Global dynamics for the two-dimensional stochastic nonlinear wave equations

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.

arXiv comment33 pages. To appear in Internat. Math. Res. Not. Minor typos corrected

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• Eyring--Kramers law for the hyperbolic ${\varphi }^4$ model
• Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation
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