Edge scaling limit of Dyson Brownian motion at equilibrium for general $beta geq 1$

For general $beta geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N to infty$. For each fixed time, this ensemble is distributed as the Airy$_beta$ random point field. We prove that the increments of the limiting process are locally Brownian. When $beta >1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{o}lder for any $r<1-(1+beta)^{-1}$. Furthermore for all $beta geq 1$ we show that the limiting process solves an SDE in a weak sense. When $beta=2$ this limiting process is the Airy line ensemble.