Fractional Fokker-Planck equation for Levy flights in nonhomogeneous environments

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Levy 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 Levy process with a simple scaling. The solution for the drift term in the form lambda sgn(x) possesses two different scales which correspond to the Levy 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.