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.

arXiv comment55 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

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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