A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation

TL;DR

The paper analyzes log-Sobolev inequalities for focusing Gibbs measures associated with the nonlinear Schrödinger equation on the circle, with an $L^2$ mass cutoff. It proves LSI for the subcritical-to-critical range $2 \le p \le 4$ by leveraging a Bakry–Émery-type approach on regularized, finite-dimensional approximations and using the Boué–Dupuis variational formula to control variational representations. For $p>4$, it establishes a Hessian lower bound that yields an unavoidable divergence of key moments, demonstrating that existing multiscale Bakry–Émery methods cannot produce LSI for the Gibbs measure in this regime. Together, the results delineate the limits of current techniques in establishing LSI for focusing NLS Gibbs measures and highlight the role of Hessian behavior in infinite-dimensional variational problems.

Abstract

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ When $2 \le p \le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p> 4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $ρ$.

Theorems & Definitions (20)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: Bakry-Émery BaEm
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4: Boué-Dupuis variational formula
• Lemma 2.5
• proof