Levy flights in confining potentials

Piotr Garbaczewski, Vladimir Stephanovich

Published 2009-04-27, updated 2009-06-22Version 2

We analyze confining mechanisms for L\'{e}vy 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 L\'{e}vy noise and those, named by us topological L\'{e}vy processes (occurring in systems with topological complexity like folded polymers or complex networks and generically in inhomogeneous media), 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 (eventually asymptotic) probability density, even if their detailed dynamical behavior look different. That generalizes and offers new solutions to a reverse engineering (e.g. targeted stochasticity) problem due to I. Eliazar and J. Klafter [J. Stat. Phys. 111, 739, (2003)]: design a L\'{e}vy process whose target pdf equals a priori preselected one. Our observations extend to a broad class of L\'{e}vy noise driven processes, like e.g. superdiffusion on folded polymers, geophysical flows and even climatic changes.