Antti Knowles

Many time-dependent nonlinear Schrödinger equations admit an invariant Gibbs measure, which is a probability measure on the space of distributions that is left invariant by the time evolution. Such measures have been extensively studied as tool to construct global solutions of time-dependent nonlinear Schrödinger equations with rough initial data. I review some recent progress on deriving these measures in dimensions 1,2,3 as high-temperature limits of many-body quantum mechanics. In one dimension, I also explain how time-dependent correlation functions of the nonlinear Schrödinger equation arise as limits of corresponding quantum many-body correlation functions.