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 (12)

• The scaling laws of human travel
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Lévy flights in random environments
• Fractional Fokker--Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises
• Levy Anomalous Diffusion and Fractional Fokker--Planck Equation
• Lévy flights in a steep potential well
• Levy Flights in Inhomogeneous Media
• Levy flights in quenched random force fields
• Steady-State Lévy Flights in a Confined Domain
• Diffusion equations for a Markovian jumping process
• 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 (20)

• Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes
• The Continuous Time Random Walk, still trendy: Fifty-year history, state of art, and outlook
• Saturation and Scaling of Multiplicity, Mean p_T and p_T Distributions from 200 GeV < sqrt{s} < 7 TeV - Addendum
• Multiplicative Lévy processes: Itô versus Stratonovich interpretation
• Stationary states in single-well potentials under symmetric Levy noises
• Nonlinear stochastic equations with multiplicative Lévy noise
• 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
• Continuous time multidimensional Markovian description of Levy walks
• Lévy flights in inhomogeneous environments and 1/f noise
• Closed-form solutions for the Lévy-stable distribution
• Escape from bounded domains driven by multi-variate $\alpha$-stable noises
• Stationary states in 2D systems driven by bi-variate Lévy noises
• Lévy on-off intermittency
• Residence time distributions for in-line chaotic mixers
• Competition between ac driving-forces and Lévy flights in a nonthermal ratchet
• The neutral heavy scalar productions associated with $Z_L$ in the littlest Higgs model at ILC and CLIC
• Equivalent continuous and discrete realizations of Levy flights: Model of one-dimensional motion of inertial particle
• Eigenvalues and eigenvectors of the Laplace operator in d-dimensional cut Fock basis