  We theoretically investigate the phase and voltage correlation dynamics,
which includes both the deterministic contribution and stochastic fluctuations,
under a current noise generated by a resistor including thermal and quantum
fluctuations in a resistively and capacitively shunted Josephson junction. An
external current is found to shift and intensify the deterministic
contributions in phase and voltage. In addition to effects of external current,
we observe the relaxation of autocorrelation functions of phase and voltage,
which includes the variances due to the current noise, to finite values in the
long-time limit. In particular, we find that the asymptotic correlations depend
on the resistance as a consequence of quantum effects. We also find an earlier
decay of coherence at a higher temperature in which thermal fluctuations
dominate over quantum ones. These theoretical predictions can be tested in the
next future experiments.
