  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 gray Brownian motion and suggests application to model
fractional anomalous diffusion in biological systems.
