Student Probability/PDE Seminar: Invariant Measures and (Discrete) Nonlinear Schrodinger Equations

Seminar | February 23 | 2:10-3:30 p.m. | 891 Evans Hall

 Kyeongsik Nam, UC Berkeley

 Department of Mathematics

The notion of invariant measure plays an important role in studying the long-time behavior of solutions to Nonlinear Schrödinger Equations (NLS). For instance, grand canonical Gibbs measures can be used to prove the almost sure well-posedness of NLS. However, it is hard to define grand canonical Gibbs measures in high dimensions. One way to remedy this is to use micro-canonical Gibbs measures.This was first considered by Sourav Chatterjee. Since it is not obvious to make sense of micro-canonical Gibbs measures in infinite dimensional function spaces, we first discretize the space $\mathbb {R^d}$ and then construct a Gibbs measure on the finite size box. We show that in the mass-subcritical NLS setting, micro-canonical Gibbs measures get close to solitions as we take an infinite volume limit and then a continuum limit. As a consequence, we prove the weak version of the soliton resolution conjecture using the ergodic theory.