Path-based measures of expansion rates and Lagrangian transport in stochastic flows
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We develop a systematic information-theoretic framework for a probabilistic characterisation of expansion rates in non-autonomous stochastic dynamical systems which are known over a finite time interval. This work is motivated by the desire to quantify uncertainty in time-dependent transport analysis and to improve Lagrangian (trajectory-based) predictions in multi-scale systems based on simplified, data-driven models. In this framework the average finite-time nonlinear expansion along system trajectories is based on a finite-time rate of information loss between evolving probability measures. We characterise properties of finite-time divergence rate fields, defined via so-called φ-divergencies, and we derive a link between this probabilistic approach and a diagnostic based on finite-time Lyapunov exponents which are commonly used in estimating expansion and identifying transport barriers in deterministic flows; this missing link is subsequently extended to evolution of path-based uncertainty in stochastic flows; the Lagrangian uncertainty quantification is a subject of a follow-up publication.
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