We present a new information-theoretic framework for trajectory-based characterisation of transport and mixing in stochastic dynamical systems which are non-autonomous and known over a finite time interval. This work is motivated by the desire to study time-dependent transport and provide Lagrangian (trajectory-based) predictions in multi-scale systems based on simplified, data-driven models with errors affecting path-based predictions in a non-local fashion. In deterministic dynamical systems techniques exploiting stable and unstable manifolds of finite-time hyperbolic trajectories or techniques based on finite-time Lyapunov exponents (FTLE) are frequently used to estimate transport barriers and to identify almost-invariant sets under the dynamics. While these techniques often give numerically compatible results in the deterministic setting, a rigorous connection between the two approaches has remained elusive. Here, we provide a rigorous link between the two approaches in the case of deterministic dynamics, and we extend the approach to deal with the evolution of path-based uncertainty in stochastic flows. In this new framework the average finite-time expansion along trajectories, and the uncertainty bounds on path-based functionals in stochastic dynamical systems are based on finite-time rates of certain pre-metrics (divergencies) on spaces of probability measures. We illustrate the results based on a simple numerical algorithm for computing the fields of finite-time divergence rates.
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