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One way to analyze complicated non-autonomous flows is through trying tounderstand their transport behavior. In a quantitative, set-oriented approachto transport and mixing, finite time coherent sets play an important role.These are time-parametrized families of sets with unlikely transport to andfrom their surroundings under small or vanishing random perturbations of thedynamics. Here we propose, as a measure of transport and mixing for purelyadvective (i.e., deterministic) flows, (semi)distances that arise undervanishing perturbations in the sense of large deviations. Analogously, forgiven finite Lagrangian trajectory data we derive a discrete-time and spacesemidistance that comes from the "best" approximation of the randomly perturbedprocess conditioned on this limited information of the deterministic flow. Itcan be computed as shortest path in a graph with time-dependent weights.Furthermore, we argue that coherent sets are regions of maximal farness interms of transport and mixing, hence they occur as extremal regions on aspanning structure of the state space under this semidistance---in fact, underany distance measure arising from the physical notion of transport. Based onthis notion we develop a tool to analyze the state space (or the finitetrajectory data at hand) and identify coherent regions. We validate ourapproach on idealized prototypical examples and well-studied standard cases.

From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data