Transport of Gaussian measures along the flow of the nonlinear Schrödinger equation.

par Alexis Knezevitch

mercredi 3 déc. 2025, 14:40 → 15:20 Europe/Paris

Fokko (ICJ)

In this talk, we provide a macroscopic description of the solutions to the nonlinear Schrödinger equation defined on the circle. In our approach, the initial data are random variables distributed according to Gaussian laws, and therefore generate random solutions as well. Our main objective is to study their distributions, in particular by comparing them with the initial Gaussian laws.

The first works on Hamiltonian PDEs with random initial data go back to Lebowitz, Rose, and Speer (1987), and were later revisited by Bourgain. In these works, the initial data are distributed according to the Gibbs measure, which can be interpreted as choosing initial data at equilibrium. These studies show that this equilibrium is preserved over time, meaning that the law of the solutions (at each instant) remains the Gibbs measure. In other words, the flow preserves the Gibbs measure.

Here, we consider Gaussian initial data, which can now be interpreted as choosing them out of equilibrium. We will see (under certain assumptions) that the laws of the solutions are no longer identical, but are in fact absolutely continuous with respect to the initial Gaussian laws. In other words, the flow transports Gaussian measures into measures that are absolutely continuous with respect to them. This is a qualitative result about the flow, but quantitative information on the Radon–Nikodym derivatives in turn yields more precise properties of the solutions. This phenomenon, called quasi-invariance, was introduced by Nikolay Tzvetkov in 2015.