We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation with initial data below . In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of the NLS equation almost surely for the initial data in the support of the canonical Gaussian measures on for each , and global well-posedness for each .
"Almost sure well-posedness of the cubic nonlinear Schrödinger equation below ." Duke Math. J. 161 (3) 367 - 414, February 2012. https://doi.org/10.1215/00127094-1507400