Mathematics 790STB - ProbabilistcMthd/NonLinDispPDE

The study of randomness in partial differential equations (PDEs) goes back more than seventy years and include examples such as the modeling of random vibrations of strings, or the scattering of waves by objects that are imbedded in random media. Nonlinear dispersive PDEs naturally appear as models describing wave phenomena in quantum mechanics, nonlinear optics, plasma physics, water waves, and atmospheric sciences. Due to their ubiquitousness they have been at the center of profound research both from the applied community as well as from the theoretical one. One way in which randomness enters the field of nonlinear dispersive PDE is via the random data Cauchy initial value problem for (deterministic) equations, such as the nonlinear Schrodinger (NLS) and the nonlinear wave equations (NLW). The interest comes from two fundamental problems: (1) invariance of measures such as Gibbs measures which are physical equilibria for these systems; arising naturally in statistical mechanics and closely related also to QFT models such as the F4, and (2) the study of generic behavior of solutions in the probabilistic sense, and how they are expected to be better than worst case (exceptional) scenarios. The study of this subject in the context of dispersive PDEs can be traced back to Lebowitz-Rose-Speer (1988, 1989) and Bourgain (1994, 1996) concerning the Gibbs measure for NLS. Since then there have been substantial developments of their ideas by many different researchers, extending them in different directions (geometric, infinite volume, other dispersive relations). In recent years, especially since 2018, this field has seen significant progress and many new ideas and methods have been introduced that go beyond the original ideas of Bourgain, including hyperbolic versions of paracontrolled calculus (Gubinelli-Koch-Oh and Bringmann), the method of random averaging operators and the theory of random tensor (both by Deng-Nahmod-Yue). These new methods have led to the resolution of several important open questions in this field, and are expected to play more important roles in future developments. The aim of this course is to provide the foundations upon which these recent developments have built upon, and in particular have a working knowledge of Bourgain's seminal works in the subject.

MATH 605 or MATH 623/624