An important physical property for a stochastic process is how it responds to an external force or spatial confinement. This paper aims to study the relaxation dynamics of a generic process confined in a harmonic potential. We find the dependence of ensemble- and time-averaged mean squared displacements of the confined process on the velocity correlation function $C(t,t+\tau )$ of the original process without any external force. Combining two kinds of scaling forms of $C(t,t+\tau )$ for small $\tau$ and large $\tau$, the stationary value and the relaxation behaviors can be immediately obtained. Our results are valid for a large amount of anomalous diffusion processes, including the ones with single-scaled velocity correlation function (such as fractional Brownian motion and scaled Brownian motion) and the multiscaled ones (like Lévy walk with a broad range of power law exponents of flight time distribution). Note that the latter includes a special case with telegraphic active noise, which could take up athermal energy from the environment.

• Received 3 July 2019
• Revised 3 March 2020
• Accepted 10 March 2020

DOI:https://doi.org/10.1103/PhysRevE.101.042105