## Periodic Reporting for period 4 - ProbDynDispEq (Probabilistic and Dynamical Study of Nonlinear Dispersive Equations)

Reporting period: 2019-09-01 to 2020-02-29

Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear Schrödinger equations (NLS) and the Korteweg-de Vries equation (KdV) appear ubiquitously as models describing wave phenomena in various branches of physics and engineering. Over the last several decades, multilinear harmonic analysis has played a crucial role in the development of the theoretical understanding of the subject. Furthermore, in recent years, a non-deterministic point of view has been incorporated into the study of nonlinear dispersive PDEs, enabling us to study typical behavior of solutions in a probabilistic manner and go beyond the limit of deterministic analysis.

The main objective of this project is to develop novel mathematical ideas and techniques, and make fundamental progress on some of the central problems related to the fundamental nonlinear dispersive PDEs such as NLS and KdV from both the deterministic and probabilistic points of view. On the one hand, such a study is of importance from the theoretical points of view since it promotes development of analytical tools, motivating intradisciplinary interactions across various fields in mathematics: harmonic analysis, PDEs, stochastic analysis, mathematical physics, etc. On the other hand, some of these problems are motivated by physics and engineering and hence they are also of importance from the applied points of view.

Many of the proposed objectives involve the so-called well-posedness questions (namely, existence, uniqueness, and stability under perturbation of solutions) for nonlinear dispersive equations. Furthermore, one of the main focus is to deepen our understanding of the behavior of solutions in the presence of “rough” random initial data and/or random external forcing, possibly incorporating recent significant development in the stochastic parabolic PDEs.

The PI also aims to study invariant Gibbs measures for nonlinear Hamiltonian PDEs. One project to study the limiting behavior of the Gibbs measure as the mass cutoff size approaches the threshold mass. The other project aims to make a progress in understanding how the invariant state is related at different times.

The main objective of this project is to develop novel mathematical ideas and techniques, and make fundamental progress on some of the central problems related to the fundamental nonlinear dispersive PDEs such as NLS and KdV from both the deterministic and probabilistic points of view. On the one hand, such a study is of importance from the theoretical points of view since it promotes development of analytical tools, motivating intradisciplinary interactions across various fields in mathematics: harmonic analysis, PDEs, stochastic analysis, mathematical physics, etc. On the other hand, some of these problems are motivated by physics and engineering and hence they are also of importance from the applied points of view.

Many of the proposed objectives involve the so-called well-posedness questions (namely, existence, uniqueness, and stability under perturbation of solutions) for nonlinear dispersive equations. Furthermore, one of the main focus is to deepen our understanding of the behavior of solutions in the presence of “rough” random initial data and/or random external forcing, possibly incorporating recent significant development in the stochastic parabolic PDEs.

The PI also aims to study invariant Gibbs measures for nonlinear Hamiltonian PDEs. One project to study the limiting behavior of the Gibbs measure as the mass cutoff size approaches the threshold mass. The other project aims to make a progress in understanding how the invariant state is related at different times.

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.

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.

One of the most notable progress beyond the state-of-art is the resolution of the first objective on the study of the limiting behavior of the Gibbs measure as the mass cutoff size approaches the critical mass threshold (with P. Sosoe and L. Tolomeo). In this problem, the PI combined tools and ideas from various fields: harmonic analysis, stochastic analysis, functional inequalities, concentration-compactness, spectral theory, etc.

Another important progress beyond the state-of-art is well-posedness theory of stochastic dispersive PDEs with singular noise. The PI made substantial progress in the study of stochastic nonlinear wave equations (SNLW). In particular, the PI implemented a paracontrolled approach to study SNLW with space-time white noise forcing (with M. Gubinelli and H. Koch). This involves a combination of multilinear harmonic analysis and stochastic analysis, in particular introducing random operators called paracontrolled operators.

Other important progress beyond the state of the art:

• The PI introduced the so-called “simultaneous renormalization” in studying transport properties of Gaussian measures under the 2-d NLW dynamics. This simultaneous renormalization was new in the following three points: (i) done at the level of smooth functions, (ii) without changing the equation, and (iii) done simultaneously at the level of the energy functional and its time derivative.

• With Z. Guo, the PI introduced a novel approach to study ill-posedness of the cubic NLS and proved non-existence of solutions to the cubic NLS in negative Sobolev spaces. After our paper, this idea was used by Kappeler-Molnar (SIAM J. Math. Anal. 2017) to prove ill-posedness of modified KdV outside L^2.

• The PI developed further the normal form approach to study PDEs, involving an infinite iteration of normal form reductions. This method was used in various context: construction of solutions, proving uniqueness, proving quasi-invariance, etc.

• New approaches to probabilistic well-posedness theory with random initial data. The PI incorporated, for the first time, a sophisticated deterministic energy-critical theory in the context of probabilistic well-posedness theory and in particular proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.

Another important progress beyond the state-of-art is well-posedness theory of stochastic dispersive PDEs with singular noise. The PI made substantial progress in the study of stochastic nonlinear wave equations (SNLW). In particular, the PI implemented a paracontrolled approach to study SNLW with space-time white noise forcing (with M. Gubinelli and H. Koch). This involves a combination of multilinear harmonic analysis and stochastic analysis, in particular introducing random operators called paracontrolled operators.

Other important progress beyond the state of the art:

• The PI introduced the so-called “simultaneous renormalization” in studying transport properties of Gaussian measures under the 2-d NLW dynamics. This simultaneous renormalization was new in the following three points: (i) done at the level of smooth functions, (ii) without changing the equation, and (iii) done simultaneously at the level of the energy functional and its time derivative.

• With Z. Guo, the PI introduced a novel approach to study ill-posedness of the cubic NLS and proved non-existence of solutions to the cubic NLS in negative Sobolev spaces. After our paper, this idea was used by Kappeler-Molnar (SIAM J. Math. Anal. 2017) to prove ill-posedness of modified KdV outside L^2.

• The PI developed further the normal form approach to study PDEs, involving an infinite iteration of normal form reductions. This method was used in various context: construction of solutions, proving uniqueness, proving quasi-invariance, etc.

• New approaches to probabilistic well-posedness theory with random initial data. The PI incorporated, for the first time, a sophisticated deterministic energy-critical theory in the context of probabilistic well-posedness theory and in particular proved the first almost sure global well-posedness of the energy-critical NLS with rough random initial data.