  We consider Levy flights subject to external force fields. This anomalous
transport process is described by two approaches, a Langevin equation with Levy
noise and the corresponding generalized Fokker-Planck equation containing a
fractional derivative in space. The cases of free flights, constant force and
linear Hookean force are analyzed in detail, and we corroborate our findings
with results from numerical simulations. We discuss the non-Gibbsian character
of the stationary solution for the case of the Hookean force, i.e. the
deviation from Boltzmann equilibrium for long times. The possible connection to
Tsallis's q-statistics is studied.
