Over the last decade, there has been a significant development in the study of stochastic dispersive PDEs, broadly interpreted with random initial data and/or additive stochastic forcing, where the difficulty comes from roughness in spatial regularity. In this talk, I consider well-posedness of stochastic dispersive PDEs with multilpicative noises, whose Ito solutions were contructed in 80's for the wave case and in 90's for the Schrödinger case, and present the first results on pathwise well-posedness for stochastic nonlinear wave equations (SNLW) and stochastic nonlinear Schrödinger equations (SNLS).
The main challenge of this problem comes from the deficiency of temporal regularities. We overcome this issue by considering operator-valued rough paths adapted to the wave/Schrödinger flow and implementing the Fourier restriction norm method (with the U^p- and V^p-space) for the setting of controlled paths.