In the past two decades, significant progress has been made in understanding random dispersive PDEs with polynomial nonlinearities. However, non-polynomial nonlinearities remain poorly understood. This talk will present recent advancements in this direction, focusing on the well-posedness for the two-dimensional damped wave equation with a sine nonlinearity, driven by additive space-time white noise.

I will introduce the physical Fourier restriction norm method, a novel framework that addresses the complexities of non-polynomial settings. This method leverages recent developments in the Fourier restriction theory for the cone to establish crucial deterministic estimates. Furthermore, I will discuss the proof of nonlinear smoothing for the imaginary Gaussian multiplicative chaos, which constitutes the main probabilistic component of our approach. This involves examining new Feynman diagrams, whose analysis extends beyond the classical Dyson power counting criterion. This is a joint work with Tadahiro Oh (Edinburgh, UK).