cited by in corpus (7)

• Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes
• Aging dynamics in interacting many-body systems
• Subdiffusion equation with Caputo fractional derivative with respect to another function
• Localization and universal fluctuations in ultraslow diffusion processes
• Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion
• Limiting distributions of continuous-time random walks with superheavy-tailed waiting times
• Statistics of bounded processes driven by Poisson white noise