Dynamics of singular stochastic nonlinear dispersive PDEs

Dispersion exists ubiquitously in nature. The most famous example of dispersion is seen in a rainbow, where dispersion effect separates the white light spatially into components of different wavelengths (different colours). Nonlinear dispersive partial differential equations (PDEs), such as nonlinear Schrodinger equations (NLS) and nonlinear wave equations (NLW), appear naturally in models describing wave phenomena in the real world.

In the past thirty years, the study of deterministic nonlinear dispersive PDEs has seen significant development, in which harmonic analysis has played a fundamental role, led by Kenig, Bourgain and Tao, among others. In recent years, a combination of deterministic analysis with probability theory has played an increasingly important role in the field.

This probabilistic perspective allows us to go beyond the limits of deterministic analysis. More importantly, it is also essential to understand the effect of stochastic perturbation in practice since such stochastic perturbation is ubiquitous.

The main objective of this research is to develop novel mathematical ideas and techniques to clarify long-standing fundamental questions in the study of stochastic nonlinear dispersive PDEs, with primary examples given by stochastic NLS and stochastic NLW. In the field of singular stochastic parabolic PDEs, significant progress has been taking place led by Hairer and Gubinelli with their collaborators.

This has enabled striking theories which are changing the landscape of the study in this field. However, their new theories are designed to handle parabolic problems, and it is not a priori clear on how to adapt them to solve dispersive equations. Despite some exciting recent progress, our understanding of stochastic dispersive PDEs is still very far from satisfactory.

In these proposed projects, the principal investigator (PI) will study several open problems in the field of stochastic dispersive PDEs. More specifically, the PI will focus on studying the properties of invariant measures and the local and global-in-time solutions to stochastic NLS and NLW in periodic domains. The PI plans to address these problems by combining tools from dispersive PDEs, stochastic analysis, probability theory and harmonic analysis with recent progress.