Fourier restriction norm method adapted to controlled paths: stochastic wave equations

We investigate the pathwise well-posedness issue of the stochastic nonlinear wave equation (SNLW) with a multiplicative noise. While the Ito solution theory (= random field solution theory) was established in the '80s, its pathwise well-posedness has remained a challenging open problem for over forty years. By building a unified framework for the Fourier restriction norm method adapted to the $U^p$- and $V^p$-spaces, due to Koch and Tataru (2007), and the Young/rough integration theory via the sewing lemma and controlled paths due to Gubinelli (2004) along with the random tensor estimate for multiple stochastic integrals with respect to (fractional) Brownian motions, we establish pathwise local well-posedness of SNLW in optimal regularity ranges. In particular, in the one-dimensional case with a white-in-time noise, our result covers the case of an almost space-time white noise, which is optimal within the framework of one-parameter rough paths.