Global well-posedness of the energy-critical stochastic nonlinear wave equations

TL;DR

This paper proves the global well-posedness of the defocusing energy-critical stochastic nonlinear wave equations on both Euclidean space and tori, even with stochastic forcing below the energy space, using probabilistic perturbation techniques.

Contribution

It extends the probabilistic perturbation method to establish global well-posedness for energy-critical SNLW with additive noise on $ ^d$ and $ ^d$, including below the energy space.

Findings

Proved global well-posedness for energy-critical SNLW on $ ^d$ and $ ^d$.

Extended probabilistic methods to stochastic PDEs with additive noise.

Achieved well-posedness results below the energy space on $ ^d$.

Abstract

We consider the Cauchy problem for the defocusing energy-critical stochastic nonlinear wave equations (SNLW) with an additive stochastic forcing on $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ with $d \geq 3$. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by B\'enyi-Oh-Pocovnicu (2015) and Pocovnicu (2017) and in the context of stochastic PDEs by Oh-Okamoto (2020), we prove global well-posedness of the defocusing energy-critical SNLW. In particular, on $\mathbb{T}^d$, we prove global well-posedness with the stochastic forcing below the energy space.