Singular Stochastic Dispersive Dynamics

Informations projet

SingStocDispDyn

N° de convention de subvention: 864138

DOI

10.3030/864138

Projet clôturé

Date de signature de la CE 15 Janvier 2020

Date de début 1 Mars 2020

Date de fin 28 Février 2026

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 1 920 968,00

Contribution de l’UE

€ 1 920 968,00

Coordonné par

THE UNIVERSITY OF EDINBURGH
United Kingdom

Periodic Reporting for period 3 - SingStocDispDyn (Singular Stochastic Dispersive Dynamics)

Période du rapport: 2023-03-01 au 2024-08-31

Résumé du contexte et des objectifs généraux du projet

Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear wave equations (NLW) and the nonlinear Schrödinger equations (NLS) appear ubiquitously as models describing wave phenomena in various branches of physics and engineering. The main objective of this proposal is to develop novel mathematical ideas and tools, and make significant progress in the analytical study of singular stochastic dispersive PDEs, broadly interpreted, with stochastic forcing and/or random initial data. Another important aspect of this project is to deepen our understanding of (invariant) Gibbs measures for such stochastic dynamics, including their construction. The models considered in this project come from physics and engineering and as such, they are of importance from the applied points of view. Moreover, this project is also of importance from the theoretical points of view, since it aims to promote development of analytical tools, motivating intradisciplinary interactions across various fields of mathematics: harmonic analysis, PDEs, stochastic analysis, mathematical physics, and so on.

Many of the proposed objectives involve the so-called well-posedness questions (namely, existence, uniqueness and stability under perturbation of solutions) for singular stochastic dispersive PDEs, while some other proposed objectives are on the construction of canonical probability measures arising in Euclidean quantum field theory, in particular in the focusing case. (In the defocusing case, an analogous research program was completed in a recent work (2021) by a 2022 Fields medalist, Duminil-Copin, and his collaborator.) In fact, many of the dynamical problems considered in this project are the so-called stochastic quantization of these probability measures, proposed by Parisi (2021 Nobel prize winner) and Wu.

In recent years, we have witnessed outstanding advances in the theory of singular stochastic parabolic PDEs, led by Hairer (2014 Fields medalist) and Gubinelli (2018 ICM speaker) with their collaborators. Due to the more challenging nature of dispersive problems, however, our understanding in the dispersive setting is rather poor as compared to the parabolic case. Over the last ten years, the PI has been one of the research leaders on the study at the interface of dispersive PDEs and stochastic analysis. In particular, the PI's recent work on the paracontrolled approach to the three-dimensional stochastic NLW with a quadratic nonlinearity has opened up a new research horizon. In this project, the PI aims to fundamentally advance our understanding of singular stochastic dispersive PDEs by working on concrete examples of challenging open problems.Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear wave equations (NLW) and the nonlinear Schrödinger equations (NLS) appear ubiquitously as models describing wave phenomena in various branches of physics and engineering. The main objective of this proposal is to develop novel mathematical ideas and tools, and make significant progress in the analytical study of singular stochastic dispersive PDEs, broadly interpreted, with stochastic forcing and/or random initial data. Another important aspect of this project is to deepen our understanding of (invariant) Gibbs measures for such stochastic dynamics, including their construction. The models considered in this project come from physics and engineering and as such, they are of importance from the applied points of view. Moreover, this project is also of importance from the theoretical points of view, since it aims to promote development of analytical tools, motivating intradisciplinary interactions across various fields of mathematics: harmonic analysis, PDEs, stochastic analysis, mathematical physics, and so on.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

The PI and team members have worked on various aspects of the project, primarily on (i) the (non-)construction of focusing Gibbs measures and (ii) well-posedness theory of singular stochastic NLW (also some singular stochastic parabolic PDEs).

