Global dynamics for the two-dimensional stochastic nonlinear wave equations

Massimilano Gubinelli, Herbert Koch, Tadahiro Oh, Leonardo Tolomeo

Research output: Contribution to journalArticlepeer-review

Abstract

We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus.
Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the I-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
Original languageEnglish
Number of pages33
JournalInternational Mathematics Research Notices
Publication statusAccepted/In press - 16 Mar 2021

Keywords

  • stochastic nonlinear wave equation
  • nonlinear wave equation
  • damped nonlinear wave equation
  • renormalization
  • white noise
  • Gibbs measure

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