Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity

TL;DR

This paper establishes local well-posedness for a three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity using paracontrolled calculus, addressing challenges unique to the wave setting.

Contribution

It introduces new techniques involving dispersion and novel random operators to handle the wave equation's complexities, extending paracontrolled calculus beyond parabolic cases.

Findings

Proves local well-posedness of the stochastic wave equation

Develops new methods for handling dispersion in stochastic PDEs

Introduces novel random operators to manage regularity issues

Abstract

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.