Diffusion equations for a Markovian jumping process

TL;DR

This paper derives and analyzes diffusion equations for a Markovian jumping process with position-dependent jump frequency, revealing various diffusion regimes and fractional dynamics, supported by analytical solutions and numerical simulations.

Contribution

It introduces a fractional diffusion equation with variable coefficients for a Markovian jumping process with power-law frequency, extending classical models.

Findings

Normal, subdiffusion, and superdiffusion regimes identified.

Fractional moments and diffusion coefficients calculated.

Numerical solutions show deviations from Lévy stable distributions at large wave numbers.

Abstract

We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency. For small steps, we derive the Fokker-Planck equation and show the presence of the normal diffusion, subdiffusion and superdiffusion. For the L\'evy distribution of the step-size, we construct a fractional equation, which possesses a variable coefficient, and solve it in the diffusion limit. Then we calculate fractional moments and define fractional diffusion coefficient as a natural extension to the cases with the divergent variance. We also solve the master equation numerically and demonstrate that there are deviations from the L\'evy stable distribution for large wave numbers.