Enhancement of stability of metastable states in the presence of L\'{e}vy noise

The barrier crossing event for superdiffusion in the form of symmetric L\'{e}vy flights is investigated. We derive from the fractional Fokker-Planck equation a general differential equation with the corresponding conditions useful to calculate the mean residence time of a particle in a fixed interval for an arbitrary smooth potential profile, in particular metastable, with a sink and a L\'{e}vy noise with an arbitrary index $\alpha$. A closed expression in quadrature of the nonlinear relaxation time for L\'{e}vy flights with the index $\alpha =1$ in cubic metastable potential is obtained. Enhancement of the mean residence time in the metastable state, analytically derived, due to L\'{e}vy noise is found.