Steady-state Levy flights in a confined domain

We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric Levy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for Levy flights is derived and solved analytically in the steady state. It is shown that Levy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified.