Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments

Abstract

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Lévy flights in a nonhomogeneous medium. For the case with the linear drift, the solution is stationary in the long-time limit and it represents the Lévy process with a simple scaling. The solution for the drift term in the form $\lambda sgn(x)$ possesses two different scales which correspond to the Lévy indexes $\mu$ and $\mu +1$ $(\mu <1)$. The former component of the solution prevails at large distances but it diminishes with time for a given $x$. The fractional moments, as a function of time, are calculated. They rise with time and the rate of this growth increases with $\lambda$.

6 pages, 2 figures

References in corpus (4)

• The scaling laws of human travel
• Steady-State Lévy Flights in a Confined Domain
• Synchronization of extended chaotic systems with long-range interactions: an analogy to Levy-flight spreading of epidemics
• Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

Cited by in corpus (9)

• Saturation and Scaling of Multiplicity, Mean p_T and p_T Distributions from 200 GeV < sqrt{s} < 7 TeV - Addendum
• A topological approximation of the nonlinear Anderson model
• A mixed SOC-turbulence model for nonlocal transport and Levy-fractional Fokker-Planck equation
• Green function of the double fractional Fokker-Planck equation: Path integral and stochastic differential equations
• Lévy flights in inhomogeneous environments and 1/f noise
• Stationary states in 2D systems driven by bi-variate Lévy noises
• Lévy on-off intermittency
• Residence time distributions for in-line chaotic mixers
• Eigenvalues and eigenvectors of the Laplace operator in d-dimensional cut Fock basis