Vedran Sohinger - Gibbs Measures as Local Equilibrium Kubo-Martin-Schwinger States for Focusing Nonlinear Schrödinger Equations

par Vedran Sohinger (University of Warwick)

lundi 24 mars 2025, 13:00 → 15:15 Europe/Paris

435 (UMPA, ENS de Lyon)

Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure global well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the classical Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equlibrium measures of the Liouville equation.  In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of Hamiltonian PDEs, including nonlinear Schrödinger equations with defocusing interactions. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari (University of Rennes).

In the second part of the talk, we study (local) Gibbs measures for focusing nonlinear Schrödinger equations. These measures have to be localized by a truncation in the mass in one dimension and in the Wick-ordered (renormalized) mass in dimensions two and three. We show that local Gibbs measures correspond to suitably localized KMS states. This is joint work with Andrew Rout and Zied Ammari (University of Rennes).