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

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