From Stochastic Transport Models to High-Dimensional Bayesian Inference

The Stochastic Transport in Upper Ocean Dynamics project introduced a new paradigm for representing uncertainty in geophysical fluid dynamics through stochastic transport models calibrated from data. This paper provides a synthesis of the principal theoretical and data assimilation developments achieved as part of the project. The central theme of the paper is the transformation of stochastic transport models from a modelling paradigm into a mathematically rigorous framework equipped with analytical, variational, calibration and Bayesian data assimilation methodologies suitable for geophysical fluid dynamics. We describe the development of the theory of stochastic transport equations, including well-posedness, regularity and approximation results, together with their geometric foundations based on stochastic variational principles, rough paths and fractional transport models. We further review methodologies for calibrating stochastic transport structures from observational and high-resolution simulation data, ranging from reduced-order stochastic models to modern generative approaches. Finally, we discuss Bayesian inference and high-dimensional data assimilation techniques developed for stochastic transport models, including particle filter methodologies and machine learning uncertainty quantification. Taken together, these developments establish a coherent framework linking stochastic transport modelling, rigorous analysis, calibration and Bayesian inference, thereby providing a foundation for uncertainty quantification and forecasting in geophysical fluid dynamics.