On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations

Forlano, Justin; Trenberth, William J.

doi:10.1016/j.anihpc.2019.07.006

Cited by 19 publications

(30 citation statements)

References 60 publications

(276 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for s > α large enough. We refer to [23] for the precise restriction on s. There is a gap between the best s and α leaving an interesting open problem. Remark 1.4.…”

mentioning

confidence: 99%

Gibbs Measure Dynamics for the Fractional NLS

Sun¹,

Tzvetkov²

2020

SIAM J. Math. Anal.

View full text Add to dashboard Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

“…for s > α large enough. We refer to [23] for the precise restriction on s. There is a gap between the best s and α leaving an interesting open problem. Remark 1.4.…”

mentioning

confidence: 99%

Gibbs Measure Dynamics for the Fractional NLS

Sun¹,

Tzvetkov²

2020

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…After the completion of this paper, the second method based on the energy estimate has been further developed in [33,16]. In a recent preprint [14], Forlano-Trenberth applied the approaches developed in [33,16] to study the cubic fractional NLS:…”

mentioning

confidence: 99%

Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion

Tzvetkov

2019

Comptes Rendus. Mathématique

View full text Add to dashboard Cite

In this paper, we consider the cubic nonlinear Schrödinger equation with third order dispersion on the circle. In the non-resonant case, we prove that the meanzero Gaussian measures on Sobolev spaces H s (T), s > 3 4 , are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.Résumé. Dans cet article, nous considérons l'équation de Schrödinger non linéaire cubique avec dispersion d'ordre trois sur le cercle. Dans le cas non résonant, nous prouvons que les mesures gaussiennes de moyenne nulle sur les espaces de Sobolev H s (T), s > 3 4 , sont quasiinvariantes par le flot. Enétablissant le résultat, nous appliquons des transformations de gauge pouréliminer la partie résonante de la dynamique et nous utilisons l'invariance des mesures gaussiennes par rapportà ces transformations de gauge.2010 Mathematics Subject Classification. 35Q55.

show abstract

“…One of the interesting problems in this direction is to investigate how Gaussian measures are transported by such nonlinear Hamiltonian flows as Korteweg-de Vries, nonlinear Schrödinger equations and others. Especially, it is natural to ask whether or not the Gaussian measure µ α is quasiinvariant, i.e., mutually absolutely continuous with respect to the transported Gaussian measure by the nonlinear Hamiltonian flow (see, e.g., [3], [9], [10], [19]- [23], [25] and [28]).…”

Section: Introduction and Theoremsmentioning

confidence: 99%

“…Nowadays, there are many papers about the quasi-invariance of Gaussian measures transported by various nonlinear dispersive equations (see [9] for the fractional NLS, [10] for the nonlinear wave equation, [19] and [21] for the fourth order NLS and [22] and [23] for NLS). In [20], Oh, the second author and Tzvetkov have showed the quasi-invariance of Gaussian measure µ α for α > 3/4 under the flow generated by the Cauchy problem of the cubic NLS with third-order dispersion by using the Kuo-Ramer theorem ( [12], [24]).…”

Section: Introduction and Theoremsmentioning

confidence: 99%

See 1 more Smart Citation

Quasi-invariance of Gaussian measures transported by the cubic NLS with third-order dispersion on T

Debussche

2021

Journal of Functional Analysis

View full text Add to dashboard Cite

We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance (1 − ∂ 2x ) −α on L 2 (T) is quasi-invariant for the associated flow for α > 1/2. This is sharp and improves a previous result obtained in [20] where the values α > 3/4 were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in [4] and [5]. The arguments are general and can be used to other Hamiltonian equations.

show abstract

On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations

Abstract: Under certain regularity conditions, we establish quasi-invariance of Gaussian measures on periodic functions under the flow of cubic fractional nonlinear Schrödinger equations on the one-dimensional torus.

Cited by 19 publications

References 60 publications

Gibbs Measure Dynamics for the Fractional NLS

Gibbs Measure Dynamics for the Fractional NLS

Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion

Quasi-invariance of Gaussian measures transported by the cubic NLS with third-order dispersion on T

Contact Info

Product

Resources

About