Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

• Published in: Fractional calculus & applied analysis Vol. 21; no. 5; pp. 1420 - 1435
• Main Authors: D’Ovidio, Mirko, Vitali, Silvia, Sposini, Vittoria, Sliusarenko, Oleksii, Paradisi, Paolo, Castellani, Gastone, Gianni, Pagnini
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 25.10.2018; De Gruyter Open; Walter de Gruyter GmbH
• Subjects: 26A33; 34A08; 60J60; 60J70; 91B70; Abstract Harmonic Analysis; Amplitudes; Analysis; Brownian motion; center of mass; Differential equations; Equivalence; Functional Analysis; Gaussian process; generalized grey Brownian motion; heterogeneous ensemble; Integral Transforms; Mathematics; Noise; non-autonomous stochastic differential equation; Operational Calculus; Ornstein–Uhlenbeck process; Primary 60G20; Random variables; randomly-scaled Gaussian process; Research Paper; Secondary 65C30; superposition; Superposition (mathematics); Time dependence; White noise; superposition; 60J60; 60J70; 26A33; non-autonomous stochastic differential equation; Secondary 65C30; 34A08; heterogeneous ensemble; center of mass; generalized grey Brownian motion; 91B70; Ornstein–Uhlenbeck process; Primary 60G20; randomly-scaled Gaussian process
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2018-0074

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.