  The Levy-flight dynamics can stem from simple random walks in a system whose
operational time (number of steps n) typically grows superlinearly with
physical time t. Thus, this processes is a kind of continuous-time random walks
(CTRW), dual to usual Scher-Montroll model, in which $n$ grows sublinearly with
t. The models in which Levy-flights emerge due to a temporal subordination let
easily discuss the response of a random walker to a weak outer force, which is
shown to be nonlinear. On the other hand, the relaxation of en ensemble of such
walkers in a harmonic potential follows a simple exponential pattern and leads
to a normal Boltzmann distribution. The mixed models, describing normal CTRW in
superlinear operational time and Levy-flights under the operational time of
subdiffusive CTRW lead to paradoxical diffusive behavior, similar to the one
found in transport on polymer chains. The relaxation to the Boltzmann
distribution in such models is slow and asymptotically follows a power-law.
