A pathological set regarding the propagation of almost sure properties of Gaussian measures

We provide a complementary result to the quasi-invariance of Gaussian measures supported on Sobolev spaces with high regularity under the dynamics of the three-dimensional periodic defocusing nonlinear wave equation from Gunaratnam-Oh-Tzvetkov-Weber (2022). Namely, given $p$ and $\sigma$ large enough, we prove the existence of dense sets of Sobolev spaces $W^{\sigma ,p}({\mathbb{T}}^3)$ which do not preserve the regularity $\sigma$ throughout the aforementioned dynamics. This is in sharp contrast with the propagation under the flow of almost sure properties.