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

TL;DR

This paper establishes local well-posedness for a three-dimensional stochastic cubic nonlinear wave equation driven by almost space-time white noise, using a novel approach that avoids paracontrolled calculus and employs multilinear dispersive smoothing.

Contribution

It introduces a new method for analyzing singular SNLW on  by avoiding paracontrolled calculus, focusing on second order expansion and multilinear dispersive smoothing.

Findings

Proved local well-posedness of the renormalized SNLW with almost space-time white noise.

Developed a new analytical approach that bypasses paracontrolled calculus.

Demonstrated the effectiveness of multilinear dispersive smoothing in this context.

Abstract

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus $\mathbb{T}^3$. 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 $\mathbb{T}^3$ by Gubinelli, Koch, and the first author (2018), Okamoto, Tolomeo, and the first author (2020), and Bringmann (2020). 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.