Following the seminal work of Lebowitz, Rose and Speer in 1988, a large number of one- and two-dimensional Hamiltonian dispersive equations such as the nonlinear Schroedinger equation and the Korteweg-de Vries equation, have been proved to leave Gibbs-type measures invariant. When an invariant measure is available, it provides a powerful tool allowing uniform in time control on solutions with high probability. Unfortunately, the existence of invariant Gibbs-type measures is a rather exceptional condition. Recently, N. Tzvetkov initiated the study of quasi-invariance of Gaussian measures under nonlinear dispersive equations, proving that the distribution of the solution of a Hamilton nonlinear equation is absolutely continuous with respect to a class of (regular) Gaussian initial data. This result was extended to a nonlinear Schroedinger equation with quartic dispersion by Oh-Tzvetkov.

I will review the known results concerning invariance and quasi-invariance of nonlinear PDEs with random initial data, and present a sharp quasi-invariance result for the quartic NLS, and show that dispersion is essential for this result to hold.

Joint work with Tadahiro Oh and Nikolai Tzvetkov.