  In this paper, we compare the solutions of Dyson Brownian motion with general
$\beta$ and potential $V$ and the associated McKean-Vlasov equation near the
edge. Under suitable conditions on the initial data and potential $V$, we
obtain the optimal rigidity estimates of particle locations near the edge for
short time $t=\text{o}(1)$. Our argument uses the method of characteristics
along with a careful estimate involving an equation of the edge. With the
rigidity estimates as an input, we prove a central limit theorem for mesoscopic
statistics near the edge which, as far as we know, have been done for the first
time in this paper. Additionally, combining with \cite{LandonEdge}, our
rigidity estimates are used to give a proof of the local ergodicity of Dyson
Brownian motion for general $\beta$ and potential at the edge, i.e. the
distribution of extreme particles converges to Tracy-Widom $\beta$ distribution
in short time.
