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

• Published in: Stochastic partial differential equations : analysis and computations Vol. 10; no. 3; pp. 898 - 963
• Main Authors: Oh, Tadahiro, Wang, Yuzhao, Zine, Younes
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2022; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematics; Mathematics and Statistics; Nonlinearity; Numerical Analysis; Partial Differential Equations; Probability Theory and Stochastic Processes; Quadratic equations; Statistical Theory and Methods; Toruses; Wave equations; Well posed problems; White noise; 60L40; Nonlinear wave equation; 35L71; Pathwise well-posedness; 60H15; Stochastic nonlinear wave equation
• ISSN: 2194-0401; 2194-041X
• DOI: 10.1007/s40072-022-00237-x

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus 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 T 3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ 3 4 -model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). 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.