  The pace of evolution of physical systems is fundamentally constrained by
quantum speed limits (QSL), which have found broad applications in quantum
science and technology. We consider the speed of evolution for quantum systems
undergoing stochastic dynamics due to continuous measurements. It is shown that
that there are trajectories for which standard QSL are violated, and we provide
estimates for the range of velocities in an ensemble of realizations of
continuous measurement records. We determine the dispersion of the speed of
evolution and characterize the full statistics of single trajectories. By
characterizing the dispersion of the Bures angle, we further show that
continuous quantum measurements induce Brownian dynamics in Hilbert space.
