Abstract
The Ornstein–Uhlenbeck process is a stationary and ergodic Gaussian process, that is fully determined by its covariance function and mean. We show here that the generic definitions of the ensemble- and time-averaged mean squared displacements fail to capture these properties consistently, leading to a spurious ergodicity breaking. We propose to remedy this failure by redefining the mean squared displacements such that they reflect unambiguously the statistical properties of any stochastic process. In particular we study the effect of the initial condition in the Ornstein–Uhlenbeck process and its fractional extension. For the fractional Ornstein–Uhlenbeck process representing typical experimental situations in crowded environments such as living biological cells, we show that the stationarity of the process delicately depends on the initial condition.
Highlights
The Ornstein–Uhlenbeck process is one of the most fundamental physical processes, originally devised to describe the velocity distribution and relaxation of a Brownian particle under the influence of a velocity-dependent friction
It is commonly assumed that asserting equilibrium initial condition is sufficient and necessary for a confined stochastic process to remain stationary at all times t 0
We here demonstrated that for the case of the fractional Ornstein–Uhlenbeck process this is not true
Summary
Keywords: Ornstein–Uhlenbeck process, stationary stochastic process, ensemble and time averaged mean squared displacement Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
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