<p>In this paper, we are concerned with the study of statistical equilibria for focusing nonlinear Schrödinger and Hartree equations on the <i>d</i>-dimensional torus <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {T}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d=1,2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {T}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d=2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> by using concentration inequalities.</p>

Gibbs measures as local equilibrium KMS states for focusing nonlinear Schrödinger equations

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2026年4月21日

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Zied Ammari,
Andrew Rout,
Vedran Sohinger

摘要

In this paper, we are concerned with the study of statistical equilibria for focusing nonlinear Schrödinger and Hartree equations on the d-dimensional torus $\mathbb {T}^d$ $T^{d}$ when $d=1,2,3$ $d = 1, 2, 3$ . 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 $\mathbb {T}^d$ $T^{d}$ with $d=2,3$ $d = 2, 3$ by using concentration inequalities.