  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.
