Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on $\mathbf{T}$

Abstract

We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance $(1-{\partial }_x^2{)}^{-\alpha }$ on $L^2(\mathbf{T})$ is quasi-invariant for the associated flow for $\alpha >1/2$. This is sharp and improves a previous result obtained in \cite{OTT} where the values $\alpha >3/4$ were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in \cite{Cruz1} and \cite{Cruz2}. The arguments are general and can be used to other Hamiltonian equations.

Cited by in corpus (1)

• Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation in negative Sobolev spaces