Lévy flights in confining potentials

We analyze confining mechanisms for Lévy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump-type processes are considered: those driven by Langevin equation with Lévy noise and those, named topological Lévy processes (occurring in systems with topological complexity such as folded polymers or complex networks), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay in affinity and may share common stationary probability density, even if their detailed dynamical behavior look different. This near-equilibrium observation seems to be generic to a broad class of Lévy noise-driven processes, such as e.g., superdiffusion on folded polymers, geophysical flows, and even climatic changes.