  Optical tweezers setup is often used to probe the motion of individual tracer
particle, which promotes the study of relaxation dynamics of a generic process
confined in a harmonic potential. We uncover the dependence of ensemble- and
time-averaged mean square displacements of confined processes on the velocity
correlation function $C(t,t+\tau)$ of the original process. With two different
scaling forms of $C(t,t+\tau)$ for small $\tau$ and large $\tau$, the
stationary value and the relaxation behaviors can be obtained immediately. The
gotten results are valid for a large amount of anomalous diffusion processes,
including fractional Brownian motion, scaled Brownian motion, and the
multi-scale L\'{e}vy walk with different exponents of running time
distribution.
