Singular Stochastic Dispersive Dynamics

ongoing

This proposal is concerned with the study of singular stochastic dispersive partial differential equations (PDEs), broadly interpreted, with stochastic forcing and / or random initial data. This is a young emerging field, attracting more and more attention. In recent years, we have witnessed outstanding advances in the theory of singular stochastic parabolic PDEs. Our understanding of the dispersive counterpart is, however, much poorer. The main objective of this proposal is to develop novel mathematical ideas and tools and fundamentally advance our understanding of singular stochastic dispersive PDEs by working on concrete examples of challenging open problems. Over the last ten years, there has been significant progress at the interface of dispersive PDEs and stochastic analysis and I have been one of the leading mathematicians in this development. In particular, my recent work on the three-dimensional stochastic nonlinear wave equation (NLW) with a quadratic nonlinearity has opened up new research horizons, which we will explore in this proposal. 1. We will investigate the well-posedness issue of the three-dimensional (damped) stochastic cubic NLW with an additive space-time white noise. The solution theory for the parabolic counterpart (the so-called stochastic quantisation equation) was settled by Hairer (2014). The corresponding question for the wave equation is one of the major open questions in this field. We will develop a paracontrolled approach to solve this challenging open problem. Moreover, we address other related problems of independent interest, including the two-dimensional hyperbolic sine-Gordon model, diffusive scaling limit of damped stochastic NLW and singular stochastic nonlinear Schrödinger dynamics. 2. We will also build a pathwise solution theory for stochastic dispersive PDEs with multiplicative noises by combining the Fourier restriction norm method and the rough path theory.