Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on ${\mathbb{T}}^2$

We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schrödinger equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{2k}u$ posed on ${\mathbb{T}}^2$. In particular, we show that the Gaussian measures with inverse covariance $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}^2$, are quasi-invariant under the flow for $s>2$. Moreover, we show that the Radon-Nykodim density belongs to every $L^p$ space, locally in space. The proof relies on the physical-space energies introduced in [52], as well as a new abstract quasi-invariance argument that allows us to combine space-time estimates, along the flow with probabilistic bounds on the support of the measure.