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Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion

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Published/Copyright: January 13, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 5

Abstract

We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized grey Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

MSC 2010: Primary 60G20; Secondary 65C30; 91B70; 60J60; 26A33; 34A08; 60J70
Key Words and Phrases: Ornstein–Uhlenbeck process; heterogeneous ensemble; superposition; center of mass; non-autonomous stochastic differential equation; randomly-scaled Gaussian process; generalized grey Brownian motion

Acknowledgements

This research is supported by the Basque Government through the BERC 2014-2017 and the BERC 2018-2021 programs, and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through project MTM2016-76016-R “MIP”. VS acknowledges BCAM for the financial support to her internship research period, and SV acknowledges the University of Bologna for the financial support through the “Marco Polo Programme” for funding her PhD research period abroad spent at BCAM.

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Received: 2018-02-03
Published Online: 2019-01-13
Published in Print: 2018-10-25

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Editorial Note
  2. FCAA related news, events and books (FCAA–volume 21–5–2018)
  3. Discussion Paper
  4. Fractional calculus’s adventures in Wonderland (Round table held at ICFDA 2018)
  5. Survey Paper
  6. On modifications of the exponential integral with the Mittag-Leffler function
  7. Research Paper
  8. Global solutions to stochastic Volterra equations driven by Lévy noise
  9. Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations
  10. Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems
  11. Regularity of solutions to space–time fractional wave equations: A PDE approach
  12. Asymptotically periodic solutions for Caputo type fractional evolution equations
  13. Complex fractional Zener model of wave propagation in ℝ
  14. Maximum principles for time-fractional Cauchy problems with spatially non-local components
  15. An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions
  16. A further extension of Mittag-Leffler function
  17. Frequency-distributed representation of irrational linear systems
  18. Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion
https://doi.org/10.1515/fca-2018-0074
Keywords for this article
Ornstein–Uhlenbeck process; heterogeneous ensemble; superposition; center of mass; non-autonomous stochastic differential equation; randomly-scaled Gaussian process; generalized grey Brownian motion

Articles in the same Issue

  1. Editorial Note
  2. FCAA related news, events and books (FCAA–volume 21–5–2018)
  3. Discussion Paper
  4. Fractional calculus’s adventures in Wonderland (Round table held at ICFDA 2018)
  5. Survey Paper
  6. On modifications of the exponential integral with the Mittag-Leffler function
  7. Research Paper
  8. Global solutions to stochastic Volterra equations driven by Lévy noise
  9. Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations
  10. Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems
  11. Regularity of solutions to space–time fractional wave equations: A PDE approach
  12. Asymptotically periodic solutions for Caputo type fractional evolution equations
  13. Complex fractional Zener model of wave propagation in ℝ
  14. Maximum principles for time-fractional Cauchy problems with spatially non-local components
  15. An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions
  16. A further extension of Mittag-Leffler function
  17. Frequency-distributed representation of irrational linear systems
  18. Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion
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