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

Tadahiro Oh*, Yuzhao Wang, Younes Zine

*Corresponding author for this work

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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 languageEnglish
Pages (from-to)898-963
JournalStochastics and Partial Differential Equations: Analysis and Computations
Issue number3
Early online date13 Apr 2022
Publication statusE-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).


  • 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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