Probabilistic well-posedness of dispersive PDEs beyond variance blowup

Over the last two decades, there has been significant
progress in probabilistic well-posedness theory of nonlinear dispersive
PDEs with random initial data.  In recent years, several examples of
"variance blowup" for equations with quadratic nonlinearities have been
observed, where the construction of basic stochastic objects breaks down
before reaching the limit of the analytical framework.  In the study of
stochastic parabolic PDEs, such a variance blowup phenomenon has been
observed for the fractional KPZ equation (with a noise rougher than a
space-time white noise) and, in a recent work (2025), Hairer introduced
a renormalization beyond variance blowup.  In this talk, I will talk
about a possible extension of probabilistic well-posedness theory of
dispersive PDEs beyond variance blowup, taking the Benjamin-Bona-Mahony
equation and the quadratic nonlinear wave equation as model examples,
and show that these equations with renormalized (rough) Gaussian initial
data converge in law to those with stochastic forcings.  If time
permits, I will discuss what happens in the KdV case.

This talk is based on joint works with Andreia Chapouto (Versailles),
Guopeng Li (Beijing Institute of Technology), Jiawei Li (Edinburgh),
Shao Liu (Bonn), and Nikolay Tzvetkov (ENS Lyon).
 

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