Some fundamental aspects of Lévy flights

We investigate the physical basis and properties of Lévy flights (LFs), Markovian random walks with a long-tailed density of jump lengths, λ(ξ) ξ-1-α, with 0<α<2. In particular, we show that non-trivial boundary conditions need to be carefully posed, and that the method of images fails due to the non-locality of LFs. We discuss the behaviour of LFs in external potentials, demonstrating the existence of multimodal solutions whose maxima do not coincide with the potential minimum. The Kramers escape of LFs is investigated, and the physical nature of the a priori diverging kinetic energy of an LF is addressed.