Path-based divergence rates and Lagrangian uncertainty in stochastic flows

We develop a probabilistic characterisation of trajectorial divergence rates in non- autonomous stochastic dynamical systems that are systematically shown to be suitable for uncer- tainty quantification in Lagrangian (trajectory-based) predictions. We construct scalar fields of finite-time divergence rates which are rooted in information theory/geometry, and we show their existence and space-time continuity for general stochastic flows. Combining these divergence rate fields with information inequalities derived in [23] allows for a principled quantification of Lagrangian uncertainty in a given dynamics, as well as a mitigation of the uncertainty in path- based observables estimated from simplified models of the truth/reference dynamics in a way that is amenable to algorithmic implementations. This approach can be utilised in information- geometric analysis of statistical estimation and inference, as well as in a data-driven machine learning of reduced order models. We also derive a link between the divergence rates and finite- time Lyapunov exponents for probability measures and for path-based observables.