Stretched-Gaussian asymptotics of the truncated L\'evy flights for the diffusion in nonhomogeneous media

TL;DR

This paper investigates the asymptotic behavior of truncated Lévy flights in nonhomogeneous media, revealing stretched Gaussian tails and the impact of truncation on the distribution's tail behavior through analytical solutions and Monte Carlo simulations.

Contribution

It introduces a fractional diffusion model with variable diffusion coefficient and analyzes the effects of exponential and power-law truncations on Lévy flights.

Findings

Stretched Gaussian tails appear in truncated Lévy flights.

The tail behavior depends on the truncation type and parameters.

Time to reach the limiting distribution varies with jumping rate.

Abstract

The L\'evy, jumping process, defined in terms of the jumping size distribution and the waiting time distribution, is considered. The jumping rate depends on the process value. The fractional diffusion equation, which contains the variable diffusion coefficient, is solved in the diffusion limit. That solution resolves itself to the stretched Gaussian when the order parameter $\mu\to2$. The truncation of the L\'evy flights, in the exponential and power-law form, is introduced and the corresponding random walk process is simulated by the Monte Carlo method. The stretched Gaussian tails are found in both cases. The time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter. When the cutoff function falls slowly, the tail of the distribution appears to be algebraic.