L\'evy processes on a generalized fractal comb

TL;DR

This paper explores how fractal structures in a comb model influence anomalous diffusion, leading to Le9vy flights and superdiffusion, with exact solutions derived using Fox H-functions and analysis of memory effects.

Contribution

It introduces a generalized fractal comb model incorporating Le9vy processes and provides exact solutions for probability distributions under various memory kernels.

Findings

Exact solutions for probability distributions using Fox H-functions.

Identification of superdiffusive regimes and their dependence on kernels.

Observation of competition between long rests and long jumps in certain kernels.

Abstract

Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structure can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the L\'evy processes (L\'evy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, L\'evy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox $H$-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractional structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. It is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.