Abstract: In groundbreaking work, Bourgain ’96 put forward a random data theory to study the existence of strong solutions on the statistical ensemble of Gibbs measures associated to dispersive equations. Despite numerous follow-up works to Bourgain’s fundamental questions remained open. How does a given initial random data get transported by the nonlinear flow? If it is Gaussian initially, how does this Gaussianity propagate? What’s the description of the solution beyond the linear evolution?

In recent work, joint with Yu Deng and Haitian Yue, we developed the theory of random tensors a powerful new framework which allows us to unravel the propagation of randomness under the nonlinear flow beyond the linear evolution of random data and answer these questions in an optimal range relative to what we define as the probabilistic scaling. In particular we establish the invariance of Gibbs measures for 2D NLS and 3D Hartree NLS equations using the method of random averaging operators, a first order approximation to the full random tensor theory.