One of the main goals is to study the limiting behavior of the Gibbs measure as the mass cutoff size approaches the critical mass threshold. By combining tools and ideas from various fields, the PI resolved the proposed problem. This answers an open question posed by Lebowitz-Rose-Speer '88.
In probability theory, the transport properties of Gaussian measures under linear and nonlinear transformations have attracted wide attention over the last fifty years. With N. Tzvetkov (Université de Cergy-Pontoise), the PI studied the transport properties of Gaussian measures under the dynamics of nonlinear Hamiltonian dispersive PDEs and in particular proved quasi-invariance of the Gaussian measures. The PI also showed that quasi-invariance property does not hold in a dispersonless model, thus showing the necessity of dispersion. This shows a striking contrast of dispersive and non-dispersive models.
One of the important problems is to study NLS with white noise. After some investigation, the PI realized that this problem is critical, which is not approachable with the current technology. In several works, the PI established well-posedness of (stochastic) NLS with higher dispersion or smoothed noises, covering the almost critical cases.
The PI and the team studied further stochastic dispersive PDEs with singular noise and made substantial progress in the study of stochastic nonlinear wave equations (SNLW). In particular, with M. Gubinelli and H. Koch, the PI implemented a paracontrolled approach to study SNLW in three dimensions. A team member, Tolomeo, also made fundamental contributions in studying ergodicity of the Gibbs measures for stochastic dispersive PDEs. During the five-year period of the grant, the PI and the team made important intradisciplinary connections to those in stochastic analysis and in mathematical physics.
The PI and the team also made further advancement on probabilistic well-posedness of nonlinear dispersive PDEs with random initial data and related Gibbs measures. In particular, the PI proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.
There are also works performed solely via deterministic analysis to study dispersive equation in the low regularity setting. These works address both well-posedness and ill-posedness (non-existence and norm inflation), employing various approaches. In particular, the PI made further development of an infinite iteration of normal form reductions, originally introduced by the PI in 2013. The PI also exploited an integrability structure of NLS and proved the first global well-posedness result of NLS in almost critical spaces.
During the five-year period of the grant, the PI and the team members published 49 papers and preprints and gave 70 talks (45 by the PI and 35 by the postdocs and Ph.D. students). The PI also gave 4 invited mini-courses and organized 5 conferences and workshops.
