Fractional Fokker-Planck equation for L\'evy flights in nonhomogeneous environments

TL;DR

This paper derives and solves a fractional Fokker-Planck equation modeling Lévy flights in nonhomogeneous environments, revealing stationary solutions, scale behaviors, and fractional moment dynamics.

Contribution

It introduces solutions to the fractional Fokker-Planck equation with variable diffusion, including cases with linear and sign-based drift, highlighting new scaling properties and moment behaviors.

Findings

Stationary solutions for linear drift case.

Presence of two scales for sign-based drift with different Lévy indexes.

Fractional moments increase over time, with growth rate depending on bb.

Abstract

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the L\'evy flights in a nonhomogeneous medium. For the case with the linear drift, the solution is stationary in the long-time limit and it represents the L\'evy process with a simple scaling. The solution for the drift term in the form $\lambda\hbox{sgn}(x)$ possesses two different scales which correspond to the L\'evy indexes $\mu$ and $\mu+1$ $(\mu<1)$. The former component of the solution prevails at large distances but it diminishes with time for a given $x$. The fractional moments, as a function of time, are calculated. They rise with time and the rate of this growth increases with $\lambda$.