Evolution of gaussian fields by the flow of the nonlinear Schrödinger equation

Speaker: Nikolay Tzvetkov

Abstract: We will discuss the nonlinear Schrödinger equation with data distributed
according to gaussian fields. These fields are invariant under the free evolution and the
question is how much the nonlinear interaction affects the invariance property. We will
present a recent result, obtained in collaboration with Chenmin Sun, showing that for the
three dimensional energy critical problem the law of the solution at any time is absolutely
continuous with respect to the law of the initial gaussian field. We are therefore in slightly
out of equilibrium regime. It should be emphasized that, thanks to the work by Aizenman
and Duminil-Copin, we do not expect to reasonably define a Gibbs measure for an
energy critical problem. Therefore the consideration of general gaussian fields becomes
even more natural. The main new idea is the use of a cancellation of (probabilistically)
resonant contributions in a modified energy estimate. We also rely on several techniques
developed in recent years in the field of probabilistic well-posedness of dispersive PDE’s.
We will overview these ideas and techniques.