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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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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.

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Authors and Affiliations

  1. Department of Basic and Applied Science for Engineering, Sapienza University of Rome, Via Antonio Scarpa 16, I–00161, Rome, Italy

    Mirko D’Ovidio

  2. DIFA - Department of Physics and Astronomy, University of Bologna, Viale B. Pichat 6/2, I–40127, Bologna, Italy

    Silvia Vitali & Gastone Castellani

  3. Institute for Physics and Astronomy University of Potsdam, Karl-Liebknecht-Strasse 24/25, D–14476, Potsdam-Golm, Germany

    Vittoria Sposini

  4. BCAM–Basque Center for Applied Mathematics, Alameda de Mazarredo 14 E–, 48009, Bilbao, Basque Country, Spain

    Vittoria Sposini, Oleksii Sliusarenko, Paolo Paradisi & Pagnini Gianni

  5. ISTI-CNR, Institute of Information Science and Technology “A. Faedo”, via Moruzzi 1, I–56124, Pisa, Italy

    Paolo Paradisi

  6. Ikerbasque–Basque Foundation for Science, Calle de María Díaz de Haro 3, E–48013, Bilbao, Basque Country, Spain

    Pagnini Gianni

Authors
  1. Mirko D’Ovidio
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  2. Silvia Vitali
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  3. Vittoria Sposini
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  4. Oleksii Sliusarenko
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  5. Paolo Paradisi
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  6. Gastone Castellani
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  7. Pagnini Gianni
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Correspondence to Pagnini Gianni.

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D’Ovidio, M., Vitali, S., Sposini, V. et al. Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion. FCAA 21, 1420–1435 (2018). https://doi.org/10.1515/fca-2018-0074

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