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Lagrangian uncertainty quantification and information inequalities for stochastic flows

We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., path-based) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated by the desire to improve Lagrangian predictions in complex dynamical systems based either on analytically simplified or data-driven models. We derive a hierarchy of general information bounds on uncertainty in estimates of statistical observables $\mathbb{E}^ν[f]$, evaluated on trajectories of the approximating dynamical system, relative to the "true'' observables $\mathbb{E}^μ[f]$ in terms of certain $φ$-divergences, $\mathcal{D}_φ(μ\|ν)$, which quantify discrepancies between probability measures $μ$ associated with the original dynamics and their approximations $ν$. We then derive two distinct bounds on $\mathcal{D}_φ(μ\|ν)$ itself in terms of the Eulerian fields. This new framework provides a rigorous way for quantifying and mitigating uncertainty in Lagrangian predictions due to Eulerian model error.