Levy flights and nonhomogenous memory effects: relaxation to a stationary state

TL;DR

This paper investigates non-Markovian Levy flight dynamics with position-dependent memory effects, deriving asymptotic stationary states and analyzing relaxation patterns in both linear and nonlinear oscillators.

Contribution

It introduces a position-dependent subordinator to model nonhomogeneous memory effects and derives the relaxation behavior and stationary states for Levy flights in complex potentials.

Findings

Asymptotic stationary states are obtained for nonlinear oscillators.

Relaxation to stationary states follows Mittag-Leffler functions in the linear case.

Density distribution satisfies a fractional Fokker-Planck equation.

Abstract

The non-Markovian stochastic dynamics involving Levy flights and a potential in the form of a harmonic and non-linear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the non-linear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisfies a fractional Fokker-Planck equation. The densities for the non-linear oscillator reveal a complex picture, qualitatively dependent on the potential strength, and the relaxation pattern is exponential at large time.