Non-Markovian Levy diffusion in nonhomogeneous media

TL;DR

This paper investigates a non-Markovian fractional diffusion equation with position-dependent coefficients, analyzing how different memory kernels influence Levy-type diffusion in nonhomogeneous media.

Contribution

It introduces a unified framework for solving non-Markovian fractional diffusion equations with variable coefficients and compares effects of different memory kernels on Levy diffusion.

Findings

Exponential kernel reduces to telegrapher's equation.

Power-law kernel maintains Levy process characteristics.

Renormalized fractional moments quantify diffusion differences.

Abstract

We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The resulting distributions have the form of the L\'evy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the system.