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We consider an ensemble of Ornstein-Uhlenbeck processes featuring apopulation of relaxation times and a population of noise amplitudes thatcharacterize the heterogeneity of the ensemble. We show that the centre-of-masslike variable corresponding to this ensemble is statistically equivalent to aprocess driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and thedensity function of such variable are driven by the population of timescalesand of noise amplitudes, respectively. Moreover, we show that this variable isequivalent in distribution to a randomly-scaled Gaussian process, i.e., aprocess built by the product of a Gaussian process times a non-negativeindependent random variable. This last result establishes a connection with theso-called generalized gray Brownian motion and suggests application to modelfractional anomalous diffusion in biological systems.

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