Applications of Rough Differential Systems: Theoretical Physics, Data Analysis, and Numerics

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Stochastic analysis is a term which usually encompasses both stochastic integration of Ito type and Malliavin calculus techniques. The latter should be seen as a way to define an analysis at the path level, and leads to deep and useful results concerning stochastic differential equations. This project is also concerned with the theory of regularity structures, which is a recent breakthrough allowing to define a wide range of singular stochastic systems previously out of reach. The regularity structures theory also gives an almost deterministic point of view on stochastic calculus, as opposed to the traditional approach which is very probabilistic in essence. The current project should be seen as a contribution in the areas mentioned above. A combination of rough paths, regularity structures and Malliavin calculus techniques will be used to analyze some physically relevant models such as the continuous parabolic Anderson model in rough environments. Thanks to those fundamental techniques, the PI will also study algebraic and geometric aspects of 2-d signatures with some data analysis applications in mind. The project provides research training opportunities for graduate and undergraduate students. This project is concerned with relevant models in theoretical physics and image processing. A significant portion of the proposal is devoted to localization properties of the parabolic Anderson model, and a special emphasis is also put on the signature method for random fields. A more specific list of the projects at stakes is the following: (1) Moments of renormalized parabolic Anderson models. (2) Level sets for non Gaussian processes. (3) Path confinement for the continuous polymer measure. (4) 2-d signatures and failure identification. The PI plans to develop probabilistic, geometric, and analytic methods and ideas that will lead to a deeper understanding of the qualitative and quantitative of the aforementioned systems. The main techniques will combine rough paths, regularity structures and Malliavin calculus techniques, together with fundamental tools in geometry, analysis and statistical learning. The proposed efforts will have sufficient novelty to open new research areas. They will also further promote the applicability of the theoretical techniques alluded to above.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.