Stochastic processes in a confining harmonic potential in the presence of static and dynamic measurement noise

Abstract

We consider the overdamped dynamics of different stochastic processes, including Brownian motion and autoregressive processes, continuous time random walks, fractional Brownian motion, and scaled Brownian motion, confined by an harmonic potential. We discuss the effect of both static and dynamic noise representing two kinds of localisation error prevalent in experimental single-particle tracking data. To characterise how such noise affects the dynamics of the pure, noise-free processes we investigate the ensemble-averaged and time-averaged mean squared displacements as well as the associated ergodicity breaking parameter. Process inference in the presence of noise is demonstrated to become more challenging, as typically the noise dominates the short-time behaviour of statistical measures, while the long time behaviour is dominated by the external confinement. In particular, we see that while static noise generally leads to a more subdiffusive apparent behaviour, dynamic noise makes the signal seem more superdiffusive. Our detailed study complements tools for analysing noisy time series and will be useful in data assimilation of stochastic data.