Global dynamics for the two-dimensional stochastic nonlinear wave equations

TL;DR

This paper establishes global well-posedness and invariance of the Gibbs measure for two-dimensional stochastic nonlinear wave equations with additive noise, using novel hybrid and invariant measure techniques.

Contribution

It introduces a hybrid argument combining the I-method and Gronwall's inequality for stochastic equations, and proves invariance of the Gibbs measure for stochastic damped nonlinear wave equations.

Findings

Global well-posedness of the renormalized cubic SNLW in 2D.

Double exponential growth bound on Sobolev norms.

Almost sure global well-posedness and Gibbs measure invariance for SdNLW.

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.