Levy flights: transitions and meta-stability

We consider Levy flights of stability index alpha epsilon (0, 2) in a potential landscape in the limit of a small noise parameter. We give a purely probabilistic description of the random dynamics on the basis of a special decomposition of the driving Levy processes into independent small jumps and compound Poisson parts. We prove that escape times from a potential well are exponentially distributed and their mean values increase as a power epsilon(-alpha) of the noise intensity epsilon. This allows us to obtain meta-stability results for a jump diffusion in a double-well potential.