Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

Tadahiro Oh; Yuzhao Wang; Younes Zine

doi:10.1007/s40072-022-00237-x

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

Tadahiro Oh^*, Yuzhao Wang, Younes Zine

^*Corresponding author for this work

Mathematics

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Abstract

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus 𝕋³. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on 𝕋³ by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ₃⁴-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

Original language	English
Pages (from-to)	898-963
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	10
Issue number	3
Early online date	13 Apr 2022
DOIs	https://doi.org/10.1007/s40072-022-00237-x
Publication status	E-pub ahead of print - 13 Apr 2022

Bibliographical note

Funding Information:
T.O. would like to thank István Gyöngy for his kind continual support since T.O.’s arrival in Edinburgh in 2013 and also for joyful chat over the daily tea break. Y.W. would like to thank István Gyöngy for his kindness and teaching during his stay at Edinburgh in 2016–2017. T.O. and Y.Z. were supported by the European Research Council (grant no. 864138 “SingStochDispDyn"). Y.W. was supported by supported by the EPSRC New Investigator Award (Grant No. EP/V003178/1). Lastly, the authors wish to thank the anonymous referee for the helpful comments.

Publisher Copyright:
© 2022, The Author(s).

Keywords

Nonlinear wave equation
Pathwise well-posedness
Stochastic nonlinear wave equation

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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10.1007/s40072-022-00237-xLicence: Creative Commons: Attribution (CC BY)

OhT2022ThreeFinal published version, 6.03 MBLicence: Creative Commons: Attribution (CC BY)

Cite this

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title = "Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise",

abstract = "We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus 핋3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on 핋3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ34-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.",

keywords = "Nonlinear wave equation, Pathwise well-posedness, Stochastic nonlinear wave equation",

author = "Tadahiro Oh and Yuzhao Wang and Younes Zine",

note = "Funding Information: T.O. would like to thank Istv{\'a}n Gy{\"o}ngy for his kind continual support since T.O.{\textquoteright}s arrival in Edinburgh in 2013 and also for joyful chat over the daily tea break. Y.W. would like to thank Istv{\'a}n Gy{\"o}ngy for his kindness and teaching during his stay at Edinburgh in 2016–2017. T.O. and Y.Z. were supported by the European Research Council (grant no. 864138 “SingStochDispDyn{"}). Y.W. was supported by supported by the EPSRC New Investigator Award (Grant No. EP/V003178/1). Lastly, the authors wish to thank the anonymous referee for the helpful comments. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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N2 - We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus 핋3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on 핋3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ34-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

AB - We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus 핋3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on 핋3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ34-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

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Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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