Heavy tailed Lévy random motions in super- and subharmonic potential wells

The Lévy motions (LMs) are random processes with stationary increments having a long-tailed stable probability distribution with divergent variance. As such, LMs are a natural generalization of the Brownian motion ensuing from the generalized central limit theorem. Despite their broad field of applications, LMs are far from being completely understood. Here, we consider ordinary LMs, that is Markovian processes with independent increments, subjected to an external non-linear potential force. The behavior is different for the steep superharmonic potential wells with the index c < 2 and for the gently sloping subharmonic ones, c < 2. For the superharmonic one-well potentials we found that the stationary probability density function (PDF) features a distinct bimodal shape and decays with the power-law tails which are steep enough to give rise to a finite variance, in contrast to a "free" LM. During the relaxation to stationarity the PDF exhibits unimodal-bimodal time bifurcations. In anharmonic one-well potentials we observe unimodal-bimodal bifurcations of the stationary PDF. For subharmonic one-well potentials the stationary solution does not exist if the Lévy index is smaller than the index of the well.

For the two-well potentials we found that the mean escape time is inversely proportional to the intensity of the Lévy motion, in sharp contrast with the known exponential behavior for the Brownian motion. Furthermore, we consider the phenomenon of generating directed current by the LM in periodic asymmetric potential and demonstrate how direction and magnitude of the current can be manipulated by changing the asymmetry of the LM and the potential asymmetry. These properties of the LMs are discussed on the basis of analytical and numerical solutions of the corresponding stochastic nonlinear differential equations as well as the kinetic equations with partial derivatives of fractional order.