Abstract: In this talk, we use the fourth moment theorem to investigate a possible extension of probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data beyond variance blowup. As a model equation, we study the Benjamin-Bona-Mahony equation (BBM) with Gaussian random initial data. By introducing a suitable vanishing multiplicative renormalization constant on the initial data, we show that solutions to BBM with the renormalized Gaussian random initial data beyond variance blowup converge in law to a solution to the stochastic BBM forced by the derivative of a spatial white noise. 

Based on the joint work with Jiawei Li & Tadahiro Oh (Edinburgh), and Nikolay Tzvetkov (ENS Lyon).