Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments.
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 lambda sgn(x) possesses two different scales which correspond to the Lévy indexes micro and micro+1 (micro<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 .
Srokowski
Tomasz
T
Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland.