  Periodically driven flows are fundamental models of chaotic behavior and the
study of their transport properties is an active area of research. A well-known
analytic construction is the augmentation of phase space with an additional
time dimension; in this augmented space, the flow becomes autonomous or
time-independent. We prove several results concerning the connections between
the original time-periodic representation and the time-extended representation,
focusing on transport properties. In the deterministic setting, these include
single-period outflows and time-asymptotic escape rates from time-parameterized
families of sets. We also consider stochastic differential equations with
time-periodic advection term. In this stochastic setting one has a
time-periodic generator (the differential operator given by the right-hand-side
of the corresponding time-periodic Fokker-Planck equation). We define in a
natural way an autonomous generator corresponding to the flow on time-extended
phase space. We prove relationships between these two generator representations
and use these to quantify decay rates of observables and to determine
time-periodic families of sets with slow escape rate. Finally, we use the
generator on the time-extended phase space to create efficient numerical
schemes to implement the various theoretical constructions. These ideas build
on the work of Froyland et al (SINUM, 2013), and no expensive time integration
is required. We introduce an efficient new hybrid approach, which treats the
space and time dimensions separately.