As for the (non-)construction of focusing Gibbs measures, with P. Sosoe (Cornell) and L. Tolomeo (Bonn/Edinburgh), the PI established normalizability of the critical focusing Gibbs measure in the 1-d case, which resolved a 30 year-old open question by Lebowitz, Rose, and Speer (1988). In the 3-d case, with M. Okamoto (Osaka) and L. Tolomeo (Bonn/Edinburgh), the PI studied the focusing Hartree model with a quartic interaction and also the Φ^3_3-model, both of which turned out to be critical and thus was very challenging (with the Coulomb potential in the Hartree case). Nonetheless, the PI successfully established phase transitions for these models. This completes the research program of over 30 years on the (non-)construction of focusing Gibbs measures for any power and any dimension. As for the well-posedness theory of singular stochastic NLW, the PI has developed a novel paracontrolled approach in the dispersive setting and proved local well-posedness of the three-dimensional stochastic NLW with a quadratic nonlinearity. This paracontrolled approach was further developed in joint works with M. Okamoto (Osaka) and L. Tolomeo (Bonn) to study the dynamical problems associated with the aforementioned focusing Hartree model and Φ^3_3-model. The PI also introduced a new globalization approach by studying the so-called extended Gibbs measure and using ideas from optimal transport theory.

The PI worked on the 3-d SNLW with a cubic nonlinearity, forced by an almost space-time white noise. and established its local well-posedness (2021).

The PI has also worked on the 2-d SNLW with a more general class of nonlinearities. This includes trigonometric functions (appearing in one of the proposed problems) and exponential functions, which is of physical importance. In particular, the PI constructed invariant Gibbs dynamics for the hyperbolic sine-Gordon model and the hyperbolic Liouville model in a certain regime. These results include both the construction of canonical probability measures as well as well-posedness of the associated dynamical problems. As for the Liouville model, the PI has also studied the parabolic Liouville model and constructed invariant Gibbs dynamics. This project led to the study of stochastic quantization of Liouville quantum gravity measure, where the PI constructed the associated dynamics in a successful manner.

Other work:

• Non-normalizability of a log-correlated focusing Gibbs measure with a quartic interaction in any dimension.

• Transport properties of Gaussian measures under nonlinear Hamiltonian dynamics. In particular, the PI proved the first quasi-invariance result for the Gaussian measure under the dynamics of a Hamiltonian PDE in negative Sobolev regularity.

• The PI studied NLS with a quadratic nonlinearity |u|^2 and proved sharp local well-posedness, which resolved an open question of thirty years.

• Over the last few years, the field has seen a significant progress and the notion of probabilistic scaling criticality was introduced by Deng, Nahmod, and Yue (2019). The PI and his Ph.D. student studied SNLW, the stochastic nonlinear heat equation, and NLS (with random initial data), and showed that when the noise (or the random initial data) is too rough, a prediction made by the probabilistic scaling fails.

• Ill-posedness in low regularity. In particular, the PI established a sharp norm inflation result (a strong form of instability) for the cubic nonlinear heat equation, which concludes the well-posedness study of this canonical equation. The team members established norm inflation for various models.

• The PI and team members have worked on various aspects of stochastic dispersive PDEs such as well-posedness, convergence problems, long time behaviour.

• The PI and team members worked on deterministic and statistical properties of the intermediate long wave equation.

Since the start of this grant, the PI and the team members published 62 papers and preprints.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

One of the most significant progress beyond the state-of-art is the resolution of one of the main objectives on the construction of the 3-d Gibbs measure with a cubic interaction (the so-called Φ^3_3-model). This problem turns out to be critical and the PI established the following phase transition for this model: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. This result, together with the 1-d result mentioned below, completes the focusing
Gibbs measure construction program for any power and any dimension.

Another important progress beyond the state-of-art is well-posedness theory of singular stochastic NLW, in particular developing a paracontrolled approach in the dispersive setting over a series of works. The PI also implemented a new globalization argument beyond Bourgain's invariant measure argument.

The PI has studied the 2-d stochastic dynamics with trigonometric and exponential nonlinearities. In particular, in the case of the exponential nonlinearity, the well-posedness approach for the wave equation is novel, exploiting the hidden positivity in the equation. The results for trigonometric and exponential nonlinearities currently hold for a limited range of parameters and the PI intends to investigate this problem further.

In the PI's ERC Starting Grant, the PI studied integrability at the optimal mass threshold of the critical focusing Gibbs measure in the 1-d case and essentially resolved the problem. This problem is part of the focusing Gibbs measure construction program mentioned above. In this reporting period, the PI devoted non-trivial amount of time to the typing of this work, making necessary modifications of the argument. This work is now complete and published in Inventiones mathematicae (2022). This work resolves an open question posed by Lebowitz, Rose, and Speer (1988).

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