Tadahiro Oh, University of Edinburgh
Over the last several decades, the well-posedness issue of stochastic dispersive PDEs with multiplicative noises has been studied extensively. However, this study was done primarily from the viewpoint of Ito solution theory, and pathwise well-posedness remained completely open. In this talk, I will present the first pathwise well-posedness results for stochastic nonlinear Schrödinger equations (SNLS) and stochastic nonlinear wave equations (SNLW) with multiplicative noises.
In the first part, I will consider SNLS with multiplicative fraction/white-in-time, smooth-in-space noise. I prove pathwise local well-posedness of SNLS by combining the operator-valued controlled rough path adapted to the Schrödinger flow together with a nonlinear smoothing established via the Fourier restriction norm method.
In the second part, I will discuss a paracontrolled approach in the bi-parameter setting and explain how this can be used to resolve a long-standing open problem of pathwise well-posedness of the 1-d SNLW with multiplicative space-time white noise.