  The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a
process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$.
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.
