Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation

Abstract

A computer algorithm for the visualization of sample paths of anomalous diffusion processes is developed. It is based on the stochastic representation of the fractional Fokker-Planck equation describing anomalous diffusion in a nonconstant potential. Monte Carlo methods employing the introduced algorithm will surely provide tools for studying many relevant statistical characteristics of the fractional Fokker-Planck dynamics.

Stochastic Modeling of Indoor Air Temperature

“…The above relation is the extension of the similar behavior of stable subordinator and its inverse. This property is fully examined in [15]. The process {Z(t)} defined in (6) is called the subdiffusion process with time-dependent force and was considered in [10] not only for the case of the tempered stable distribution but for all infinitely divisible distributions of the subordinator.…”

Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

“…In the last decade the anomalous diffusion processes were analyzed by various number of authors in many disciplines. For example the subordinated Brownian motion driven by inverse Lévy-stable subordinator was considered in [11,22,24,25], the inverse tempered stable subordinator was examined in [23,[26][27][28] while inverse gamma process was mentioned in [29]. The general case of Lévy processes that can play a role of inverse subordinators were explored for example in [30][31][32].…”

“…In the domain of anomalous diffusion the typical approach is based on continuous time random walk (CTRW), [20,21], and subordinated Lévy processes can be treat as a limit in distribution of CTRW, [22]. The key issue in the framework of CTRW as well as in subordination technique is the waiting-times distribution corresponding to observed constant time periods [23].…”

Clustered continuous-time random walks: diffusion and relaxation consequences

“…However, the derivation of the HN function (1.2), based on the fractional Fokker-Planck equation (Kalmykov et al 2004) and the CTRW (Weron et al 2005;Jurlewicz et al 2008), does not lead to values of g > 1. Only recently, by employing the subordination approach Weron et al 2010), developed in the last decade in the theory of anomalous diffusion (Sokolov 2001(Sokolov , 2002Meerschaert et al 2002;Meerschaert & Scheffler 2004;Piryatinska et al 2005;Magdziarz & Weron 2006;Sokolov & Klafter 2006;Magdziarz et al 2007Magdziarz et al , 2008Lubelski et al 2008), has the effective picture underlying all typical and less typical relaxation patterns been found. In contrast to earlier studies (Weron & Kotulski 1996;Metzler et al 1999) where the anomalous-diffusion approach (based on a decoupled CTRW) has led to CC relaxation only, the compound subordination framework allows one to derive rigorously Weron et al 2010) the original HN function as well as the novel version…”