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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 particlelocations; that is, with overwhelming probability, the particles are close totheir classical locations with an almost-optimal error estimate. Under thecondition that the density of states of the initial data is bounded below andabove down to the scale $\eta_* \ll t \ll 1$, we prove a mesoscopic centrallimit theorem for linear statistics at all scales $N^{-1}\ll\eta\ll t$

Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential