Gibbs measures as local equilibrium KMS states for focusing nonlinear Schr\"odinger equations

TL;DR

This paper investigates local Gibbs measures as equilibrium states for focusing nonlinear Schrödinger equations, establishing their properties as KMS states and characterizing all such equilibria under certain conditions using advanced probabilistic and functional analysis techniques.

Contribution

It introduces a novel characterization of local Gibbs measures as KMS states for focusing NLS equations, employing Malliavin calculus and Dirichlet form theory.

Findings

Local Gibbs measures are stationary solutions satisfying KMS conditions.

All local KMS equilibrium states are characterized as local Gibbs measures.

Revisits Bourgain's normalizability proof using concentration inequalities.

Abstract

In this paper, we are concerned with the study of statistical equilibria for focusing nonlinear Schr\"odinger and Hartree equations on the d-dimensional torus when d=1,2,3. Due to the focusing nature of the nonlinearity in these PDEs, Gibbs measures have to be appropriately localized. First, we show that these local Gibbs measures are stationary solutions for the Liouville probability density equation and that they satisfy a local equilibrium Kubo-Martin-Schwinger (KMS) condition. Secondly, under some natural assumptions, we characterize all possible local KMS equilibrium states for these PDEs as local Gibbs measures. Our methods are based on Malliavin calculus in Gross-Stroock Sobolev spaces and on a suitable Gaussian integration by parts formula. To handle the technical problems due to localization, we rely on the works of Aida and Kusuoka on irreducibility of Dirichlet forms over infinite-dimensional domains. This leads us to the study of sublevel sets of the renormalized mass and their connectedness properties. In this paper, we also revisit Bourgain's proof of the normalizability of the local Gibbs measure for the focusing Hartree equation on the d-dimensional torus with d=2,3 by using concentration inequalities.