Measures of path-based nonlinear expansion rates and Lagrangian uncertainty in stochastic flows

TL;DR

This paper introduces a probabilistic framework for quantifying and analyzing the expansion rates and uncertainty in stochastic flows over finite time intervals, with applications in uncertainty quantification, inference, and machine learning.

Contribution

It develops a novel divergence-based method to measure trajectorial expansion rates and links these to finite-time Lyapunov exponents, enabling improved uncertainty quantification in stochastic systems.

Findings

Existence and continuity of divergence rate fields in stochastic flows.

A new link between divergence rates and finite-time Lyapunov exponents.

Algorithmic approaches for uncertainty mitigation in path-based observables.

Abstract

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.