  We study Dyson Brownian motion with general potential $V$ and for general
$\beta \geq 1$. For short times $t = o (1)$ and under suitable conditions on
$V$ we obtain a local law and corresponding rigidity estimates on the particle
locations; that is, with overwhelming probability, the particles are close to
their classical locations with an almost-optimal error estimate. Under the
condition that the density of states of the initial data is bounded below and
above down to the scale $\eta_* \ll t \ll 1$, we prove a mesoscopic central
limit theorem for linear statistics at all scales $N^{-1}\ll\eta\ll t$
