  Random quantum processes play a central role both in the study of fundamental
mixing processes in quantum mechanics related to equilibration, thermalisation
and fast scrambling by black holes, as well as in quantum process design and
quantum information theory. In this work, we present a framework describing the
mixing properties of continuous-time unitary evolutions originating from local
Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the
unitary group. The induced stochastic time evolution is shown to converge to a
unitary design. As a first main result, we present bounds to the mixing time.
By developing tools in representation theory, we analytically derive an
expression for a local k-th moment operator that is entirely independent of k,
giving rise to approximate unitary k-designs and quantum tensor product
expanders. As a second main result, we introduce tools for proving bounds on
the rate of decoupling from an environment with random quantum processes. By
tying the mathematical description closely with the more established one of
random quantum circuits, we present a unified picture for analysing local
random quantum and classes of Markovian dissipative processes, for which we
also discuss applications.
