B09 – Large scale modeling of non-linear microscopic dynamics via singluar SPDEs
Singular Stochastic Partial Differential equations (SSPDEs) arise as scaling limits of microscopic stochastic dynamics. They resume the details of the microscopic model in a small number of parameters which enters a canonical macroscopic description of the large scale behaviour of the system. This description usually features irregular functions and non-linear partial differential relations among them. Rigorous mathematical understanding of these relations has recently witnessed substantial progress due to the introduction of various robust approaches for their analysis: Lyons' rough path theory (RPT), then more recently Hairer's theory of regularity structures and, in more restricted contexts, energy solutions and paracontrolled calculus.
The long term goal of this project is to further advance the analysis of SSPDEs and their relation with microscopic models. We will focus on improving the paracontrolled approach to allow for hyperbolic or dispersive equations with noise and non-linearities and to develop the qualitative theory by investigating global in time and space behaviour and metastable dynamics. We will investigate generalisations of the notion of energy solutions to non-stationary situations and have a general convergence result for a large class of microscopic models. Moreover for the Kardar-Parisi-Zhang equation we will investigate alternative notions of solutions that do not rely on the parabolicity of the equation: for example, formulations inspired by viscosity formulation of fully non-linear PDEs.