Smooth Calculus for Low Regularity Random Data Partial Differential Equations – Smooth

There has been in the last thirteen years considerable progresses in our understanding of some classes of random partial differential equations (PDEs) after the seminal 2008 work of Burq & Tzvetkov on supercritical wave equations with random initial conditions and the groundbreaking 2014 works of Hairer and Gubinelli, Imkeller & Perkowski on singular stochastic elliptic and parabolic PDEs. These fundamental developments have laid the basis of a robust solution theory for equations that are classically ill-posed, due to a low regularity random data in the problem. While the regularity structures and paracontrolled approaches to stochastic PDEs have their roots in T. Lyons’ theory of rough paths, the progresses in the dispersive side have their roots in Bourgain’s works on invariant Gibbs measures for the nonlinear Schro?dinger equation in the mid nineties. In recent years, it became obvious that some of the conceptual ideas introduced in echo the strategy adopted to study singular stochastic PDEs. For instance, in both situations, any potential solution is decomposed as the sum of a ‘finite dimensional’ singular part enjoying regularity properties beyond the deterministic analysis, and a remainder part that is more regular and can be treated using purely analytical tools adapted to the equation – Bourgain spaces, regularity or paracontrolled structures. This project aims at a deeper clarification of the common features and key differences between the analysis of dispersive PDEs with random initial data and the analysis of singular stochastic PDEs. The objective will also consist in identifying new possible interactions between these domains, as well as developing specific tools for these classes of equations.