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 the desire to improve complex based either on analytically simplified or data-driven models. derive hierarchy general information bounds uncertainty estimates statistical observables $\mathbb{E}^{\nu}[f]$, evaluated trajectories approximating system, relative "true'' $\mathbb{E}^{\mu}[f]$ terms certain $\varphi$-divergences, $\mathcal{D}_\varphi(\mu\|\nu)$, quantify discrepancies between probability measures $\mu$ associated with original dynamics their approximations $\nu$. then two distinct $\mathcal{D}_\varphi(\mu\|\nu)$ itself Eulerian new provides rigorous way quantifying mitigating due model error.