Almost sure global nonlinear smoothing for the 2D NLS

In this article, we prove an almost-sure global in time nonlinear smoothing effect for NLS on the two-dimensional torus. For deterministic data, this phenomenon was proved for the NLS on the circle by Erdoğan–Tzirakis, which remains unknown on multidimensional torus. Our argument is based on a quantitative quasi-invariance of Gaussian measures with covariance operator $(1-\mathrm{\Delta }{)}^{-s}$ for $s>2$.