Strong Convergence of Unitary Brownian Motion

The Brownian motion $(U_t^N{)}_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t{)}_{t\ge 0}$ as $N\to \mathrm{\infty }$. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.