Lévy flights in nonhomogeneous media: Distributed-order fractional equation approach

Abstract

A jumping process, defined in terms of the Lévi distributed jumping size and the Poissonian, position-dependent waiting time with the algebraic jumping rate, is discussed on the assumption that parameters of both distributions are themselves random variables which are determined from given probability distributions. The fractional equation for the distributed Lévy order parameter mu is derived and solved. The solution is of the form of a combination of the Fox functions and simple scaling is lacking. The problem of accelerated diffusion is also discussed. The case of the distributed waiting time parameter theta is similarly solved and the solution offers a possibility to manage processes which are characterized by more general forms of the jumping rate, not only algebraic. Moreover, we mention a possibility that the parameters mu and theta are mutually dependent.