Focusing Gibbs measures with harmonic potential

Tristan Robert; Kihoon Seong; Leonardo Tolomeo; Yuzhao Wang

doi:10.1214/23-AIHP1435

Focusing Gibbs measures with harmonic potential

Tristan Robert
, Kihoon Seong
, Leonardo Tolomeo
, Yuzhao Wang

Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

In this paper, we study the Gibbs measures associated to the focusing nonlinear Schrödinger equation with harmonic potential on Euclidean spaces. We establish a dichotomy for normalizability vs non-normalizability in the one dimensional case, and under radial assumption in the higher dimensional cases. In particular, we complete the programs of constructing Gibbs measures in the presence of a harmonic potential initiated by Burq–Thomann–Tzvetkov (2013) in dimension one and Deng (2012) in dimension two with radial assumption.

Original language	English
Pages (from-to)	571-598
Number of pages	28
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	61
Issue number	1
DOIs	https://doi.org/10.1214/23-AIHP1435
Publication status	Published - 11 Feb 2025

Keywords

Focusing Gibbs measure
Nonlinear Schrödinger equation
Harmonic potential
Normalizability
Variational approach

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10.1214/23-AIHP1435

https://www.research.ed.ac.uk/en/publications/focusing-gibbs-measures-with-harmonic-potential

Dynamics of singular stochastic nonlinear dispersive PDEs
Wang, Y. (Principal Investigator)
Engineering & Physical Science Research Council
1/04/21 → 31/03/24
Project: Research Councils

Cite this

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abstract = "In this paper, we study the Gibbs measures associated to the focusing nonlinear Schr{\"o}dinger equation with harmonic potential on Euclidean spaces. We establish a dichotomy for normalizability vs non-normalizability in the one dimensional case, and under radial assumption in the higher dimensional cases. In particular, we complete the programs of constructing Gibbs measures in the presence of a harmonic potential initiated by Burq–Thomann–Tzvetkov (2013) in dimension one and Deng (2012) in dimension two with radial assumption.",

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author = "Tristan Robert and Kihoon Seong and Leonardo Tolomeo and Yuzhao Wang",

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AU - Robert, Tristan

AU - Seong, Kihoon

AU - Tolomeo, Leonardo

AU - Wang, Yuzhao

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N2 - In this paper, we study the Gibbs measures associated to the focusing nonlinear Schrödinger equation with harmonic potential on Euclidean spaces. We establish a dichotomy for normalizability vs non-normalizability in the one dimensional case, and under radial assumption in the higher dimensional cases. In particular, we complete the programs of constructing Gibbs measures in the presence of a harmonic potential initiated by Burq–Thomann–Tzvetkov (2013) in dimension one and Deng (2012) in dimension two with radial assumption.

AB - In this paper, we study the Gibbs measures associated to the focusing nonlinear Schrödinger equation with harmonic potential on Euclidean spaces. We establish a dichotomy for normalizability vs non-normalizability in the one dimensional case, and under radial assumption in the higher dimensional cases. In particular, we complete the programs of constructing Gibbs measures in the presence of a harmonic potential initiated by Burq–Thomann–Tzvetkov (2013) in dimension one and Deng (2012) in dimension two with radial assumption.

KW - Focusing Gibbs measure

KW - Nonlinear Schrödinger equation

KW - Harmonic potential

KW - Normalizability

KW - Variational approach

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DO - 10.1214/23-AIHP1435

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JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 1

ER -

Focusing Gibbs measures with harmonic potential

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Dynamics of singular stochastic nonlinear dispersive PDEs

Cite this