Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

Arpad Benyi; Tadahiro Oh; Oana Pocovnicu

doi:10.1007/978-3-319-20188-7

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

Arpad Benyi, Tadahiro Oh, Oana Pocovnicu

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter (peer-reviewed) › peer-review

Abstract / Description of output

We consider a randomization of a function on ℝ^d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on ℝ^4 is almost surely locally well-posed with respect to randomized initial data below the energy space.

Original language	English
Title of host publication	Excursions in Harmonic Analysis
Subtitle of host publication	The February Fourier Talks at the Norbert Wiener Center
Editors	R. Balan, M. Begue, J.J. Benedetto, W. Czaja, K.A. Okoudjou
Publisher	Basel: Birkhauser Verlag
Pages	3-25
Number of pages	20
Volume	4
ISBN (Electronic)	978-3-319-20188-7
ISBN (Print)	978-3-319-20187-0
DOIs	https://doi.org/10.1007/978-3-319-20188-7
Publication status	Published - Oct 2015

Publication series

Name	Applied and Numerical Harmonic Analysis
ISSN (Print)	2296-5009

Keywords / Materials (for Non-textual outputs)

nonlinear Schrödinger equation
almost sure well-posedness
modulation space
Wiener decomposition
Strichartz estimate
Fourier restriciton norm method

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10.1007/978-3-319-20188-7

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLSAccepted author manuscript, 232 KB

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Benyi, A., Oh, T., & Pocovnicu, O. (2015). Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. In R. Balan, M. Begue, J. J. Benedetto, W. Czaja, & K. A. Okoudjou (Eds.), Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center (Vol. 4, pp. 3-25). (Applied and Numerical Harmonic Analysis). Basel: Birkhauser Verlag. https://doi.org/10.1007/978-3-319-20188-7

Benyi, Arpad ; Oh, Tadahiro ; Pocovnicu, Oana. / Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. editor / R. Balan ; M. Begue ; J.J. Benedetto ; W. Czaja ; K.A. Okoudjou. Vol. 4 Basel: Birkhauser Verlag, 2015. pp. 3-25 (Applied and Numerical Harmonic Analysis).

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abstract = "We consider a randomization of a function on ℝ^d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schr{\"o}dinger equation. As an example, we also show that the energy-critical cubic nonlinear Schr{\"o}dinger equation on ℝ^4 is almost surely locally well-posed with respect to randomized initial data below the energy space. ",

keywords = "nonlinear Schr{\"o}dinger equation, almost sure well-posedness, modulation space, Wiener decomposition, Strichartz estimate, Fourier restriciton norm method",

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Benyi, A, Oh, T & Pocovnicu, O 2015, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. in R Balan, M Begue, JJ Benedetto, W Czaja & KA Okoudjou (eds), Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. vol. 4, Applied and Numerical Harmonic Analysis, Basel: Birkhauser Verlag, pp. 3-25. https://doi.org/10.1007/978-3-319-20188-7

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. / Benyi, Arpad; Oh, Tadahiro; Pocovnicu, Oana.
Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. ed. / R. Balan; M. Begue; J.J. Benedetto; W. Czaja; K.A. Okoudjou. Vol. 4 Basel: Birkhauser Verlag, 2015. p. 3-25 (Applied and Numerical Harmonic Analysis).

Research output: Chapter in Book/Report/Conference proceeding › Chapter (peer-reviewed) › peer-review

TY - CHAP

T1 - Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

AU - Benyi, Arpad

AU - Oh, Tadahiro

AU - Pocovnicu, Oana

PY - 2015/10

Y1 - 2015/10

N2 - We consider a randomization of a function on ℝ^d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on ℝ^4 is almost surely locally well-posed with respect to randomized initial data below the energy space.

AB - We consider a randomization of a function on ℝ^d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on ℝ^4 is almost surely locally well-posed with respect to randomized initial data below the energy space.

KW - nonlinear Schrödinger equation

KW - almost sure well-posedness

KW - modulation space

KW - Wiener decomposition

KW - Strichartz estimate

KW - Fourier restriciton norm method

U2 - 10.1007/978-3-319-20188-7

DO - 10.1007/978-3-319-20188-7

M3 - Chapter (peer-reviewed)

SN - 978-3-319-20187-0

VL - 4

T3 - Applied and Numerical Harmonic Analysis

SP - 3

EP - 25

BT - Excursions in Harmonic Analysis

A2 - Balan, R.

A2 - Begue, M.

A2 - Benedetto, J.J.

A2 - Czaja, W.

A2 - Okoudjou, K.A.

PB - Basel: Birkhauser Verlag

ER -

Benyi A, Oh T, Pocovnicu O. Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. In Balan R, Begue M, Benedetto JJ, Czaja W, Okoudjou KA, editors, Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. Vol. 4. Basel: Birkhauser Verlag. 2015. p. 3-25. (Applied and Numerical Harmonic Analysis). doi: 10.1007/978-3-319-20188-7

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

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Keywords / Materials (for Non-textual outputs)

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