Fractional Fokker-Planck equation with tempered $\ensuremath{\alpha}$-stable waiting times: Langevin picture and computer simulation

Fractional Fokker-Planck equation with tempered $α$ -stable waiting times: Langevin picture and computer simulation

Janusz Gajda and Marcin Magdziarz

Phys. Rev. E 82, 011117 – Published 14 July 2010

Abstract

In this paper we introduce a Langevin-type model of subdiffusion with tempered $α$ -stable waiting times. We consider the case of space-dependent external force fields. The model displays subdiffusive behavior for small times and it converges to standard Gaussian diffusion for large time scales. We derive general properties of tempered anomalous diffusion from the theory of tempered $α$ -stable processes, in particular we find the form of the fractional Fokker-Planck equation corresponding to the tempered subdiffusion. We also construct an algorithm of simulation of sample paths of the introduced process. We apply the algorithm to approximate solutions of the fractional Fokker-Planck equation and to study statistical properties of the tempered subdiffusion via Monte Carlo methods.