The problem of analytical calculation of barrier crossing characteristics for Lévy flights

By using the backward fractional Fokker–Planck equation we investigate the barrier crossing event in the presence of Lévy noise. After briefly reviewing recent results obtained with different approaches on the time characteristics of the barrier crossing, we derive a general differential equation useful to calculate the nonlinear relaxation time. We obtain analytically the nonlinear relaxation time for free Lévy flights and a closed expression in quadrature of the same characteristics for the cubic potential.