I will introduce a (non Gaussian) measure on functions on the circle considered by Lebowitz Rose and Speer in the 1980s. This measure was shown by Bourgain in the 1990s to be invariant for the cubic nonlinear Schroedinger equation, a deterministic PDE.

In the limit where the radius of the circle increases to infinity, it was shown by Rider that the measure collapses to a delta function on the zero path. I will explain this result in terms of the optimizers of a certain Sobolev inequality, and describe a new result with Kihoon Seong identifying the fluctuations about this trivial limit as white noise.