Time characteristics of one-dimensional and two-dimensional stationary Lévy flights in different potential profiles

We consider the anomalous diffusion in the form of Lévy flights in one-dimensional and two-dimensional potentials. For one-dimensional case the correlation time of the steady-state Lévy flights in the bistable symmetric quartic potential potential is explored. We have found that the dependence of the correlation time on the barrier height for sufficiently high barriers obeys a power law unlike an exponential dependence in the case of normal diffusion. Further, for two-dimensional diffusion the general Kolmogorov equation for the joint probability density function of particle coordinates is obtained by functional methods directly from two Langevin equations with statistically independent noise sources. We analyze in detail the Brownian diffusion and Lévy flights in potentials with radial symmetry. As shown, the radial symmetry property of the steady-state joint probability distribution available for normal diffusion is broken for Lévy flights.