Résumé : Gibbs measures associated with nonlinear Schrödinger equations are fundamental objects used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. In joint work with J. Fröhlich, A. Knowles, and B. Schlein, we study how these measures arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. Furthermore, in one-dimension, we study the corresponding problem for time-dependent correlations.
