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

Understand

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.