Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential

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 Levy 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.