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Using ideas from paracontrolled calculus, we prove local well-posedness of arenormalized version of the three-dimensional stochastic nonlinear waveequation with quadratic nonlinearity forced by an additive space-time whitenoise on a periodic domain. There are two new ingredients as compared to theparabolic setting. (i) In constructing stochastic objects, we have to carefullyexploit dispersion at a multilinear level. (ii) We introduce novel randomoperators and leverage their regularity to overcome the lack of smoothing ofusual paradifferential commutators.

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