Levy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of Levy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place. To this end we employ two model patterns of dynamical behavior: Langevin-driven and (Levy-Schrodinger) semigroup-driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Levy oscillators), with respect to both dynamical mechanisms. We discuss the links of the Levy semigroup motion scenario with that of random searches in spatially inhomogeneous media. (C) 2010 Elsevier B.V. All rights reserved.