Transport of gaussian measures under the flow of Hamiltonian PDE's

The transport of gaussian measures under transformations in an infinite dimensional space is a delicate issue as shown by the classical Cameron-Martin theorem (1944). We will show that a natural class of gaussian measures, invariant under the flow of linear Hamiltonian PDE's, remain quasi-invariant under nonlinear (Hamiltonian) perturbations. We will also identify the Radon-Nikodym derivative by using some hidden cancellations. We will make an attempt to keep the presentation at a non technical level by probably sacrificing a little rigor.