Stochastic representation of subdiffusion processes with time-dependent drift
type: Item
instance of: scholarly article
title: Stochastic representation of subdiffusion processes with time-dependent drift
zbMATH Open document ID: 1180.60051
author: Marcin Magdziarz
published in: Stochastic Processes and their Applications
publication date: 2009-10-13T00:00:00Z
review text: The paper deals with a class of real valued stochastic processes \(X(t)\), \(t\geq 0\) whose variance has the power-law form, namely, \(\text{Var}[X(t)]\sim ct^\alpha\) as \(t\to\infty\) with some constant \(\alpha\in(0,1)\). If \(\alpha= 1\), this reminds the behavior of a process like the Brownian motion. Thus the terms subdiffusion processes, or slow diffusions, are natural. The author has used fractional Fokker-Planck equations to describe the probability density functions of subdiffusion processes. Since these equations usually do not have explicit solutions, a well converging approximations are constructed. Other interesting results are derived for the moments of stochastic integrals driven by the inverse \(\alpha\)-stable subordinator. These moments determine uniquely a distribution and this fact is effectively used to involve a specific fractional Fokker-Planck equation and its solution. This well written paper shows that the subdiffusion processes are not only interesting from theoretical point of view, but they are closely related to important applied areas. The reference list represent well both theoretical studies and applications.
reviewed by: Jordan Stoyanov
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 60G52
zbMATH DE Number: 5614614
zbMATH Keywords: subdiffusion
zbMATH Keywords: inverse subordinator
zbMATH Keywords: first-passage time
zbMATH Keywords: \(\alpha \)-stable distribution
zbMATH Keywords: fractional Fokker-Planck equation
MaRDI profile type: Publication
full work available at URL: j.spa.2009.05.006
OpenAlex ID: W2130253079
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DOI: 10.1016/J.SPA.2009.05.006
description: scientific article; zbMATH DE number 5614614