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

TL;DR

This paper introduces a novel approach to modeling heterogeneous Ornstein-Uhlenbeck processes through a center-of-mass like variable, linking it to non-autonomous SDEs and fractional diffusion, with implications for biological systems.

Contribution

It demonstrates that a collective variable of heterogeneous OU processes can be represented by a non-autonomous SDE and a randomly-scaled Gaussian process, connecting to fractional diffusion models.

Findings

Center-of-mass variable is equivalent to a non-autonomous SDE.

The variable can be represented as a product of Gaussian and random variables.

Application potential for modeling fractional anomalous diffusion in biology.

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