Fractional dynamics from the ordinary Langevin equation

TL;DR

This paper derives a fractional Fokker-Planck equation from a Langevin equation with a time change based on a self-similar Markov process, revealing non-Markovian dynamics with finite moments and preserved physical relations.

Contribution

It introduces a novel approach to fractional dynamics by replacing the internal time in the Langevin equation with a first passage time of a self-similar Markov process, leading to new non-Markovian models.

Findings

The resulting process satisfies the fluctuation-dissipation relation.

All moments of the process are finite.

The process obeys the H-theorem.

Abstract

We consider the usual Langevin equation depending on an internal time. This parameter is substituted by a first passage time of a self-similar Markov process. Then the Gaussian process is parent, and the hitting time process is directing. The probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes. The corresponding master equation becomes the fractional Fokker-Planck equation. We show that the resulting process has non-Markovian properties, all its moments are finite, the fluctuation-dissipation relation and the H-theorem hold.