Transport of Gaussian measures under Hamiltonian and stochastic dynamics.

We consider the evolution of Gaussian measures under infinite-dimensional dynamics, in both Hamiltonian and dissipative settings.

In the Hamiltonian case, we consider the flow of Gaussian measures under the Szego equation, a toy model for weakly-dispersive systems. We identify a sharp transition between quasi-invariance of measures above a certain regularity and a singular regime below, where the laws of solutions are immediately singular with respect to the initial distribution. These regimes are determined by the instantaneous change in the energy profile of solutions.

In the dissipative case, we consider the 2d incompressible Navier--Stokes equations with Gaussian forcing that is white-in-time and coloured-in-space. Using a nonlinear time-shifted Girsanov method, we show that the time marginals of solutions are equivalent to those of the associated Ornstein--Uhlenbeck process. We exploit the skew-symmetric structure of the Navier--Stokes nonlinearity to construct a modified system that leaves the Gaussian measure invariant while remaining comparable to the Navier--Stokes dynamics.

This talk is based on joint work with Martin Hairer and Leonardo Tolomeo.