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Fractional Calculus and Applied Analysis

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The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion

Caibin Zeng
  • School of Sciences, South China University of Technology, Guangzhou, 510640, China
  • Center for Self-Organizing and Intelligent Systems Department of Electrical and Computer Engineering, Utah State University, Logan, Utah, 84322, USA
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/ YangQuan Chen
  • Center for Self-Organizing and Intelligent Systems Department of Electrical and Computer Engineering, Utah State University, Logan, Utah, 84322, USA
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Published Online: 2012-06-30 | DOI: https://doi.org/10.2478/s13540-012-0034-z

Abstract

This paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

MSC: Primary 60G22; Secondary 26A33, 35R60, 60H15, 35Q84

Keywords: fractional Brownian motion; Ornstein-Uhlenbeck process; anomalous diffusion; Fokker-Planck equation

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About the article

Published Online: 2012-06-30

Published in Print: 2012-09-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-012-0034-z.

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© 2012 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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