Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on T
Résumé
We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance (1 − ∂ 2 x) −α on L 2 (T) is quasi-invariant for the associated flow for α > 1/2. This is sharp and improves a previous result obtained in [20] where the values α > 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 [4] and [5]. The arguments are general and can be used to other Hamiltonian equations.
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