Comment on Fractional Fokker–Planck Equation with Space and Time Dependent Drift and Diffusion

Marcin Magdziarz,Tomasz Zorawik,Janusz Gajda

doi:10.1007/s10955-014-0919-9

Abstract

The purpose of this paper is to correct errors presented recently in the paper [Lv et al. J Stat Phys 149:619–628 (2012)], where the authors analyzed Fractional Fokker–Planck equation (FFPE) with space–time dependent drift $$F(x,t)=F(x)f(t)$$ and diffusion $$D(x,t)=D(x)\tilde{d}(t)$$ coefficients in the factorized form. We show an important drawback in the derivation of the stochastic representation of FFPE presented in the aforementioned paper, which makes the whole proof wrong. Moreover, we present a correct proof of their result in even more general case, when both drift and diffusion can have any, not necessarily factorized, form.

Highlights

In recent paper by Lv et al [1] authors studied the question of stochastic representation of anomalous diffusion with space and time dependent drift and diffusion coefficient
We point out some drawbacks in the derivation of the main result in [1] and present the correct proof
Applying the method from [6], we derive the stochastic representation of Fractional Fokker– Planck equation (FFPE) with arbitrary drift and diffusion parameters

Summary

Introduction

In recent paper by Lv et al [1] authors studied the question of stochastic representation of anomalous diffusion with space and time dependent drift and diffusion coefficient. Since in every moment of jump we have Uα(Sα(t)) = t (see [3]), the time-dependent biases acting on the particle are equal to f (Uα(Sα(t))) = f (t) and d(Uα(Sα(t))) = d(t) This means that whenever the particle does not rest in the trap, the coefficients f (t) and d(t) are not modified by the subordinator Sα(t). This fulfills the physical requirement that the external time-dependent force cannot be influenced by the environment and corresponds to the fact that the fractional operator 0 Dt1−α in (1) appears to the right of the Fokker–Planck operator. Applying the method from [6], we derive the stochastic representation of FFPE with arbitrary drift and diffusion parameters

Problem in the Proof

Stochastic Representation of Generalized FFPE

Conclusions