Lévy flights and nonhomogenous memory effects: Relaxation to a stationary state

Abstract

The non-Markovian stochastic dynamics involving Lévy 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.

Escape process in systems characterized by stable noises and position-dependent resting times

“…The system of equations (8) corresponds, in 1D case, to a fractional Fokker-Planck equation with a variable diffusion coefficient [31,32],…”

“…Eq. ( 8) corresponds to a fractional Fokker-Planck equation where the fractional derivatives are taken over both time and position [32]. The first passage time problem resolves itself to a solution of that equation with appropriate boundary conditions.…”