Levy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

We consider Levy flights subject to external force fields. This anomalous t ransport process is described by two approaches, a Langevin equation with L evy noise and the corresponding generalized Fokker-Planck equation containi ng a fractional derivative in space. The cases of foe flights, constant for ce, and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbs ian character of the stationary solution for the case of the Hookean force, i.e., the deviation from Boltzmann equilibrium for long times. The possibl e connection to Tsallis's q statistics is studied. [S1063-651X(99)09903-1].