Stochastic nonlinear wave equation with rougher than white noise
Xue-Mei Li, Xianfeng Ren
TL;DR
The paper develops a pathwise local well-posedness theory for the 2D stochastic nonlinear wave equation on the torus with a cubic nonlinearity and rough Matérn forcing of regularity parameter α, proving well-posedness for 0<α<3/8 after Wick renormalization. Central to the method is a two-tier decomposition of the solution into a rough stochastic part and a regular remainder, with a refined, range-dependent analysis of nonlinear interactions using Bourgain spaces and multilinear estimates. The work introduces and systematically leverages random tensor operators, higher-order stochastic symbols (e.g., 30, 31, 32, 70, 90), and a hierarchy of renormalized remainder equations to accommodate increasing roughness, culminating in a two-regime local theory and a complete set of limit objects. These results extend the white-noise and quadratic cases, provide a robust framework for handling rough stochastic forcing in dispersive SPDEs, and offer tools potentially applicable to broader singular SPDEs with dispersive dynamics.
Abstract
We study the singular stochastic wave equation on $\mathbb T^2$, with a cubic nonlinearity and Gaussian rough Matérn forcing (a Fourier multiplier of order $α>0$ applied to space-time white noise) and establish local well-posedness for $α< \tfrac{3}{8}$. This extends [GKO18] beyond white noise and strengthens the quadratic-case result [OO21] ($α<\tfrac 12$). Our argument develops new trilinear estimates in Bourgain spaces together with sharp, case-specific cubic counting estimates.