The work in this line of research was initiated in the articles:

N. Tzvetkov, Quasi-invariant Gaussian measures for one dimensional Hamiltonian PDEs, Forum Math. Sigma 3 (2015).
T. Oh, N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schroedinger equation, PTRF (2017).
T. Oh, N. Tzvetkov, Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation, JEMS 22 (2020).

We consider Hamiltonian PDE's with random data distributed according to a gaussian field and we show that the (infinite dimensional) law of the solution at any time is absolutely continuous with respect to the initial law. It seems that such results cannot be obtained via Malliavin calculus methods which only applies to finite dimensional valued random variables. We instead use PDE methods which exploit in an intricate way both the random and the time oscillations presented in the problem. These works break the rigidity in the choice of the distribution of the initial data imposed when considering Gibbs distributed initial data only.

There has been several remarkable works by other authors on this set of problems. Debussche-Tsutsumi where the first to find a way to identify the associated Radom-Nikodym derivatives. Oh-Seong were the first to establish such quasi-invariance results for gaussian field living in negative Sobolev spaces. In a very promising work by Forlano-Tolomeo the quasi-invariance property is used in order to construct the dynamics building thus a kind of DiPerna-Lions theory in the context of dispersive PDE (the methods are however completely different).

As shown in another work by Forlano-Tolomeo, the tools developed in this line of research have the potential to identify the nature of the invariant measures in the context of stochastic PDE.