  We consider the edge statistics of Dyson Brownian motion with deterministic
initial data. Our main result states that if the initial data has a spectral
edge with rough square root behavior down to a scale $\eta_* \geq N^{-2/3}$ and
no outliers, then after times $t \gg \sqrt{ \eta_*}$, the statistics at the
spectral edge agree with the GOE/GUE. In particular we obtain the optimal time
to equilibrium at the edge $t = N^{\varepsilon} / N^{1/3}$ for sufficiently
regular initial data. Our methods rely on eigenvalue rigidity results similar
to those appearing in [Lee-Schnelli], the coupling idea of
[Bourgade-Erd\H{o}s-Yau-Yin] and the energy estimate of
[Bourgade-Erd\H{o}s-Yau].
