  Much recent experimental effort has focused on the realization of exotic
quantum states and dynamics predicted to occur in periodically driven systems.
But how robust are the sought-after features, such as Floquet topological
surface states, against unavoidable imperfections in the periodic driving? In
this work, we address this question in a broader context and study the dynamics
of quantum systems subject to noise with periodically recurring statistics. We
show that the stroboscopic time evolution of such systems is described by a
noise-averaged Floquet superoperator. The eigenvectors and -values of this
superoperator generalize the familiar concepts of Floquet states and
quasienergies and allow us to describe decoherence due to noise efficiently.
Applying the general formalism to the example of a noisy Floquet topological
chain, we re-derive and corroborate our recent findings on the noise-induced
decay of topologically protected end states. These results follow directly from
an expansion of the end state in eigenvectors of the Floquet superoperator.
