Quasi-invariant Gaussian measures for the cubic nonlinear Schr\"odinger equation with third order dispersion

In this paper, we consider the cubic nonlinear Schr"odinger equation with third order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac 34$, are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.