Asymmetric Levy flights in nonhomogeneous environments

Abstract

We consider stochastic systems involving general -- non-Gaussian and asymmetric -- stable processes. The random quantities, either a stochastic force or a waiting time in a random walk process, explicitly depend on the position. A fractional diffusion equation corresponding to a master equation for a jumping process with a variable jumping rate is solved in a diffusion limit. It is demonstrated that for some model parameters the equation is satisfied in that limit by the stable process with the same asymptotics as the driving noise. The Langevin equation containing a multiplicative noise, depending on the position as a power-law, is solved; the existing moments are evaluated. The motion appears subdiffusive and the transport depends on the asymmetry parameter: it is fastest for the symmetric case. A special case of the one-sided distribution is discussed.