  In this paper we study mesoscopic fluctuations for Dyson's Brownian motion
with $\beta=2$. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the
invariant measure for this stochastic evolution and conjectured that, when
starting from a generic configuration of initial points, the time that is
needed for the GUE statistics to become dominant depends on the scale we look
at: The microscopic correlations arrive at the equilibrium regime sooner than
the macrosopic correlations. In this paper we investigate the transition on the
intermediate, i.e. mesoscopic, scales. The time scales that we consider are
such that the system is already in microscopic equilibrium (sine-universality
for the local correlations), but we have not yet reached equilibrium at the
macrosopic scale. We describe the transition to equilibrium on all mesoscopic
scales by means of Central Limit Theorems for linear statistics with
sufficiently smooth test functions. We consider two situations: deterministic
initial points and randomly chosen initial points. In the random situation, we
obtain a transition from the classical Central Limit Theorem for independent
random variables to the one for the GUE.
