Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments

Abstract

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Lévy 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 Lévy process with a simple scaling. The solution for the drift term in the form $λ\hbox{sgn}(x)$ possesses two different scales which correspond to the Lévy indexes $μ$ and $μ+1$ $(μ<1)$. 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 $λ$.