Overcoming criticality for PDEs with randomness

Description du projet

Répondre aux questions sur les EDP et les équations dispersives

Les équations aux dérivées partielles sont fondamentales dans de nombreux domaines, car elles permettent d’effectuer des calculs et des analyses critiques. L’étude des équations aux dérivées partielles (EDP), en particulier des équations dispersives, suscite un intérêt croissant en raison de leur potentiel dans des domaines en pleine évolution tels que la théorie quantique des champs. Malheureusement, malgré de récentes avancées dans la théorie des EDP paraboliques stochastiques singulières, de nombreuses questions clés sur les équations dispersives restent sans réponse. Le projet CritPDEsRand, financé par le CER, vise à développer des outils et des concepts mathématiques pour explorer ces problèmes fondamentaux. Plus précisément, le projet s’appuiera sur les progrès réalisés dans le domaine des probabilités et sur l’interface entre les deux domaines pour déduire des informations essentielles et surmonter les difficultés rencontrées jusqu’à présent.

Objectif

"This proposal is concerned with the study of the dynamics of partial differential equations (PDEs), broadly interpreted, in the presence of randomness, with a particular focus on dispersive equations. This is a young but promising emerging field, and it has deep connections with the more established field of constructive quantum field theory.
In recent years, we have witnessed outstanding advances in the theory of singular stochastic parabolic PDEs, and while several breakthroughs have been obtained for the dispersive counterpart, many fundamental questions are still open. In particular, many of these questions are ""critical"" or ""supercritical"" according to our current understanding. The main goal of this proposal is to develop novel mathematical ideas and tools to break this barrier of criticality, and provide a resolution to these fundamental problems.
Over the last ten years, there has been significant progress at the interface of dispersive PDEs and probability. In my short career,
I have been one of the leading figures of this field and I have achieved significant breakthroughs. Particularly, my works on phase transitions for focusing Gibbs measures, ergodicity results for stochastic wave equations and global well posedness result for fractional NLS in negative regularity have opened the door to new exciting possibilities.
In this proposal,

In this proposal,
1. I will work on the Φ^p_d quantum field theories on R^d, both in the focusing and defocusing regime. This will answer major open questions related to the soliton resolution conjecture and in constructive quantum field theory.
2. I will improve our understanding of how Gaussian measures are transported by the flow of nonlinear PDEs.
3. I will develop a theory of (regular) Lagrangian flows for dispersive PDEs, and use it to break the barrier of criticality for these equations."