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 Erdo\u{g}an--Tzirakis, which remains unknown on multidimensional torus. Our argument is based on a quantitative quasi-invariance of Gaussian measures with covariance operator $(1-\Delta)^{-s}$ for $s>2$.