  The study of transport and mixing processes in dynamical systems is
particularly important for the analysis of mathematical models of physical
systems. Barriers to transport, which mitigate mixing, are currently the
subject of intense study. In the autonomous setting, the use of transfer
operators (Perron-Frobenius operators) to identify invariant and
almost-invariant sets has been particularly successful. In the nonautonomous
(time-dependent) setting, \emph{coherent sets}, a time-parameterised family of
minimally dispersive sets, are a natural extension of almost-invariant sets.
The present work introduces a new analytic transfer operator construction that
enables the calculation of \emph{finite-time} coherent sets (sets are that
minimally dispersive over a finite time interval). This new construction also
elucidates the role of diffusion in the calculation and we show how properties
such as the spectral gap and the regularity of singular vectors scale with
noise amplitude. The construction can also be applied to general Markov
processes on continuous state space.
