  We develop a probabilistic characterisation of trajectorial expansion rates
in non-autonomous stochastic dynamical systems that can be defined over a
finite time interval and used for the subsequent uncertainty quantification in
Lagrangian (trajectory-based) predictions. These expansion rates are quantified
via certain divergences (pre-metrics) between probability measures induced by
the laws of the stochastic flow associated with the underlying dynamics. We
construct scalar fields of finite-time divergence/expansion rates, show their
existence and space-time continuity for general stochastic flows. Combining
these divergence rate fields with our 'information inequalities' derived in
allows for quantification and mitigation of the uncertainty in path-based
observables estimated from simplified models in a way that is amenable to
algorithmic implementations, and it can be utilised in information-geometric
analysis of statistical estimation and inference, as well as in a data-driven
machine/deep learning of coarse-grained models. We also derive a link between
the divergence rates and finite-time Lyapunov exponents for probability
measures and for path-based observables.
