We analyze a specific class of random systems that, while being driven by a symmetric Lévy stable noise, asymptotically set down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ∗(x)∼exp[−Φ(x)]. This behavior needs to be contrasted with the...

We show that, under suitable confinement conditions, the ordinary Fokker–Planck equation may generate non-Gaussian heavy-tailed probability density functions (pdfs) (like, for example, Cauchy or more general Lévy stable distributions) in its long-time asymptotics. In fact, all heavy-tailed...

We analyze confining mechanisms for Lévy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the...

We investigate an undamped random phase-space dynamics in deterministic external force fields (conservative and magnetic ones). By employing the hydrodynamical formalism for those stochastic processes we analyze microscopic kinetic-type “collision invariants” and their relationship to local...

We study the long time asymptotics of probability density functions (pdfs) of Lévy 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...