Rui Liang, Random averaging operator, random tensor, and fractional NLS

Rui Liang, University of Massachusetts, Amherst

In this talk, we will consider the Schrödinger equation with cubic nonlinearity on the circle, with initial data distributed according to the Gibbs measure.  We will discuss the challenges and strategies involved in establishing the Poincaré recurrence property with respect to the Gibbs measure in the full dispersive range. This work, using the theory of the random averaging operator developed by Deng-Nahmod-Yue '19, addresses an open question proposed by Sun-Tzvetkov '21. We will also explain why the Gibbs dynamics for the full dispersive range is sharp in some sense. Additionally, we will see how the theory of random tensors works for extending this work to multi-dimensional settings.