  Coherent sets are time-dependent regions in the physical space of
nonautonomous flows that exhibit little mixing with their neighborhoods,
robustly under small random perturbations of the flow. They thus characterize
the global long-term transport behavior of the system. We propose a framework
to extract such time-dependent families of coherent sets for nonautonomous
systems with an ergodic driving dynamics and (small) Brownian noise in physical
space. Our construction involves the assembly and analysis of an operator on
functions over the augmented space of the associated skew product that, for
each fixed state of the driving, propagates distributions on the corresponding
physical-space fibre according to the dynamics. This time-dependent operator
has the structure of a semigroup (it is called the Mather semigroup), and we
show that a spectral analysis of its generator allows for a trajectory-free
computation of coherent families, simultaneously for all states of the driving.
Additionally, for quasi-periodically driven torus flows, we propose a tailored
Fourier discretization scheme for this generator and demonstrate our method by
means of three examples of two-dimensional flows.
