  We theoretically investigate the phase and voltage correlation dynamics under
a current noise including thermal and quantum fluctuations in a resistively and
capacitively shunted Josephson (RCSJ) junction. Within the linear regime, an
external current is found to shift and intensify the deterministic
contributions in phase and voltage. In addition to the deterministic
contribution, we observe the relaxation of autocorrelation functions of phase
and voltage to finite values due to the current noise. We also find an earlier
decay of coherence at a higher temperature in which thermal fluctuations
dominate over quantum ones.
