In this talk, I will discuss recent progress on constructing Gibbs measure and their associated dynamics for nonlinear partial differential equations.
In the first part of the talk (first hour), I will review how this subject has evolved to its current shape since the seminal work of Bourgain in 1990s. Bourgain's introduction of the Fourier restriction norm method, rigorous construction of the focusing Gibbs measure, and the invention of the invariant measure argument to globalise local dynamics have been foundational. Recent advancements have built upon these pillars, with Barashkov and Gubinelli's variational argument, the concept of para-controlled distributions, and Deng-Nahmod-Yue's random average operator and random tensor arguments further pushing the boundaries of understanding in this field.
In the second part of the talk (second hour), I will discuss our recent work on constructing singular Gibbs measures, using the radial Phi4_4 measure (measure supports on radial functions) as an example. Utilising Barashkov-Gubinelli's variational approach, alongside innovative ideas by exploiting the Markov property of the Gaussian free field and Green function estimates, allows us to overcome the singularities appeared at the origin.
