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

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.

55 pages. Expanded Remark 1.10. Published in Stoch. Partial Differ. Equ. Anal. Comput. (2022). Special issue dedicated to Professor István Gyöngy on the occasion of his seventieth birthday

References in corpus (3)

• Random tensors, propagation of randomness, and nonlinear dispersive equations
• Focusing ${\Phi }_3^4$-model with a Hartree-type nonlinearity
• Stochastic quantization of the ${\Phi }_3^3$-model

Cited by in corpus (2)

• Well-posedness of a gauge-covariant wave equation with space-time white noise forcing
• Global well-posedness of the energy-critical stochastic nonlinear wave equations