Seminar| Institute of Mathematical Sciences
Time: Aug. 13rd, Tuesday 10:00-12:00; Aug. 14th, Wednesday 10:00-12:00; Aug. 15th, Thursday 10:00-12:00
Speaker: Rui Liang, University of Massachusetts Amherst
Abstrct: In this minicourse, we will revisit Bourgain's contributions to Hamiltonian equations with random initial data, followed by an exploration of the main ideas in the theory of random averaging operators and random tensors.
As an example, we will consider the fractional Schrödinger equation with cubic nonlinearity on the circle, considering initial data distributed via the Gibbs measure. We will discuss the challenges and strategies involved in establishing its Poincaré recurrence property with respect to the Gibbs measure in some full dispersive range. This work, using the theory of the random averaging operators developed by Deng-Nahmod-Yue '24, 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 the almost sure local-in-time existence for the half-wave equation with random data above some probabilistically critical scaling.