Overcoming criticality for PDEs with randomness

Descrizione del progetto

Rispondere a domande su equazioni differenziali parziali ed equazioni dispersive

Le equazioni differenziali sono fondamentali in molti campi, poiché consentono calcoli e approfondimenti critici. Lo studio delle equazioni differenziali parziali (PDE), in particolare delle equazioni dispersive, ha visto crescere l’attenzione e l’interesse per il loro potenziale in campi avanzati quali la teoria quantistica dei campi. Purtroppo, nonostante i recenti progressi nella teoria delle PDE paraboliche stocastiche singolari, molte domande centrali sulle equazioni dispersive rimangono senza risposta. Il progetto CritPDEsRand, finanziato dal CER, intende sviluppare strumenti e concetti matematici per esaminare questi problemi fondamentali. In particolare, il progetto si avvarrà dei progressi della probabilità e l’interfaccia dei due campi per dedurre approfondimenti fondamentali e superare le sfide precedenti.

Obiettivo

"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."