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 finitetime 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.

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

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We consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.